diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzznmsk" "b/data_all_eng_slimpj/shuffled/split2/finalzznmsk" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzznmsk" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction} \\label{sec:intro}\n\nWe consider the extension of a defect-based estimator for the local error\nof self-adjoint time-stepping schemes of even order $p$, which was introduced in~\\cite{auzingeretal18a}\nfor the linear time-independent case, to nonlinear evolution equations\n(we set $ t_0=0 $),\n\\begin{equation} \\label{y'=F(y)}\n\\dt u(t) = F(u(t)), \\quad u(0)=u_0.\n\\end{equation}\nWe define a symmetrized version of the defect to serve as the basis for the construction\nof a local error estimator in the nonlinear case, thus representing an extension of~\\cite{auzingeretal18a}. The error\nestimator is derived from a representation\nof the local error in terms of the symmetrized defect,\nbased on a modified nonlinear variation-of-constant formula.\nIts deviation from the exact local error is one order in the step-size more precise than an analogous error estimator\nbased on the classical defect, for the latter see for instance~\\cite{auzingeretal12a,auzingeretal13a,auzingeretal13b,auzingeretal14a}.\nOur theoretical analysis is based on the assumption that the problem is smooth\n(the right-hand side is bounded and differentiable with bounded derivatives as required in\nthe analysis) with a unique, smooth solution. In this sense, our treatment is formal\nand in practical applications with unbounded right-hand side, different techniques\nare required to deduce the required regularity assumptions in order to establish\nhigh-order convergence, see for instance~\\cite{auzingeretal13b}.\n\nWe also point out that in addition to the practical merit of providing a\nmore precise estimator enabling a better choice of adaptive time-steps\nand a higher-order corrected solution if desired, the approach\nhas potential advantages for theoretical purposes.\n\nIn the analysis of local errors and error estimators for self-adjoint schemes,\nthe representation of the local error in terms\nof the symmetrized defect can be rewritten in a way such that\nits analysis can be based on an asymptotic expansion\nin even powers of the stepsize.\nApplications of this type of analysis will be reported elsewhere.\n\n\\paragraph{Outline}\nIn Section~\\ref{sec:def} we introduce the notions `classical defect'\nand the new `symmetrized defect' associated with\none-step integrators for nonlinear evolution equations\nin the autonomous\\footnote{The extension to nonautonomous\n problems is deferred to Section~\\ref{sec:nonaut}.}\nform~\\eqref{y'=F(y)}.\nA well-known integral representation of the local error in terms of the classical defect\nis obtained from the nonlinear variation-of-constant\nformula (V.O.C., also referred to as Gr{\\\"o}bner-Alexeev-Lemma~\\cite{haireretal87}),\nthis is recapitulated in~Theorem~\\ref{th:GAL}. Then,\nin Theorem~\\ref{th:GALS} we present a modified nonlinear V.O.C.~formula\nleading to an integral representation of the local error\nin terms of the symmetrized defect.\n\nAn Hermite-type quadrature approximation to the ensuing\nintegral representation provides a computable\ndefect-based local error estimator, see Section~\\ref{sec:locerrest}.\nIn particular, Theorem~\\ref{th:Sdach} shows that the\nsymmetrized error estimator is asymptotically correct,\nand for the case of a self-adjoint scheme\nit is of an improved asymptotic quality compared with the analogous\nclassical estimator.\nHere the required regularity of the problem data and of the exact solution is tacitly assumed.\n\nIn Sections~\\ref{sec:appl-auto} and~\\ref{sec:nonaut} we study\nthe application of these ideas to particular examples of self-adjoint schemes.\nIn Section~\\ref{subsec:imr}, the results are particularized to the implicit midpoint rule\nto show a concrete example of an implicit one-step method.\nIn Section~\\ref{subsec:strang},\nStrang splitting is discussed, and the algorithmic realization for general splitting\nmethods is given in Section~\\ref{subsec:algsplit}.\n\nIn Section~\\ref{sec:nonaut}, the nonautonomous case is considered.\nIn order to illustrate the extension of our ideas to this case,\nwe give details for linear problems with a $ t $\\,-\\,dependent\nright-hand side.\nSection~\\ref{subsec:emr} shows the realization for the exponential midpoint rule, and\nSection~\\ref{subsec:cfm} contains the algorithmic implementation for general commutator-free\nMagnus-type and classical Magnus methods.\n\nIn Section~\\ref{sec:num}, numerical examples for a splitting\napproximation to a cubic nonlinear Schr{\\\"o}dinger equation and Magnus-type exponential\nintegrators applied to a time-dependent Rosen--Zener model \\rev{support} the theoretical results,\n\\rev{and adaptive time-stepping based on the new error estimator is illustrated}.\n\n\n\\paragraph{Notation and preliminaries}\nThe flow associated with~\\eqref{y'=F(y)}\nis denoted by $ \\nE(t,u) $, such that the solution of~\\eqref{y'=F(y)}\nis $ u(t) = \\nE(t,u_0) $.\nBy $ \\pdone\\nE(t,u_0) $ and $ \\pdtwo\\nE(t,u_0) $ we denote\nthe derivatives of $ \\nE $ with respect to its first and second\narguments, respectively.\nBy definition, $ \\nE(t,u_0) $ satisfies\n\\begin{equation*}\n\\pdone\\nE(t,u_0) = F(\\nE(t,u_0)), \\quad \\nE(0,u_0) = u_0.\n\\end{equation*}\nWe will repeatedly make use of the following fundamental\nidentity.\\footnote{For the nonautonomous case see Lemma~\\ref{lem:fin}\n in Section~\\ref{sec:nonaut}.}\n\n\\rev{\n\\begin{lemma} \\label{lem:fi}\n\\begin{equation} \\label{fi}\n[\\,\\pdone\\nE(t,u_0) =\\,]~\nF(\\nE(t,u_0)) = \\pdtwo\\nE(t,u_0) \\cdot F(u_0).\n\\end{equation}\n\\end{lemma}\n{\\em Proof.}\n\\eqref{fi} is a consequence of the first-order variational equation\nfor $ \\nE(t,u) $,\nsee~\\cite[Theorem I.14.3]{haireretal87},~\\cite[Appendix~A]{auzingeretal13b}.\nThe simple direct proof given in~\\cite[(3.7)]{DescombesThalhammerLie}\nproceeds from the identity\n\\begin{equation*}\n\\nE(t+s,u_0) = \\nE(t,\\nE(s,u_0)).\n\\end{equation*}\nDifferentiation with respect to~$ s $ gives\n\\begin{align*}\n\\pds\\nE(t+s,u_0)\n&= \\pdone\\nE(t+s,u_0), \\\\\n\\pds\\nE(t+s,u_0)\\big|_{s=0}\n&= \\pdone\\nE(t,u_0) = F(\\nE(t,u_0)),\n\\end{align*}\nand on the other hand,\n\\begin{align*}\n\\pds\\nE(t,\\nE(s,u_0))\n&= \\pdtwo\\nE(t,\\nE(s,u_0)) \\cdot \\pdone\\nE(s,u_0), \\\\\n\\pds\\nE(t,\\nE(s,u_0))\\big|_{s=0}\n&= \\pdtwo\\nE(t,u_0) \\cdot \\pdone\\nE(0,u_0) = \\pdtwo\\nE(t,u_0) \\cdot F(u_0), \\\\\n\\end{align*}\nwhich completes the proof.\n\\hfill $ \\square $\n}\n\n\\section{Classical and symmetrized defects for one-step integrators}\n\\label{sec:def}\nConsider an approximation to the given problem~(\\ref{y'=F(y)})\ndefined by the flow\n\\begin{equation} \\label{S(t_0,u_0)}\n\\nS(t,u_0) \\approx \\nE(t,u_0), \\quad \\nS(0,u_0) = u_0,\n\\end{equation}\nof a consistent one-step scheme with stepsize $ t $,\nstarting at $ (0,u_0) $.\nWe assume that the scheme has order~$ p $,\ni.e., the local error\n\\begin{equation} \\label{L(t_0,u_0)}\n\\nL(t,u_0) = \\nS(t,u_0) - \\nE(t,u_0)\n\\end{equation}\nsatisfies $ \\nL(t,u_0) = \\Order(t^{p+1}) $.\n\nWe call\n\\begin{equation} \\label{cdnonlin}\n\\nDc(t,u) = \\pdone\\nS(t,u) - F(\\nS(t,u)) = \\Order(t^p)\n\\end{equation}\nthe {\\em classical defect}\\, associated with $ \\nS(t,u) $.\nThe local error can be represented in terms of the classical defect via\nthe well-known nonlinear variation-of-constant formula,\nthe so-called Gr{\\\"o}bner-Alekseev Lemma.\nFor convenience we restate this in a form required in our context\nand also include the\nproof following\\footnote{See~\\cite[Figure~I.14.1]{haireretal87},\n {\\em Lady Windermere's Fan, Act 2.}}~\\cite[Theorem~I.14.5]{haireretal87}\n(see also~\\cite[Theorem~3.3]{DescombesThalhammerLie}).\nWe formulate it in a concise way making direct use of~\\eqref{fi}.\n\n\\begin{theorem}\\label{th:GAL}\nIn terms of the classical defect~\\eqref{cdnonlin},\nthe local error satisfies the integral representation\n\\begin{equation} \\label{GAL-identity}\n\\nL(t,u_0) =\n\\int_0^t \\pdtwo\\nE(t-s,\\nS(s,u_0)) \\cdot \\nDc(s,u_0)\\,{\\mathrm d} s.\n\\end{equation}\n\\end{theorem}\n{\\em Proof.}\nFor fixed $ t $, let\n\\begin{align*}\ny(s) &= \\nS(s,u_0), \\\\\nz(s) &= \\nE(t-s,y(s)).\n\\end{align*}\nIn this notation, we have\n\\begin{align*}\nz(s) &= \\nE(t-s,\\nS(s,u_0)), \\\\\n\\text{satisfying} \\quad z(0) &= \\nE(t,u_0), ~~z(t) = \\nS(t,u_0).\n\\end{align*}\nThus,\n\\begin{equation} \\label{Lint-1}\n\\nL(t,u_0) = \\nS(t,u_0) - \\nE(t,u_0) = \\int_0^t \\ds z(s) \\,{\\mathrm d} s,\n\\end{equation}\nwith\n\\begin{equation*}\n\\ds z(s) = -F(z(s)) + \\pdtwo\\nE(t-s,y(s)) \\cdot \\ds y(s).\n\\end{equation*}\nNow, using~\\eqref{fi}\\footnote{{\\em Mutatis mutandis:} $ s,t-s $ and $ y(s) $\n play the role of $ 0,t $ and $ u_0 $\n from~\\eqref{fi}.}\nthis can be rewritten in the form\n\\begin{align*}\n\\ds z(s)\n&= \\ub{-F(\\nE(t-s,y(s))) + \\pdtwo\\nE(t-s,y(s)) \\cdot F(y(s))}{=\\,0} \\\\\n& \\quad {} + \\pdtwo\\nE(t-s,y(s)) \\cdot\n \\big( \\ds y(s) - F(y(s)) \\big) \\\\\n&= \\pdtwo\\nE(t-s,y(s)) \\cdot \\nDc(s,u_0),\n\\end{align*}\nand together with~\\eqref{Lint-1},\nidentity~\\eqref{GAL-identity} immediately follows.\n\\hfill $ \\square $\n\n\\begin{remark}\nDue to~\\eqref{fi}, an alternative, plausible way to define the defect is\n\\begin{equation} \\label{cdnonlin-alt}\n\\nD(t,u) = \\pdone\\nS(t,u) - \\pdtwo\\nS(t,u) \\cdot F(u).\n\\end{equation}\nThen,\n\\begin{equation*}\n\\nL(t,u_0) =\n\\int_0^t \\ds \\nS(s,\\nE(t-s,u_0))\\,{\\mathrm d} s =\n\\int_0^t \\nD(s,\\nE(t-s,u_0))\\,{\\mathrm d} s.\n\\end{equation*}\n\\end{remark}\n\n\\begin{remark} \\label{rem:DcD}\nWe can express the modified defect~\\eqref{cdnonlin-alt}\nin terms of $ \\nDc(t,u) $ plus a higher-order perturbation,\n\\begin{align*}\n&\\pdone\\nS(t,u) - \\pdtwo\\nS(t,u) \\cdot F(u) \\\\\n&= \\big( \\pdone\\nS(t,u) - F(\\nS(t,u)) \\big)\n + \\big( F(\\nS(t,u) - \\pdtwo\\nS(t,u) \\cdot F(u) \\big) \\\\\n&= \\nDc(t,u)\n + \\ub{\\big( F(\\nE(t,u)) - \\pdtwo\\nE(t,u) \\cdot F(u) \\big)}{=\\,0} \\\\\n& \\quad {} + \\ub{\\big( F(\\nS(t,u)) - F(\\nE(t,u)) \\big)}\n {=\\,\\Order(t^{p+1})}\n + \\ub{\\big( \\pdtwo\\nS(t,u) - \\pdtwo\\nE(t,u) \\big)}\n {=\\,\\Order(t^{p+1})} \\cdot\\,F(u) \\\\\n&= \\nDc(t,u) + \\Order(t^{p+1}).\n\\end{align*}\n\\end{remark}\n\nAlso, e.g., a convex combination of~\\eqref{cdnonlin}\nand~\\eqref{cdnonlin-alt} represents a plausible defect.\nIn particular, we will consider the arithmetic mean of~\\eqref{cdnonlin}\nand~\\eqref{cdnonlin-alt} (see~\\eqref{sdnonlin} below),\nand we will introduce a symmetrized variant of Theorem~\\ref{th:GAL},\nsee Theorem~\\ref{th:GALS} below.\n\n\\subsection{Symmetrization}\nThe following considerations are relevant for the case\nwhere the approximate flow $ \\nS $ is self-adjoint\n(symmetric, time-reversible),\\footnote{Definition~\\eqref{sdnonlin} and\n the assertion of Theorem~\\ref{th:GALS}\n are independent of this assumption.\n However, our results derived later on\n essentially depend on it, in particular\n Theorem~\\ref{th:Sdach}.}\ni.e.,\n\\begin{equation} \\label{Ssymm}\n\\nS(-t,\\nS(t,u)) = u.\n\\end{equation}\nSelf-adjoint schemes have an even order $ p $,\nsee~\\cite[Theorem~II.3.2]{haireretal02}.\n\nThe identity\\footnote{\n In the terminology of Lie calculus\n (cf.~for instance~\\cite{haireretal02}), with\n \\begin{equation*}\n (D_F\\,G)(u) := G'(u) \\cdot F(u) =\n \\tfrac{{\\mathrm d}}{{\\mathrm d} t} G(\\nE(t,u))|_{t=0},\n \\end{equation*}\n and\n \\begin{equation*}\n {\\mathrm e}^{t D_F} G(u) := G(\\nE(t,u)),\n \\end{equation*}\n we have (set $ G=\\Id $ and $ G=F $, respectively)\n \\begin{equation*}\n F(\\nE(t,u)) = F({\\mathrm e}^{t D_F} u) = {\\mathrm e}^{t D_F} F(u).\n \\end{equation*}\n In this formalism,~\\eqref{quaxi} assumes a more `symmetric flavour',\n as in the linear case (see~\\cite{auzingeretal18a}),\n \\begin{equation*}\n \\pdone\\nE(t,u)\n = \\th \\big( F({\\mathrm e}^{t D_F}u) + {\\mathrm e}^{t D_F} F(u) \\big).\n \\end{equation*}\n \\rev{However, in the present context this formalism is\n of little practical use,\n and we stick to explicit, classical notation.}\n}\n\\begin{equation} \\label{quaxi}\n\\pdone\\nE(t,u)\n= \\th \\big( F(\\nE(t,u)) + \\pdtwo\\nE(t,u) \\cdot F(u) \\big),\n\\end{equation}\nwhich is valid due to~\\eqref{fi},\nmotivates the definition of the {\\em symmetrized defect}\n\\begin{equation} \\label{sdnonlin}\n\\nDs(t,u) =\n\\pdone\\nS(t,u) - \\th \\big( F(\\nS(t,u)) + \\pdtwo\\nS(t,u) \\cdot F(u) \\big),\n\\end{equation}\nsatisfying $ \\nDs(t,u) = \\nDc(t,u) + \\Order(t^{p+1}) $\n(see Remark~\\ref{rem:DcD}).\n\n\\begin{theorem}\\label{th:GALS}\nIn terms of the symmetrized defect~\\eqref{sdnonlin},\nthe local error has the integral representation\n\\begin{equation} \\label{GALS-identity}\n\\nL(t,u_0) =\n\\int_0^t \\pdtwo\\nE(\\tfrac{t-s}{2},\\nS(s,\\nE(\\tfrac{t-s}{2},u_0)))\n \\cdot \\nDs(s,\\nE(\\tfrac{t-s}{2},u_0))\\,{\\mathrm d} s.\n\\end{equation}\n\\end{theorem}\n\\begin{figure}[!ht]\n\\begin{center}\n\\quad\\includegraphics[width=0.9\\textwidth]{GALS.pdf}\n\\caption{Lady Windermere's Fan, Act 2\\,\\nicefrac{1}{2}\n\\label{fig:GALS}}\n\\end{center}\n\\end{figure}\n{\\em Proof.}\nWe reason in a similar way as in the proof of Theorem~\\ref{th:GAL},\nbut now in the spirit of Figure~\\ref{fig:GALS}.\nFor fixed $ t $, let\n\\begin{align*}\nx(s) &= \\nE(\\tfrac{t-s}{2},u_0), \\\\\ny(s) &= \\nS(s,x(s)), \\\\\nz(s) &= \\nE(\\tfrac{t-s}{2},y(s)).\n\\end{align*}\nIn this notation, we have\n\\begin{align*}\nz(s) &= \\nE(\\tfrac{t-s}{2},\\nS(s,\\nE(\\tfrac{t-s}{2},u_0))), \\\\\n\\text{satisfying} \\quad z(0) &= \\nE(t,u_0), ~~z(t) = \\nS(t,u_0).\n\\end{align*}\nThus,\n\\begin{equation} \\label{Lint-2}\n\\nL(t,u_0) = \\nS(t,u_0) - \\nE(t,u_0) = \\int_0^t \\ds z(s) \\,{\\mathrm d} s,\n\\end{equation}\nwith\n\\begin{equation*}\n\\ds z(s) = -\\th F(z(s)) + \\pdtwo\\nE(\\tfrac{t-s}{2},y(s)) \\cdot \\ds y(s).\n\\end{equation*}\nNow, using~\\eqref{fi}\\footnote{{\\em Mutatis mutandis:}\n $ s,\\frac{t-s}{2} $ and $ y(s) $\n play the role of $ 0,t $ and $ u_0 $\n from~\\eqref{fi}.}\nthis can be rewritten in the form\n\\begin{subequations}\n\\begin{equation} \\label{GALS-identity-proof-2}\n\\begin{aligned}\n\\ds z(s)\n&= \\th \\ub{\\big( -F(\\nE(\\tfrac{t-s}{2},y(s))) + \\pdtwo\\nE(\\tfrac{t-s}{2},y(s)) \\cdot F(y(s)) \\big)}{=\\,0} \\\\\n& \\quad {} + \\pdtwo\\nE(\\tfrac{t-s}{2},y(s)) \\cdot \\big( \\ul{\\ds y(s) - \\th F(y(s))} \\big).\n\\end{aligned}\n\\end{equation}\nFurthermore, from the definition~\\eqref{sdnonlin}\nof $ \\nDs(s,u) $, with $ u=x(s) $ we obtain\n\\begin{equation} \\label{GALS-identity-proof-3}\n\\begin{aligned}\n&\\ul{\\ds y(s) - \\th F(y(s))} \\\\\n&= \\pdone\\nS(s,x(s)) +\n \\pdtwo\\nS(s,x(s)) \\cdot \\big( -\\th F(x(s)) \\big) - \\th F(y(s)) \\\\\n&= \\pdone\\nS(s,x(s)) -\n \\th \\big( F(y(s)) + \\pdtwo\\nS(s,x(s)) \\cdot F(x(s)) \\big) \\\\\n&= \\nDs(s,x(s)).\n\\end{aligned}\n\\end{equation}\n\\end{subequations}\nAfter inserting~\\eqref{GALS-identity-proof-3} into~\\eqref{GALS-identity-proof-2},\ntogether with~\\eqref{Lint-2} we obtain~\\eqref{GALS-identity}.\n\\hfill $ \\square $\n\n\\section{Classical and symmetrized defect-based local error estimation}\n\\label{sec:locerrest}\n\n\\paragraph{Defect-based local error estimate}\nThe idea is due to~\\cite{auzingeretal18a,auzingeretal13b}.\nLet $ \\nD(t,u)=\\nDc(t,u) $ or $ \\nDs(t,u) $, respectively,\nand denote the integrands in~\\eqref{GAL-identity} respectively~\\eqref{GALS-identity},\ngenerically by $ \\Theta(s) $. Due to order~$ p $ we have\n$ \\nD(s,u) = \\Order(s^p ) $ and $ \\Theta(s) = \\Order(s^p) $, whence\n\\begin{equation} \\label{LDest}\n\\begin{aligned}\n\\nL(t,u_0) =\n\\int_0^t \\Theta(s)\\,{\\mathrm d} s\n&\\approx\n\\int_0^t \\tfrac{s^p}{p!}\\,\\Theta^{(p)}(0)\\,{\\mathrm d} s =\n\\tfrac{t^{p+1}}{(p+1)!}\\,\\Theta^{(p)}(0) \\\\\n&\\approx \\tfrac{t}{p+1}\\,\\Theta(t) =\n\\tfrac{t}{p+1}\\nD(t,u_0).\n\\end{aligned}\n\\end{equation}\nHere, `$ \\approx $' means asymptotic approximation at the level\n$ \\Order(t^{p+2}) $.\n\\rev{\nThis approximation can be interpreted as an Hermite-type quadrature\nof order $ p+1 $ for the local error integral,\nwhere the quadrature error depends on\n$ \\tfrac{\\partial^{p+1}}{\\partial s^{p+1}}\\nD(s,u_0) = \\Order(1) $\ndue to $ \\nD(s,u_0) = \\Order(s^p) $, whence\n\\begin{equation*}\n\\nL(t,u_0) = \\tfrac{t}{p+1}\\,\\nD(t,u_0) + \\Order(t^{p+2})\n\\quad \\text{for $ \\nD = \\nDc $\\, or \\,$ \\nD = \\nDs $.}\n\\end{equation*}\n}\nFor a precise analysis of the resulting quadrature error based on its Peano representation\nfor the classical case in concrete applications,\nsee for instance~\\cite{auzingeretal18a,auzingeretal13a,auzingeretal13b}.\n\nNext we show that for the self-adjoint case and using the\nsymmetrized defect~\\eqref{sdnonlin} we even\nhave\\footnote{For the linear constant coefficient case\n see~\\cite[Theorem~1]{auzingeretal18a}.}\n\\begin{equation*}\n\\nL(t,u_0) = \\tfrac{t}{p+1} \\nDs(t,u_0) + \\Order(t^{p+3}).\n\\end{equation*}\nTo this end we consider the corrected scheme\n\\begin{equation} \\label{Sdach-nonlin}\n\\nSD(t,u) = \\nS(t,u) - \\tfrac{t}{p+1} \\nDs(t,u),\n\\end{equation}\nand we show that it is of (global) order $ p+2 $.\n\n\\begin{theorem} \\label{th:Sdach}\nConsider a self-adjoint one-step scheme of (even) order $ p \\geq 2 $,\nrepresented by its flow $ \\nS(t,u) $ satisfying~\\eqref{Ssymm},\napplied to an evolution equation~\\eqref{y'=F(y)}.\nThen the corrected scheme~\\eqref{Sdach-nonlin} is almost self-adjoint,\ni.e.,\n\\begin{subequations} \\label{Sdach-results}\n\\begin{equation} \\label{Sdach-almost-selfadjoint}\n\\nSD(-t,\\nSD(t,u_0)) = u_0 + \\Order(t^{2p+2}).\n\\end{equation}\nMoreover, the local error $ \\nLD(t,u) = \\nSD(t,u) - \\nE(t,u) $\nof the corrected scheme satisfies\n\\begin{equation} \\label{Sdach-highord-general-nonlin}\n\\nLD(t,u_0) = \\Order(t^{p+3}),\n\\end{equation}\n\\end{subequations}\ni.e., $ \\nSD $ has even order $ p+2 $.\n\\end{theorem}\n{\\em Proof.}\nWe consider\n\\begin{align*}\n\\nSD(-t,\\nSD(t,u_0))\n&= \\nS\\big(\\!-\\!t,\\nSD(t,u_0)) + \\tfrac{t}{p+1}\\,\\nDs(-t,\\nSD(t,u_0)) \\\\\n&= \\nS\\big(\\!-\\!t,\\nS(t,u_0) - \\tfrac{t}{p+1}\\,\\nDs(t,u_0)\\big) \\\\\n& \\quad {} + \\tfrac{t}{p+1}\\,\\nDs\\big(\\!-\\!t,\\nS(t,u_0) - \\tfrac{t}{p+1}\\,\\nDs(t,u_0) \\big),\n\\end{align*}\napply Taylor expansion, and make use of the assumption that\n$ \\nS $ is self-adjoint, and the fact that $ t \\nDs(t,u_0) = \\Order(t^{p+1}) $:\n\\begin{align} \\label{SdachSdach}\n&\\nSD(-t,\\nSD(t,u_0))\n = \\ub{\\nS(-t,\\nS(t,u_0))}{=\\,u_0} \\\\\n& \\quad {} + \\pdtwo\\nS(-t,\\nS(t,u_0))\n \\,\\cdot\\,\\big(\\!-\\!\\tfrac{t}{p+1}\\,\\nDs(t,u_0)\\big) + \\Order(t^{2p+2}) \\notag \\\\\n& \\quad {} + \\tfrac{t}{p+1}\\,\\nDs(-t,\\nS(t,u_0)) + \\Order(t^{2p+2}) \\notag \\\\\n&= u_0 - \\tfrac{t}{p+1}\\,\n \\ub{\\big( \\pdtwo\\nS(-t,\\nS(t,u_0)) \\cdot \\nDs(t,u_0) -\n \\nDs(-t,\\nS(t,u_0)) \\big)}\n {\\text{\\small\\bf\\em critical term}} \\notag\n+\\, \\Order(t^{2p+2}).\n\\end{align}\nNow we collect the contributions to the $ {\\text{\\bf\\em critical term}} $.\nFirst, from~\\eqref{Ssymm}\nwe have\\footnote{Here,\n $ \\pdt\\nS(-t,\\nS(t,u_0)) $ means $ \\pdt{\\tilde\\nS}(t,u_0) $\n with $ {\\tilde\\nS}(t,u_0) = \\nS(-t,\\nS(t,u_0)) $.}\n\\begin{equation*}\n0 = \\pdt\\ub{\\nS(-t,\\nS(t,u_0))}{=\\,u_0}\n= -\\pdone\\nS(-t,\\nS(t,u_0))\n + \\pdtwo\\nS(-t,\\nS(t,u_0))\\,\\cdot\\,\\pdone\\nS(t,u_0).\n\\end{equation*}\nThis implies\n\\begin{align*}\n&\\nDs(-t,\\nS(t,u_0)) = \\\\\n&= \\pdone\\nS(-t,\\nS(t,u_0))\n - \\th \\big( F(\\ub{\\nS(-t,\\nS(t,u_0))}{=\\,u_0}) + \\pdtwo\\nS(-t,\\nS(t,u_0))\n \\cdot F(\\nS(t,u_0)) \\big) \\\\\n&= \\pdtwo\\nS(-t,\\nS(t,u_0)) \\cdot \\pdone\\nS(t,u_0)\n - \\th F(u_0) - \\th \\pdtwo\\nS(-t,\\nS(t,u_0)) \\cdot F(\\nS(t,u_0)) \\\\\n&= \\pdtwo\\nS(-t,\\nS(t,u_0)) \\cdot\n \\big( \\pdone\\nS(t,u_0) - \\th F(\\nS(t,u_0)) \\big) - \\th F(u_0).\n\\end{align*}\nSummarizing and collecting terms gives\n\\begin{align*}\n&\\text{\\normalsize\\bf\\em critical term} \\,= \\\\\n&= \\pdtwo\\nS(-t,\\nS(t,u_0)) \\cdot \\nDs(t,u_0)\n - \\nDs(-t,\\nS(t,u_0)) \\\\\n&= \\pdtwo\\nS(-t,\\nS(t,u_0)) \\cdot\n \\Big( \\canc{\\,\\pdone\\nS(t,u_0)\n - \\th F(\\nS(t,u_0))}\n - \\th \\pdtwo\\nS(t,u_0) \\cdot F(u_0) \\Big) \\\\\n& \\quad {} -\n \\pdtwo\\nS(-t,\\nS(t,u_0)) \\cdot\n \\Big( \\canc{\\,\\pdone\\nS(t,u_0) - \\th F(\\nS(t,u_0))}\\,\\Big)\n - \\th F(u_0) \\\\\n&= -\\th \\big( \\pdtwo\\nS(-t,\\nS(t,u_0)) \\cdot \\pdtwo\\nS(t,u_0)\n - \\Id \\big) \\cdot F(u_0) \\\\\n&= -\\th \\big( \\ub{\\pdunot\\,\\nS(-t,\\nS(t,u_0))}{=\\;\\Id} - \\,\\Id \\big) \\cdot F(u_0) = 0.\n\\end{align*}\nThus,~\\eqref{SdachSdach} indeed simplifies to~\\eqref{Sdach-almost-selfadjoint},\n\\begin{equation*}\n\\nSD(-t,\\nSD(t,u_0)) = u_0 + \\Order(t^{2p+2}).\n\\end{equation*}\nThe proof of~\\eqref{Sdach-highord-general-nonlin}\nnow works in the same way as for the linear case~\\cite[proof of Theorem 1]{auzingeretal18a},\nfollowing the argument from~\\cite[Theorem~II.3.2]{haireretal02}.\n\\hfill $ \\square $\n\nAssertion~\\eqref{Sdach-highord-general-nonlin} is equivalent\nto the fact that the symmetrized defect-based local error estimator\naccording to~\\eqref{LDest},\n\\begin{equation} \\label{improved-errest}\n\\nLLtildes(t,u_0) := \\tfrac{t}{p+1} \\nDs(t,u_0)\n\\end{equation}\nis indeed of a better asymptotic quality than the classical defect,\nwith a deviation\n\\begin{equation} \\label{p+3}\n\\nLLtildes(t,u_0) - \\nL(t,u_0) = \\Order(t^{p+3}),\n\\end{equation}\nand not only $ \\Order(t^{p+2}) $.\n\nIn the following sections we present some examples of self-adjoint\nmethods and show how to evaluate the symmetrized defect $ \\nDs(t,u_0) $\nas the basis for evaluating the local error estimator~\\eqref{improved-errest}.\n\n\\section{Examples for the autonomous case} \\label{sec:appl-auto}\n\n\\subsection{Example: Implicit midpoint rule} \\label{subsec:imr}\nWe illustrate the defect computation for the simplest example\nof a self-adjoint implicit one-step integrator.\nThe flow of the second order implicit midpoint rule\nis defined by the relation\n\\begin{equation*}\n\\nS(t,u) = u + t\\,F(\\th(u+\\nS(t,u))).\n\\end{equation*}\nWith\n\\begin{equation}\\label{weh}\nw = \\nS(t,u)\n\\end{equation}\nwe obtain\n\\begin{subequations} \\label{dS-imr}\n\\begin{equation*}\n\\pdone\\nS(t,u)\n= \\ub{F(\\th(u+w))}{=\\,(w-u)\/t}\n +\\, t\\,F'\\big( \\th(u+w) \\big) \\cdot \\th \\pdone\\nS(t,u).\n\\end{equation*}\nThus, $ x = \\pdone\\nS(t,u) $ is obtained by solving the linear system\n\\begin{equation} \\label{dS-imr-t}\n\\big( \\Id - \\tfrac{t}{2}F'(\\th(u+w)) \\big) \\cdot x = F(\\th(u+w)).\n\\end{equation}\nFurthermore,\n\\begin{align*}\n\\pdtwo\\nS(t,u)\n&= \\Id + t F'(\\th(u+\\nS(t,u))) \\cdot \\big(\\th (\\Id + \\pdtwo\\nS(t,u)) \\big) \\\\\n&= \\Id + \\tfrac{t}{2} F'(\\th(u+w))\n + \\tfrac{t}{2} F'(\\th(u+w)) \\cdot \\pdtwo\\nS(t,u),\n\\end{align*}\nwhence\n\\begin{equation*}\n\\big( \\Id - \\tfrac{t}{2}F'(\\th(u+w)) \\big) \\cdot \\pdtwo\\nS(t,u)\n= \\big( \\Id + \\th F'(\\th(u+w)) \\big).\n\\end{equation*}\nThus, $ y = \\pdtwo\\nS(t,u) \\cdot F(u) $ is obtained by solving the linear system\n\\begin{equation} \\label{dS-imr-u}\n\\big( \\Id - \\tfrac{t}{2}F'(\\th(u+w)) \\big) \\cdot y\n= \\big( \\Id + \\tfrac{t}{2} F'(\\th(u+w)) \\big) \\cdot F(u),\n\\end{equation}\n\\end{subequations}\nwith the same matrix as in~\\eqref{dS-imr-t}.\n\nThis gives the following defect representations.\n\\begin{itemize}\n\\item Classical defect:\n\\begin{equation*}\n\\nDc(t,u) = x - F(w),\n\\end{equation*}\nwhere $ x = \\pdone\\nS(t,u) $ is the solution of~\\eqref{dS-imr-t}\nand with $w$ from~\\eqref{weh}.\n\\item Symmetrized defect:\n\\begin{equation*}\n\\nDs(t,u) =\nx - \\th (F(w) + y),\n\\end{equation*}\nwhere $ x = \\pdone\\nS(t,u) $ is the solution of~\\eqref{dS-imr-t},\nand $ y = \\pdtwo\\nS(t,u) \\cdot F(u) $ is the solution\nof~\\eqref{dS-imr-u}. This can also be\nwritten in the form\n\\begin{equation*}\n\\nDs(t,u) = z - \\th F(w),\n\\end{equation*}\nwhere $ z = x - \\th y $ is the solution of\n\\begin{align*}\n\\big( \\Id - \\tfrac{t}{2}F'(\\th(u+w)) \\big) \\cdot z\n&= F(\\th(u+w)) - \\th F(u)\n - \\tfrac{t}{4} F'(\\th(u+w)) \\cdot F(u).\n\\end{align*}\nThus, the computation of the symmetrized defect requires only one additional\nevaluation of $F$ as compared to the classical version.\n\\end{itemize}\n\n\\subsection{Example: Strang splitting applied to a semilinear evolution equation}\\label{subsec:strang}\nWe consider a semilinear problem of the form\n\\begin{equation*}\n\\dt u(t) = F(u(t)) = A u(t) + B(u(t)), \\quad u(0) = u_0.\n\\end{equation*}\nDenoting the flow of the nonlinear part by $ \\nE_B(t,u) $,\nthe second order self-adjoint Strang splitting scheme is given by\n\\begin{equation*}\n\\nS(t,u) = {\\mathrm e}^{\\tth A} \\nE_B\\big(t,{\\mathrm e}^{\\tth A} u\\big).\n\\end{equation*}\nLet\n\\begin{equation*}\nv_1 = {\\mathrm e}^{\\tth A} u,\n\\quad\nv_2 = \\nE_B(t,v_1),\n\\quad\nw = {\\mathrm e}^{\\tth A} v_2 = \\nS(t,u).\n\\end{equation*}\nThen,\n\\begin{align*}\n\\pdone\\nS(t,u)\n&= \\th A \\nS(t,u)\n + {\\mathrm e}^{\\tth A} \\big( \\pdone\\nE_B(t,v_1)\n + \\pdtwo\\nE_B(t,v_1)\n (\\th A v_1)\n \\big) \\\\\n&= \\th A w\n + {\\mathrm e}^{\\tth A} \\big( B(v_2) + \\th \\pdtwo\\nE_B(t,v_1) (A v_1)\n \\big),\n\\end{align*}\nand\n\\begin{equation*}\n\\pdtwo\\nS(t,u) (\\xi)\n = {\\mathrm e}^{\\tth A} \\pdtwo\\nE_B(t,v_1) \\big( {\\mathrm e}^{\\tth A} \\xi \\big).\n\\end{equation*}\nThis gives the following defect representations.\n\\begin{itemize}\n\\item Classical defect:\n\\begin{align}\n\\nDc(t,u)\n&= \\pdone\\nS(t,u) - F(\\nS(t,u)) \\notag \\\\\n&= {\\mathrm e}^{\\tth A}\n \\big( B(v_2)\n + \\th \\pdtwo\\nE_B(t,v_1) \\cdot (A v_1)\n \\big)\n - \\th A w - B(w). \\label{dc-strang}\n\\end{align}\n\\item Symmetrized defect:\n\\begin{align} \\label{ds-strang}\n&\\nDs(t,u)\n= \\pdone\\nS(t,u) - \\th \\big( F(\\nS(t,u))\n + \\pdtwo\\nS(t,u) \\cdot F(u) \\big) \\notag \\\\\n&= \\th \\canc{A w}\n + {\\mathrm e}^{\\tth A}\n \\big( B(v_2)\n + \\th \\pdtwo\\nE_B(t,v_1) \\cdot (A v_1)\n \\big) \\notag \\\\\n& \\quad {} - \\th \\Big(\n \\canc{A w} + B(w) +\n {\\mathrm e}^{\\tth A}\n \\pdtwo\\nE_B(t,v_1)\n \\cdot \\big( {\\mathrm e}^{\\tth A} (A u + B(u)) \\big)\n \\Big) \\notag \\\\\n&= {\\mathrm e}^{\\tth A} B(v_2)\n + \\canc{\\th {\\mathrm e}^{\\tth A} \\pdtwo\\nE_B(t,v_1)\n (A v_1)} \\notag \\\\\n& \\quad {}\n - \\th B(w)\n - \\canc{\\th {\\mathrm e}^{\\tth A} \\pdtwo\\nE_B(t,v_1) (A v_1)}\n - \\th {\\mathrm e}^{\\tth A}\\pdtwo\\nE_B(t,v_1)\n \\big( {\\mathrm e}^{\\tth A} B(u) \\big) \\notag \\\\[\\jot]\n&= {\\mathrm e}^{\\tth A} \\left( B(v_2)\n - \\th \\pdtwo\\nE_B(t,v_1)\n \\big( {\\mathrm e}^{\\tth A} B(u) \\big) \\right)\n - \\th B(w).\n\\end{align}\n\\end{itemize}\nThus,~\\eqref{dc-strang} resp.~\\eqref{ds-strang} require\none evaluation of $ \\pdtwo\\nE_B(t,v_1) \\cdot (\\,\\cdot\\,)$, and\neither one or two evaluations of $ {\\mathrm e}^{\\tth A} (\\,\\cdot\\,) $,\nrespectively.\n\n\\subsection{Algorithmic realization for higher order splitting methods}\\label{subsec:algsplit}\n\nIn Figure~\\ref{fig:algssplit}, we give pseudocodes for the economical algorithmic\nrealization of the symmetrized defect when it is employed in the context\nof splitting methods involving an arbitrary number of $J$ compositions.\nIf we denote the subflow of the nonlinear operator\nby $\\nE_B(t,u_0)$, an $n$-stage splitting approximation is defined by a composition of the two subflows,\n\\begin{equation*}\n\\nS(t,u_0) = \\nE_B(b_J t, \\cdots {\\mathrm e}^{a_2 t A}\\nE_B(b_1 t, {\\mathrm e}^{a_1 t A}u_0)\\cdots ).\n\\end{equation*}\nAn optimized fourth order method we will use in Section~\\ref{subsec:nls} has the coefficient tableau given in Table~\\ref{splitcoeffs}.\n\\begin{table}[ht]\n\\begin{center}\n\\begin{tabular}{|r||r|r|}\n\\hline\n\\multicolumn{1}{|c||}{$i$} & \\multicolumn{1}{c|}{$a_i$} & \\multicolumn{1}{c|}{$b_i$} \\\\ \\hline\\hline\n1 & 0.267171359000977615 & $-$0.361837907604416033 \\\\\n2 & $-$0.033827909669505667 & 0.861837907604416033 \\\\\n3 & 0.533313101337056104 & 0.861837907604416033 \\\\\n4 & $-$0.033827909669505667 & $-$0.361837907604416033 \\\\\n5 & 0.267171359000977615 & 0 \\\\\n\\hline\n\\end{tabular}\n\\caption{Coefficients of the self-adjoint splitting method from~\\cite[\\texttt{Emb 4\/3 AK s}]{splithp}.\\label{splitcoeffs}}\n\\end{center}\n\\end{table}\n\nThe algorithms in Figure~\\ref{fig:algssplit} have\nthe splitting approximation~$ u=\\nS(t,u_0) $ and the symmetrized\ndefect $d=\\nDs(t,u_0)$ as the output; for efficiency, $ u $ and $ d $ are\nevaluated simultaneously. The left algorithm refers to the situation\nwhere the operator $ A $ is linear, and\non the right the general nonlinear case is elaborated.\n\n\\begin{figure}[h!]\n{\\small\n\\begin{tabular}{cc}\n&\n\\begin{boxedminipage}{0.72\\textwidth}\n\\begin{tabbing}\n \\qquad\\= \\kill\n $u=u_0$ \\\\\n $d=-\\tfrac{1}{2}B(u)$ \\\\\n {\\bf for} $j=1:J-1$ \\\\\n \\>$d=d+\\left\\{\n \\begin{array}{rl}\n (a_j-\\tfrac{1}{2})Au,&j=1\\\\\n a_j Au,&j>1\n \\end{array}\n \\right.$\\\\\n \\>$d={\\mathrm e}^{a_j t A}d$\\\\\n \\>$u={\\mathrm e}^{a_j t A}u$\\\\\n \\>$d=d+b_j B(u)$\\\\\n \\>$d=\\partial_2\\nE_B(b_j t,u)\\cdot d$\\\\\n \\>$u=\\nE_B(b_j t,u)$\\\\\n {\\bf end}\\\\\n $d=d+(a_J-\\tfrac{1}{2})Au$\\\\\n $d={\\mathrm e}^{a_J t A}d$\\\\\n $u={\\mathrm e}^{a_J t A}u$\\\\\n $d=d-\\frac{1}{2}B(u)\n\n\\end{tabbing}\n\\end{boxedminipage}\n\\quad\n\\begin{boxedminipage}{0.72\\textwidth}\n\\begin{tabbing}\n \\qquad\\= \\kill\n $u=u_0$ \\\\\n $d=-\\tfrac{1}{2}B(u)$ \\\\\n {\\bf for} $j=1:J-1$ \\\\\n \\>$d=d+\\left\\{\n \\begin{array}{rl}\n (a_j-\\tfrac{1}{2})A(u),&j=1\\\\\n a_j A(u),&j>1\n \\end{array}\n \\right.$\\\\\n \\>$d=\\partial_2\\nE_A(a_j t,u)\\cdot d$\\\\\n \\>$u=\\nE_A(a_j t,u)$\\\\\n \\>$d=d+b_j B(u)$\\\\\n \\>$d=\\partial_2\\nE_B(b_j t,u)\\cdot d$\\\\\n \\>$u=\\nE_B(b_j t,u)$\\\\\n {\\bf end}\\\\\n $d=d+(a_J-\\tfrac{1}{2})A(u)$\\\\\n $d=\\partial_2\\nE_A(a_J t,u)\\cdot d$\\\\\n $u=\\nE_A(a_J t,u)$\\\\\n $d=d-\\frac{1}{2}B(u)\n\n\\end{tabbing}\n\\end{boxedminipage}\n\\end{tabular}\n}\n\\caption{Algorithmic realization of the symmetrized defect for splitting methods.\\newline\nLeft: semilinear case. Right: nonlinear case.\\label{fig:algssplit}}\n\\end{figure}\n\n\\section{The nonautonomous case, with examples} \\label{sec:nonaut}\nThe results from Sections~\\ref{sec:def} and~\\ref{sec:locerrest}\ncarry over to nonautonomous evolution equations\n\\begin{subequations} \\label{y'=F(t,y)}\n\\begin{equation} \\label{y'=F(y)-t}\n\\dt u(t) = F(t,u(t)), \\quad u(t_0)=u_0.\n\\end{equation}\nFor our purpose it is notationally more favorable to\nintroduce the~`local' variable~$ \\tau $, such that\n$ t = t_0 + \\tau $, and reformulate~\\eqref{y'=F(y)-t} in the form\n\\begin{equation} \\label{y'=F(y)-ttau}\n\\dtau u(t_0+\\tau) = F(t_0+\\tau,u(t_0+\\tau)), \\quad u(t_0)=u_0.\n\\end{equation}\n\\end{subequations}\nThe exact flow associated with~\\eqref{y'=F(t,y)}\nis denoted by $ \\nE(\\tau,t_0,u) $.\nIt satisfies\\footnote{Again, $ \\pdone\\nE(\\tau,t_0,u_0) $ denotes\n $ \\dtau\\nE(\\tau,t_0,u_0) $,\n and $ \\pdtwo,\\pdthree $ are defined analogously.}\n\\begin{equation*}\n\\pdone\\nE(\\tau,t_0,u_0) =\nF(t_0+\\tau,\\nE(\\tau,t_0,u_0)),\n\\quad \\nE(0,t_0,u_0) = u_0.\n\\end{equation*}\n\\rev{\nTo infer the appropriate definition of the symmetrized\ndefect in this case there are two approaches, which we\nboth discuss for the sake of completeness. The first\none relies on a direct extension of the\nfundamental identity~\\eqref{fi} (Lemma~\\ref{lem:fi}),\nsee Lemma~\\ref{lem:fin} below. The other approach is based\non reformulating~\\eqref{y'=F(t,y)} in autonomous form in the usual way,\nleading to the same conclusion and showing that the theoretical\nbackground based on Theorems~\\ref{th:GALS} and~\\ref{th:Sdach}\ndirectly carries over to the nonautonomous case.\n\\begin{lemma} \\label{lem:fin}\n\\begin{equation} \\label{fin}\n\\begin{aligned}\n&[\\,\\pdone\\nE(\\tau,t_0,u_0) =\\,] \\\\\n& \\quad F(t_0+\\tau,\\nE(\\tau,t_0,u_0))\n= \\pdtwo\\nE(\\tau,t_0,u_0) + \\pdthree\\nE(\\tau,t_0,u_0) \\cdot F(t_0,u_0).\n\\end{aligned}\n\\end{equation}\n\\end{lemma}\n{\\em Proof.}\nThe idea is the same as in the proof of Lemma~\\ref{lem:fi}.\nWe proceed from the identity\n\\begin{equation*}\n\\nE(\\tau+\\sig,t_0,u_0) = \\nE(\\tau,t_0+\\sig,\\nE(\\sig,t_0,u_0)).\n\\end{equation*}\nDifferentiation with respect to~$ \\sig $ gives\n\\begin{align*}\n\\pdsig\\nE(\\tau+\\sig,t_0,u_0)\n&= \\pdone\\nE(\\tau+\\sig,t_0,u_0), \\\\\n\\pdsig\\nE(\\tau+\\sig,t_0,u_0)\\big|_{\\sig=0}\n&= \\pdone\\nE(\\tau,t_0,u_0) = F(t_0+\\tau,\\nE(\\tau,t_0,u_0)),\n\\end{align*}\nand on the other hand,\n\\begin{align*}\n&\\pdsig\\nE(\\tau,t_0+\\sig,\\nE(\\sig,t_0,u_0)) \\\\\n&~~= \\pdtwo\\nE(\\tau,t_0+\\sig,\\nE(\\sig,t_0,u_0))\n + \\pdthree\\nE(\\tau,t_0+\\sig,\\nE(\\sig,t_0,u_0))\n \\cdot \\pdone\\nE(\\sig,t_0,u_0), \\\\\n&\\pdsig\\nE(\\tau,t_0+\\sig,\\nE(\\sig,t_0,u_0))\\big|_{\\sig=0} \\\\\n&~~= \\pdtwo\\nE(\\tau,t_0,\\nE(0,t_0,u_0))\n + \\pdthree\\nE(\\tau,t_0,\\nE(0,t_0,u_0))\n \\cdot \\pdone\\nE(0,t_0,u_0) \\\\\n&~~= \\pdtwo\\nE(\\tau,t_0,u_0)\n + \\pdthree\\nE(\\tau,t_0,u_0)\n \\cdot F(t_0,u_0),\n\\end{align*}\nwhich completes the proof.\n\\hfill $ \\square $\n}\n\n\\rev{\nAlternatively, we can reformulate~\\eqref{y'=F(y)-ttau} in autonomous form,\ndefining\n\\begin{equation*}\nU = \\left\\lgroup \\begin{array}{c}\n t_0+\\tau \\\\\n u\n \\end{array} \\right\\rgroup,\n\\quad\n{\\bm F}(U) = \\left\\lgroup \\begin{array}{c}\n 1 \\\\ F(t_0+\\tau,u)\n \\end{array} \\right\\rgroup\n\\end{equation*}\nwhence\n\\begin{equation*}\n\\dtau U(\\tau) = {\\bm F}(U(\\tau)),\n\\quad\nU(0) = \\left\\lgroup \\begin{array}{c}\n t_0 \\\\ u_0\n \\end{array} \\right\\rgroup,\n\\end{equation*}\nand with the flow\n\\begin{equation*}\n{\\bm \\nE}(\\tau,U) =\n{\\bm \\nE}(\\tau,t_0,u) =\n\\left\\lgroup \\begin{array}{c}\n t_0+\\tau \\\\ \\nE(\\tau,t_0,u)\n\\end{array} \\right\\rgroup\n\\end{equation*}\nsatisfying the fundamental identity according to Lemma~\\ref{lem:fi},\n\\begin{equation} \\label{fit}\n[\\,{\\pdone{\\bm \\nE}}(\\tau,U) =\\,]~\\;\n{\\bm F}({\\bm \\nE}(\\tau,U)) = {\\pdtwo{\\bm \\nE}}(\\tau,U) \\cdot {\\bm F}(U).\n\\end{equation}\nWith $ U_0 = (t_0,u_0) $ we have\n\\begin{equation*}\n{\\pdone{\\bm \\nE}}(\\tau,U_0) = {\\bm F}({\\bm \\nE}(\\tau,U_0))\n = \\left\\lgroup \\begin{array}{c}\n 1 \\\\\n F(t_0+\\tau,\\nE(\\tau,t_0,u_0))\n \\end{array} \\right\\rgroup,\n\\quad {\\bm \\nE}(0,U_0) = U_0,\n\\end{equation*}\nand\n\\begin{equation*}\n{\\pdtwo{\\bm \\nE}}(\\tau,U_0) =\n\\left\\lgroup \\begin{array}{cc}\n1 & 0 \\\\\n\\pdtwo\\nE(\\tau,t_0,u_0) &\n\\pdthree\\nE(\\tau,t_0,u_0)\n\\end{array} \\right\\rgroup.\n\\end{equation*}\nUsing~\\eqref{fit} and evaluating the second component\nagain gives~\\eqref{fin}.\n}\n\n\\rev{\nFor a one-step approximation represented by\n$ \\nS(\\tau,t_0,u_0) \\approx \\nE(\\tau,t_0,u_0) $,\nrelation~\\eqref{fin} again motivates the definition of the symmetrized defect\n\\begin{align}\n&\\nDs(\\tau,t_0,u_0)\n= \\pdone\\nS(\\tau,t_0,u_0) \\notag \\\\\n& \\quad {} - \\th\\big( F(t_0+\\tau,\\nS(\\tau,t_0,u_0))\n + \\pdtwo\\nS(\\tau,t_0,u_0)\n + \\pdthree\\nS(\\tau,t_0,u_0)F(t_0,u_0)\n \\big) \\label{dst-n} \\\\\n&= \\big( \\pdone - \\th\\pdtwo \\big) \\nS(\\tau,t_0,u_0)\n - \\th\\big( F(t_0+\\tau,\\nS(\\tau,t_0,u_0)) + \\pdthree\\nS(\\tau,t_0,u_0)F(t_0,u_0) \\big).\n\\notag\n\\end{align}\n}\n\n\\paragraph{The linear nonautonomous case}\nNow we consider the case of a linear time-dependent problem\n\\begin{equation} \\label{y'=A(t)y}\n\\dtau u(t_0+\\tau) = A(t_0+\\tau) u(t_0+\\tau), \\quad u(t_0) = u_0.\n\\end{equation}\nSince in the present case the flow\nis linear in $ u_0 $, we write it in the simplified\nform\\footnote{\\eqref{flowdet-magnus}\n is a minor abuse of notation.\n Note that $ \\nE(\\tau,t_0) $ can be expressed\n as a matrix exponential\n via the so-called Magnus expansion,\n\n see for instance~\\cite{auzingeretal18a,blanesetal08b}.}\n\\begin{equation} \\label{flowdet-magnus}\n\\nE(\\tau,t_0,u_0) =: \\nE(\\tau,t_0) u_0,\n\\end{equation}\nsatisfying\n\\begin{equation*}\n\\pdone\\nE(\\tau,t_0) =\nA(t_0+\\tau)\\,\\nE(\\tau,t_0),\n\\quad \\nE(0,t_0) = \\Id.\n\\end{equation*}\nNote that\n\\begin{equation} \\label{nonaut-timereversed}\n\\nE(-\\tau,t_0+\\tau)\\,\\nE(\\tau,t_0) = \\Id.\n\\end{equation}\nA one-step approximation $ \\nS(\\tau,t_0,u_0) \\approx \\nE(\\tau,t_0,u_0) $,\nis also typically linear in~$ u_0 $,\n\\begin{equation*}\n\\nS(\\tau,t_0,u_0) =: \\nS(\\tau,t_0) u_0 \\approx \\nE(\\tau,t_0) u_0.\n\\end{equation*}\nIn particular, we again focus on self-adjoint schemes which are\ncharacterized by the identity (cf.~\\eqref{nonaut-timereversed})\n\\begin{equation} \\label{nonaut-symmscheme}\n\\nS(-\\tau,t_0+\\tau)\\,\\nS(\\tau,t_0) = \\Id.\n\\end{equation}\nFor $ {\\nS(\\tau,t_0)u_0} $\nwe obtain the following defect representations.\n\\begin{itemize}\n\\item\nClassical defect:\n\\begin{equation*}\n\\nDc(\\tau,t_0,u_0) =: \\nDc(\\tau,t_0)u_0,\n\\end{equation*}\nwith\n\\begin{equation} \\label{dct}\n\\nDc(\\tau,t_0) =\n\\pdone\\nS(\\tau,t_0) - A(t_0+\\tau) \\nS(\\tau,t_0).\n\\end{equation}\n\\item\nSymmetrized defect~\\eqref{dst-n}:\n\\begin{equation*}\n\\nDs(\\tau,t_0,u_0) =: \\nDs(\\tau,t_0)u_0,\n\\end{equation*}\nwith\n\\begin{align}\n&\\nDs(\\tau,t_0) =\n\\pdone\\nS(\\tau,t_0) -\n\\th\\big( A(t_0+\\tau) \\nS(\\tau,t_0)\n + \\pdtwo\\nS(\\tau,t_0)\n + \\nS(\\tau,t_0)A(t_0)\n \\big) \\notag \\\\\n& \\quad = \\big( \\pdone - \\th\\pdtwo \\big) \\nS(\\tau,t_0)\n- \\th\\big( A(t_0+\\tau) \\nS(\\tau,t_0) + \\nS(\\tau,t_0) A(t_0) \\big). \\label{dst}\n\\end{align}\n\\end{itemize}\n\n\\subsection{Example: Exponential midpoint rule}\\label{subsec:emr}\nThe self-adjoint second order exponential midpoint rule\napplied to~\\eqref{y'=A(t)y} is given by\n\\begin{equation*}\nS(\\tau,t_0) = {\\mathrm e}^{\\tau A(t_0+\\frac{\\tau}{2})}.\n\\end{equation*}\nLet\n\\begin{equation*}\n\\nR(\\tau,t_0)(\\,\\cdot\\,) =\n\\tfrac{{\\mathrm d}}{{\\mathrm d}\\Omega} {\\mathrm e}^{\\Omega}\\big|_{\\Omega=\\tau A(t_0+\\frac{\\tau}{2})}(\\,\\cdot\\,),\n\\end{equation*}\nwhere $ \\tfrac{{\\mathrm d}}{{\\mathrm d}\\Omega} {\\mathrm e}^\\Omega $ denotes the Fr{\\'e}chet\nderivative of the matrix exponential, see~\\eqref{mimi} below. Then,\n\\begin{align*}\n\\pdone \\nS(\\tau,t_0)\n&= \\nR(\\tau,t_0) \\big( \\pdtau(\\tau A(t_0+\\tfrac{\\tau}{2}) \\big) \\\\\n&= \\nR(\\tau,t_0) \\big( A(t_0+\\tfrac{\\tau}{2})\n + \\th\\tau A'(t_0+\\tfrac{\\tau}{2}) \\big), \\\\\n\\pdtwo \\nS(\\tau,t_0)\n&= \\nR(\\tau,t_0) \\big( \\pdtnot(\\tau A(t_0+\\tfrac{\\tau}{2}) \\big) \\\\\n&= \\nR(\\tau,t_0) \\big( \\tau A'(t_0+\\tfrac{\\tau}{2}) \\big).\n\\end{align*}\nThis gives the following defect representations.\n\\begin{itemize}\n\\item\nClassical defect~\\eqref{dct}:\n\\begin{equation} \\label{dcA(t)}\n\\nDc(\\tau,t_0) =\n\\nR(\\tau,t_0) \\big( A(t_0+\\tfrac{\\tau}{2})\n + \\th\\tau A'(t_0+\\tfrac{\\tau}{2}) \\big)\n- A(t_0+\\tau) \\nS(\\tau,t_0).\n\\end{equation}\n\\item\nSymmetrized defect~\\eqref{dst}:\n\\begin{align}\n&\\nDs(\\tau,t_0) =\n\\nR(\\tau,t_0) \\big( A(t_0+\\tfrac{\\tau}{2})\n + \\canc{\\th\\tau A'(t_0+\\tfrac{\\tau}{2})} \\big) \\notag \\\\\n& \\qquad\\qquad {} - \\th\\big( A(t_0+\\tau) \\nS(\\tau,t_0)\n + \\nR(\\tau,t_0) \\big( \\canc{\\tau A'(t_0+\\tfrac{\\tau}{2})}\n - \\nS(\\tau,t_0) A(t_0)\n \\big) \\notag \\\\\n& \\quad {} = \\nR(\\tau,t_0) \\big( A(t_0+\\tfrac{\\tau}{2}) \\big)\n - \\th \\big( A(t_0+\\tau) \\nS(\\tau,t_0) + \\nS(\\tau,t_0) A(t_0) \\big). \\label{dsA(t)}\n\\end{align}\n\\end{itemize}\nHere, the explicit representation\n\\begin{align} \\label{mimi}\n\\nR(\\tau,t_0) \\big( V \\big)\n&= \\int_0^1 {\\mathrm e}^{\\sig\\tau A(t_0+\\frac{\\tau}{2})}\n V\n {\\mathrm e}^{(1-\\sig)\\tau A(t_0+\\frac{\\tau}{2})}\\,{\\mathrm d}\\sig \\notag \\\\\n&= \\int_0^1 {\\mathrm e}^{\\sig\\tau A(t_0+\\frac{\\tau}{2})}\n V\n {\\mathrm e}^{-\\sig\\tau A(t_0+\\frac{\\tau}{2})}\\,{\\mathrm d}\\sig\n \\cdot \\nS(\\tau,t_0)\n\\end{align}\nfollows from~\\cite[(10.15)]{higham08}.\nFor evaluating~\\eqref{dcA(t)}, a sufficiently accurate quadrature\napproximation for the integral according to~\\eqref{mimi} is required.\nThis involves evaluation of $ A' $ and the commutator $ [A,A'] $,\nsee~\\cite{auzingeretal18a}.\nIn contrast, the relevant term from~\\eqref{dsA(t)} simplifies to\n\\begin{align*}\n\\nR(\\tau,t_0) \\big( A(t_0+\\tfrac{\\tau}{2}) \\big)\n&=\n\\int_0^1 {\\mathrm e}^{\\sig\\tau A(t_0+\\frac{\\tau}{2})}\n A(t_0+\\tfrac{\\tau}{2})\\,\n {\\mathrm e}^{-\\sig\\tau A(t_0+\\frac{\\tau}{2})}\\,{\\mathrm d}\\sig\n \\cdot \\nS(\\tau,t_0) \\\\\n&= A(t_0+\\tfrac{\\tau}{2}) \\nS(\\tau,t_0)\n = \\nS(\\tau,t_0) A(t_0+\\tfrac{\\tau}{2}),\n\\end{align*}\nwhence the symmetrized defect~\\eqref{dsA(t)} can be evaluated exactly,\n\\begin{equation} \\label{dsA(t)-1}\n\\begin{aligned}\n\\nDs(\\tau,t_0)\n&= \\big( A(t_0+\\tfrac{\\tau}{2}) - \\th A(t_0+\\tau) \\big) \\nS(\\tau,t_0)\n - \\th \\nS(\\tau,t_0) A(t_0) \\\\\n&= \\nS(\\tau,t_0) \\big( A(t_0+\\tfrac{\\tau}{2}) - \\th A(t_0) \\big)\n - \\th A(t_0+\\tau) \\nS(\\tau,t_0).\n\\end{aligned}\n\\end{equation}\nThis involves an additional application of $ \\nS(\\tau,t_0) $,\nbut it does not require evaluation of the derivative $ A' $\nor of a commutator expression.\nWe also note that the applications of\n$\\nS$ from left and right can be evaluated in parallel.\n\n\\subsection{Algorithmic realization for higher order Magnus-type methods} \\label{subsec:cfm}\n\nThe integrators which we consider for the\nnumerical approximation of~\\eqref{zener}\nare commutator-free Magnus-type methods (CFM) and classical Magnus integrators.\n\nIn contrast to the special case of the exponential midpoint rule,\nfor practical evaluation the defect needs to be approximated\nin an asymptotically correct way.\nTo this end we require an approximation scheme\nwhich preserves the desired order $ p+2 $ of the corrected\nscheme~\\eqref{Sdach-nonlin}, or equivalently,\nthe asymptotic quality~\\eqref{p+3}\nof the local error estimator is not affected by such an approximation.\n\nVarious versions of the resulting classical defect-based error estimators for\nthese exponential integrators are presented in~\\cite{auzingeretal18b}.\nWe now follow two of these approaches.\nTo keep the presentation self-contained within reason,\nwe briefly recapitulate the underlying material\nfrom~\\cite[Section~3]{auzingeretal18b},\nand we introduce the corresponding symmetrized defect approximations.\n\n\\subsubsection{Commutator-free Magnus-type integrators}\nAs the basic integrator we consider a\n\\emph{commutator-free Magnus-type (CFM)} method~\\cite{alvfeh11},\n\\begin{subequations} \\label{cfm}\n\\begin{equation} \\label{cfmS}\n\\nS(\\tau,t_0) = \\nS_J(\\tau,t_0)\\,\\cdots\\,S_1(\\tau,t_0),\n\\end{equation}\nwhere\n\\begin{equation} \\label{cfmSB}\n\\begin{aligned}\n&S_j(\\tau,t_0) = {\\mathrm e}^{\\Omega_j(\\tau,t_0)} = {\\mathrm e}^{\\tau B_j(\\tau,t_0)}, \\\\\n&\\text{with}~~ B_j(\\tau,t_0) = \\sum_{k=1}^{K} a_{jk}\\,A(t_0+c_k\\tau),\n\\end{aligned}\n\\end{equation}\n\\end{subequations}\nwhere the coefficients $ c_k $ and $ a_{jk} $\nare chosen in such a way that\na desired order of consistency is obtained.\nNote that the assumption of symmetry of the scheme also implies symmetry of the\ncoefficients in the following sense,\n\\begin{subequations}\\label{symmetriesgack}\n\\begin{equation}\nc_k-\\tfrac{1}{2} = \\tfrac{1}{2}-c_{K+1-k},\n\\quad k=1,\\ldots,K,\n\\end{equation}\nand\n\\begin{equation}\na_{jk} = a_{J+1-j,K+1-k}, \\quad j=1,\\ldots,J, ~~ k=1,\\ldots,K.\n\\end{equation}\n\\end{subequations}\n\nOur construction involves evaluation of the derivatives\n\\begin{equation*}\n\\pdtau{\\mathrm e}^{\\Omega_j(\\tau,t_0)}\n= \\Gamma_{\\tau,j}(\\tau,t_0)\\,{\\mathrm e}^{\\Omega_j(\\tau,t_0)},\\quad\n\\tfrac{\\partial}{\\partial t_0}{\\mathrm e}^{\\Omega_j(\\tau,t_0)}\n= \\Gamma_{t_0,j}(\\tau,t_0)\\,{\\mathrm e}^{\\Omega_j(\\tau,t_0)},\n\\end{equation*}\nwhere\n\\begin{equation*}\n\\Gamma_{\\tau,j}(\\tau,t_0)\n= B_j(\\tau,t_0) +\n\\sum_{m \\geq 0}\\tfrac{1}{(m+1)!}\n \\tau^{m+1}\\mathrm{ad}^m_{B_j(\\tau,t_0)}\n (\\pdtau B_j(\\tau,t_0)),\n\\end{equation*}\nand\n\\begin{equation*}\n\\Gamma_{t_0,j}(\\tau,t_0)\n= \\sum_{m \\geq 0}\\tfrac{1}{(m+1)!}\n \\tau^{m+1}\\mathrm{ad}^m_{B_j(\\tau,t_0)}\n (\\pdtwo B_j(\\tau,t_0)).\n\\end{equation*}\n{Applying the product rule\nto $ \\nS(\\tau,t_0) $ defined in~\\eqref{cfm}\nwe see that the symmetrized} defect~\\eqref{dst}\nof the numerical approximation\nis an expression involving the derivatives\n\\begin{equation} \\label{wadlstrumpf}\n\\big( \\pdone-\\tfrac{1}{2}\\,\\pdtwo \\big) \\nS_j(\\tau,t_0)\n= \\Gamma_{j}(\\tau,t_0)\\,\\nS_j(\\tau,t_0),\n\\end{equation}\nwith\n\\begin{align}\n\\Gamma_j(\\tau,t_0)\n&=\\Gamma_{\\tau,j}(\\tau,t_0)-\\tfrac{1}{2}\\Gamma_{t_0,j}(\\tau,t_0) \\label{Gammaj-series} \\\\\n&=B_j(\\tau,t_0)\n +\\sum_{m \\geq 0}\\tfrac{1}{(m+1)!}\n \\tau^{m+1}\\mathrm{ad}^m_{B_j(\\tau,t_0)}\n ( \\Bcheck_j(\\tau,t_0)), \\notag\n\\end{align}\nwhere we have defined\n\\begin{equation*}\n\\Bcheck_j(\\tau,t_0)\n= \\big( \\pdone - \\tfrac{1}{2} \\pdtwo \\big) B_j(\\tau,t_0) =\n\\sum_{k=1}^{K} {a_{jk}(c_k-\\tfrac{1}{2})}A'(t_0+c_k\\tau).\n\\end{equation*}\nOne possible computable approximation is obtained by truncating the series (\\ref{Gammaj-series});\nwe will refer to the resulting procedure as \\emph{Taylor variant}.\nThe procedure in conjunction with the classical defect is given in detail\nin~\\cite[Section~3]{auzingeretal18b}.\n\nWe remark at this point that symmetry of the basic CFM integrator\nimplies that truncation of the series~\\eqref{Gammaj-series} at $m=p$,\ni.e., approximating $ \\Gamma_j(\\tau,t_0) $\nby\\footnote{A~priori one would expect that it is required to include\n the term of degree~$ p+1 $ also.}\n\\begin{equation} \\label{Gammaj-series-truncated}\n{\\widetilde\\Gamma}_j(\\tau,t_0)\n= B_j(\\tau,t_0)\n + \\sum_{m=0}^{p-1} \\tfrac{1}{(m+1)!}\n \\tau^{m+1}\\mathrm{ad}^m_{B_j(\\tau,t_0)}\n (\\Bcheck_j(\\tau,t_0))\n\\end{equation}\nis already sufficient to obtain a defect approximation of accuracy $p+2$,\nas is demonstrated in the following.\n\n\\begin{proposition}\\label{prop:symm}\nLet $\\nDs$ be the symmetrized defect of a self-adjoint CFM integrator of\norder $p$, and $\\nDstilde$ its approximation constructed via the\ntruncated Taylor variant\naccording to~\\eqref{Gammaj-series-truncated}. Then,\n\\begin{equation*}\n\\nDs(\\tau,u_0) - \\nDstilde(\\tau,u_0) = \\Order(\\tau^{p+2}).\n\\end{equation*}\n\\end{proposition}\n{\\em Proof.}\nObserve that\n\\begin{equation*}\nB_j(\\tau,t_0) = X_j A(t_0) + \\Order(\\tau),\n\\quad\n\\Bcheck_j(\\tau,t_0)\n= Y_j A'(t_0) + \\Order(\\tau),\n\\end{equation*}\nwhere\n\\begin{equation*}\nX_j = \\sum_{k=1}^K a_{jk}, \\quad\nY_j = \\sum_{k=1}^K a_{jk}(c_k-\\tfrac{1}{2}).\n\\end{equation*}\nThus,\n\\begin{equation*}\n\\Gamma_j(\\tau,t_0)-{\\widetilde\\Gamma}_j(\\tau,t_0) =\n\\tfrac{1}{(p+1)!}\\tau^{p+1} X_j^p Y_j\\,\\mathrm{ad}_{A(t_0)}^{p}(A'(t_0))\n+ \\Order(\\tau^{p+2}).\n\\end{equation*}\nInserting this in the computational algorithm given in Figure~\\ref{fig:algscfm} (left)\nand taking into account that\n\\begin{equation*}\n{\\mathrm e}^{\\tau B_j(\\tau,t_0)} = \\mathrm{Id} + \\Order(\\tau),\n\\end{equation*}\nthe total error resulting from substitution of the exact defect $\\nDs$ by the\ntruncated Taylor approximation of $\\Gamma_j$ is\n\\begin{equation*}\n\\nDs(\\tau,t_0)-\\nDstilde(\\tau,t_0) = \\tfrac{1}{(p+1)!} \\tau^{p+1} Z\\,\n\\mathrm{ad}_{A(t_0)}^{p}(A'(t_0)) + \\Order(\\tau^{p+2}),\n\\end{equation*}\nwith\n\\begin{equation*}\nZ = \\sum_{j=1}^{J} X_j^p \\, Y_j.\n\\end{equation*}\nTo establish the assertion of the proposition we now show $Z=0$:\nFrom~\\eqref{symmetriesgack},\n\\begin{equation*}\nX_j^p\\,Y_j = -X_{J+1-j}^p Y_{J+1-j}, ~~j=1,\\ldots,J,\n\\quad X_{\\lfloor J\/2 \\rfloor +1}^p Y_{\\lfloor J\/2 \\rfloor +1}\n= 0 \\ \\text{ if } J \\text{ is odd},\n\\end{equation*}\nwhence\n\\begin{equation*}\nZ = \\sum_j^{\\lfloor J\/2 \\rfloor} (X_j^p Y_j + X_{J+1-j}^p Y_{J+1-j})\n~~\\big[+ X_{\\lfloor J\/2 \\rfloor +1}^p Y_{\\lfloor J\/2 \\rfloor +1}\n\\ \\text{ if }\nJ \\text{ is odd} \\big] = 0,\n\\end{equation*}\nwhich completes the proof. \\hfill $ \\square $\n\nAs an alternative to the series representation~\\eqref{Gammaj-series},\nwe may use the integral {representation} which follows from~\\cite[(10.15)]{higham08},\n\\begin{equation*} \\label{Gammaj-int}\n\\Gamma_j(\\tau,t_0) = B_j(\\tau,t_0)+\n\\int_0^\\tau {\\mathrm e}^{\\sig B_{j}(\\tau,t_0)} \\Bcheck_j(\\tau,t_0)\\,{\\mathrm e}^{-\\sig B_{j}(\\tau,t_0)}\\, {\\mathrm d}\\sig,\n\\end{equation*}\nand apply a {$ p $-th order two-sided Hermite-type quadrature}\n(see~\\cite[Section~3]{auzingeretal18b})\nto approximate the integral.\nWe will refer to the resulting procedure as \\emph{Hermite variant}.\nThe procedure in conjunction with the classical defect\nwas also introduced in~\\cite[Section~3]{auzingeretal18b}.\nSimilarly as for the Taylor variant,\nit can be shown that quadrature of order $p$ is sufficient\nto obtain a defect approximation of order $p+2$.\n\nThese two sketched strategies result in the\nprocedures given as pseudocode in Figure~\\ref{fig:algscfm} where\nthe defect $d=\\nDs(\\tau,t_0)u_0$ is computed as the output\n{along with the basic approximation $ u = \\nS(\\tau,t_0)u_0 $}.\nThen, for order $p=4$, for instance, for the Taylor variant we have\n\\begin{align*}\n{\\widetilde\\Gamma}_j(\\tau,t_0)\n&= B_j(\\tau,t_0) + \\tau\\Bcheck_j(\\tau,t_0)\n+\\tfrac{1}{2}\\tau^2[B_j(\\tau,t_0),\\Bcheck_j(\\tau,t_0)] \\notag \\\\\n& \\quad {} +\\tfrac{1}{6}\\tau^3[B_j(\\tau,t_0),[B_j(\\tau,t_0),\\Bcheck_j(\\tau,t_0)]] \\\\\n& \\quad {} +\\tfrac{1}{24}\\tau^4[B_j(\\tau,t_0),[B_j(\\tau,t_0),[B_j(\\tau,t_0),\\Bcheck_j(\\tau,t_0)]]],\n\\end{align*}\nand for the Hermite variant,\n\\begin{equation*}\nC^{\\pm}_j(\\tau,t_0)=\\tfrac{1}{2}\\big(B_j(\\tau,t_0)+\\tau\\Bcheck_j(\\tau,t_0)\\big)\n\\pm\\tfrac{1}{12}\\tau^2[B_j(\\tau,t_0),\\Bcheck_j(\\tau,t_0)].\n\\end{equation*}\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\begin{tabular}{cc}\n\\begin{boxedminipage}{0.72\\textwidth}\n\\begin{tabbing}\n \\qquad\\= \\kill\n $u=u_0$ \\\\\n $d=-\\tfrac{1}{2}A(t_0)u$ \\\\\n {\\bf for} $j=1:J$\\\\\n \\>$u={\\mathrm e}^{\\tau B_j(\\tau,t_0)}u$\\\\\n \\>$d={\\mathrm e}^{\\tau B_j(\\tau,t_0)}d$\\\\\n \\>$d=d+\\widetilde{\\Gamma}_j(\\tau,t_0)u$\\\\\n {\\bf end}\\\\\n $d = d-\\tfrac{1}{2}A(t_0+\\tau)u$\n\\end{tabbing}\n\\end{boxedminipage}\\quad\n&\n\\quad \\begin{boxedminipage}{0.72\\textwidth}\n\\begin{tabbing}\n \\qquad\\=\\qquad\\= \\kill\n $u=u_0$ \\\\\n $d=-\\tfrac{1}{2}A(t_0)u$\\\\\n {\\bf for} $j=1:J$\\\\\n \\>$d=d+C^{-}_j(\\tau, t_0)u$\\\\\n \\>$u={\\mathrm e}^{\\tau B_j(\\tau,t_0)}u$\\\\\n \\>$d={\\mathrm e}^{\\tau B_j(\\tau,t_0)}d$\\\\\n \\>$d=d+C^{+}_j(\\tau, t_0)u$\\\\\n {\\bf end}\\\\\n $d=d-\\tfrac{1}{2}A(t_0+\\tau)u$\n\\end{tabbing}\n\\end{boxedminipage}\n\\end{tabular}\n\\caption{Algorithmic realization of the symmetrized defect for CFM methods.\\newline\nLeft:~Taylor variant.\nRight:~Hermite variant. \\label{fig:algscfm}}\n\\end{center}\n\\end{figure}\n\n\\subsubsection{Classical Magnus integrators}\n\nAs an example we consider the classical fourth order\nMagnus integrator based on quadrature at Gaussian points\n(see~\\cite{auzingeretal18b}),\n\\begin{subequations} \\label{MC-4}\n\\begin{equation} \\label{MC-4a}\n\\nS(\\tau,t_0) = {\\mathrm e}^{\\Omega(\\tau,t_0)} = {\\mathrm e}^{\\tau B(\\tau,t_0)},\n\\end{equation}\nwhere $ {\\Omega(\\tau,t_0)} = \\tau B(\\tau,t_0)$\napproximates the Magnus series $ {\\bm\\Omega}(\\tau,t_0) $,\n\\begin{align}\nB(\\tau,t_0)\n&= \\tfrac{1}{2}\\big(A(t_0+c_1\\tau)+A(t_0+c_2\\tau)\\big)\n-\\tfrac{\\sqrt{3}}{12}\\tau[A(t_0+c_1\\tau),A(t_0+c_2\\tau)], \\notag \\\\\nc_{1,2}&=\\tfrac{1}{2}\\pm\\tfrac{\\sqrt{3}}{6}. \\label{MC4-b}\n\\end{align}\n\\end{subequations}\nFollowing~\\cite[Section~3]{auzingeretal18b} for\nthe classical defect, the symmetrized defect~\\eqref{dst} is now given by\n\\begin{equation*}\n\\nDs(\\tau,t_0)\n= \\big(\\Gamma(\\tau,t_0) -\n\\tfrac{1}{2}A(t_0+\\tau)\\big)\\,\\nS(\\tau,t_0)-\\tfrac{1}{2}\\nS(\\tau,t_0)A(t_0),\n\\end{equation*}\nwhere $ \\Gamma(\\tau,t_0) $ has a series representation\nanalogous to~\\eqref{Gammaj-series}.\nTo approximate $ \\nDs(\\tau,t_0) $ in an asymptotically correct way,\nwe again truncate the series defining $ \\Gamma(\\tau,t_0) $ and obtain\nthe \\emph{Taylor variant}\n\\begin{equation*}\n\\nDs(\\tau,t_0)\n\\approx \\big(\\widetilde{\\Gamma}(\\tau,t_0)\n -\\tfrac{1}{2}A(t_0+\\tau)\\big)\\nS(\\tau,t_0)-\\tfrac{1}{2}\\nS(\\tau,t_0)A(t_0),\n\\end{equation*}\nwhere\n\\begin{align*}\n{\\widetilde{\\Gamma}}(\\tau,t_0)\n&= B(\\tau,t_0) + \\tau\\Bcheck(\\tau,t_0)\n+\\tfrac{1}{2}\\tau^2[B(\\tau,t_0),\\Bcheck_j(\\tau,t_0)] \\\\\n& \\quad {} +\\tfrac{1}{6}\\tau^3[B(\\tau,t_0),[B(\\tau,t_0),\\Bcheck(\\tau,t_0)]] \\\\\n& \\quad {} +\\tfrac{1}{24}\\tau^4[B(\\tau,t_0),[B(\\tau,t_0),[B(\\tau,t_0),\\Bcheck(\\tau,t_0)]]],\n\\end{align*}\nwith\n\\begin{equation} \\label{tildeBMagnus}\n\\begin{aligned}\n\\Bcheck(\\tau,t_0)\n&= \\big( \\pdone - \\tfrac{1}{2} \\pdtwo \\big)B(\\tau,t_0) \\\\\n&= \\tfrac{1}{2}\\big((c_1-\\tfrac{1}{2})A'(t_0+c_1\\tau)+(c_2-\\tfrac{1}{2})A'(t_0+c_2\\tau)\\big) \\\\\n& \\quad {} -\\tfrac{\\sqrt{3}}{12}[A(t_0+c_1\\tau),A(t_0+c_2\\tau)] \\\\\n& \\quad {} -\\tfrac{\\sqrt{3}}{12}(c_1-\\tfrac{1}{2})\\tau\\,[A'(t_0+{c_1}\\tau),A(t_0+c_2\\tau)] \\\\\n& \\quad {} -\\tfrac{\\sqrt{3}}{12}(c_2-\\tfrac{1}{2})\\tau\\,[A(t_0+c_1\\tau),A'(t_0+c_2\\tau)].\n\\end{aligned}\n\\end{equation}\nDue to $c_1+c_2=1$ it follows by expansion in $\\tau$ that\n$\\Bcheck(\\tau,t_0)=\\Order(\\tau)$. Thus, truncation after $p=4$ again yields\na sufficiently accurate approximation.\nAlternatively, application of fourth order two-sided Hermite quadrature\nfor the approximation of $\\Gamma(\\tau,t_0)$ yields the \\emph{Hermite variant}\n\\begin{equation*}\n\\nDs(\\tau,t_0) \\approx\n\\big(C^{+}(\\tau,t_0)-\\tfrac{1}{2}A(t_0+\\tau)\\big)\\nS(\\tau,t_0)+\\nS(\\tau,t_0)\\big(C^{-}(\\tau,t_0)-\\tfrac{1}{2}A(t_0)\\big),\n\\end{equation*}\nwhere\n\\begin{equation*}\nC^{\\pm}(\\tau,t_0)=\\tfrac{1}{2}\\big(B(\\tau,t_0)+\\tau\\Bcheck(\\tau,t_0)\\big)\n\\pm\\tfrac{1}{12}\\tau^2[B(\\tau,t_0),\\Bcheck(\\tau,t_0)],\n\\end{equation*}\nwith $\\Bcheck(\\tau,t_0)$ as in~\\eqref{tildeBMagnus}.\n\n\\section{Numerical examples} \\label{sec:num}\nWe illustrate the theoretical analysis of the deviation of the symmetrized\nerror estimator by showing the orders \\rev{of the error} of the basic\nintegrator and of the deviation of the error estimator\nfrom \\rev{the true error}. We will consider\nsplitting methods for a cubic nonlinear Schr{\\\"o}dinger equation\nand commutator-free and classical Magnus-type integrators for a Rosen--Zener model.\n\n\\subsection{Cubic Schr{\\\"o}dinger equation}\\label{subsec:nls}\n\nWe solve the cubic nonlinear Schr{\\\"o}dinger equation on the real line $ x \\in {{\\mathbb{R}}\\vphantom{|}} $\n\\begin{equation} \\label{eq:cnls}\n\\begin{aligned}\n{\\mathrm i}\\,\\partial_t \\psi(x,t)\n&= -\\,\\tfrac{1}{2}\\partial_x^2\\,\\psi(x,t) -|\\psi(x,t)|^2\\,\\psi(x,t), \\quad t>0, \\\\\n\\psi(x,0) &= \\psi_0(x)\n\\end{aligned}\n\\end{equation}\nby splitting methods. Here, a soliton solution exists,\n\\begin{equation*}\n\\psi(x,t) = 2\\,{\\mathrm e}^{{\\mathrm i}(\\frac{3}{2}t-x)}\\,\\text{sech}(2(t+x))\n\\end{equation*}\nOur initial condition is chosen commensurate with this solution,\nand we truncate the spatial domain to $ x \\in [-16,16] $ and impose periodic boundary\nconditions. Spectral collocation at 512 equidistant mesh points leads to an ODE system of the form\n\\begin{equation*}\n\\dt \\Psi(t) = F(\\Psi(t)) = A \\Psi(t) + B(\\Psi(t)), \\quad \\Psi(0) = \\Psi_0,\n\\end{equation*}\nwith $ A \\Psi \\sim \\tfrac{{\\mathrm i}}{2} \\partial_x^2\\,\\psi $ and $ B(\\Psi) \\sim {\\mathrm i}\\, |\\psi|^2 \\psi. $\nWe solve this by the second order Strang splitting\nand by the self-adjoint fourth-order method\nrepresented by the higher-order method in the embedded pair referred to as \\texttt{Emb 4\/3 AK s}\nin the collection~\\cite{splithp}, recapitulated for easy reference in Table~\\ref{splitcoeffs}\nin Section~\\ref{subsec:algsplit}.\nThe $ A $-part is solved via [I]FFT, while the $ B $-part can\nbe integrated directly on the given mesh.\n\nIn Table~\\ref{tab:conv1}, we give the local error of the Strang splitting and the\nerror of our symmetrized error\nestimator as compared to the exact errors.\n\\rev{Table~\\ref{tab:conv1b} shows the global errors on the interval $[0,1\/8]$ of the basic integrator\nand of the solution corrected by adding the error estimate.\nIn accordance with our theory, we observe local orders three and five, respectively,\nand the expected orders two and four for the global errors. Likewise, Table~\\ref{tab:conv2} shows\norders five and seven for the local errors of the fourth order integrator from~\\cite[\\texttt{Emb 4\/3 AK s}]{splithp},\nand Table~\\ref{tab:conv2b} shows the matching global errors}.\n\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{||r||r|r||c|r||}\n\\hline\n\\multicolumn{1}{||c||}{$\\tau$} & \\multicolumn{1}{c|}{$\\|\\nL(\\tau,u_0)\\|_2$} & \\multicolumn{1}{|c||}{order} &\n\\multicolumn{1}{c|}{$\\|{\\nLLtildes}(\\tau,u_0) - \\nL(\\tau,u_0)\\|_2$} & \\multicolumn{1}{|c||}{order} \\\\ \\hline\n1.563e$-$02 & 3.791e$-$05 & 2.98 & 3.377e$-$07 & 4.59 \\\\\n7.813e$-$03 & 4.753e$-$06 & 3.00 & 1.161e$-$08 & 4.86 \\\\\n3.906e$-$03 & 5.946e$-$07 & 3.00 & 3.726e$-$10 & 4.96 \\\\\n1.953e$-$03 & 7.434e$-$08 & 3.00 & 1.172e$-$11 & 4.99 \\\\\n9.766e$-$04 & 9.293e$-$09 & 3.00 & 3.669e$-$13 & 5.00 \\\\\n4.883e$-$04 & 1.162e$-$09 & 3.00 & 1.160e$-$14 & 4.98 \\\\\n\\hline\n\\end{tabular}\n\\caption{Local error and deviation of the symmetrized defect-based error\nestimator for the second order Strang splitting applied to~\\eqref{eq:cnls}.\\label{tab:conv1}}\n\\end{center}\n\\end{table}\n\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{||r||r|r||c|r||}\n\\hline\n\\multicolumn{1}{||c||}{$\\tau$} & \\multicolumn{1}{c|}{global error} & \\multicolumn{1}{|c||}{order} &\n\\multicolumn{1}{c|}{error of corrected solution} & \\multicolumn{1}{|c||}{order} \\\\ \\hline\n1.563e$-$02 & 2.539e$-$04 & 1.99 & 5.703e$-$07 & 4.00 \\\\\n7.813e$-$03 & 6.354e$-$05 & 2.00 & 3.634e$-$08 & 3.97 \\\\\n3.906e$-$03 & 1.589e$-$05 & 2.00 & 2.283e$-$09 & 3.99 \\\\\n1.953e$-$03 & 3.972e$-$06 & 2.00 & 1.428e$-$10 & 4.00 \\\\\n9.766e$-$04 & 9.931e$-$07 & 2.00 & 8.928e$-$12 & 4.00 \\\\\n4.883e$-$04 & 2.483e$-$07 & 2.00 & 5.611e$-$13 & 3.99 \\\\\n\\hline\n\\end{tabular}\n\\caption{\\rev{Global error and corrected solution for the second order Strang splitting applied to~\\eqref{eq:cnls}.}\\label{tab:conv1b}}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{||r||r|r||c|r||}\n\\hline\n\\multicolumn{1}{||c||}{$\\tau$} & \\multicolumn{1}{c|}{$\\|\\nL(\\tau,u_0)\\|_2$} & \\multicolumn{1}{|c||}{order} &\n\\multicolumn{1}{c|}{$\\|{\\nLLtildes}(\\tau,u_0) - \\nL(\\tau,u_0)\\|_2$} & \\multicolumn{1}{|c||}{order} \\\\ \\hline\n3.125e$-$02 & 7.017e$-$06 & 4.69 & 3.420e$-$07 & 6.36 \\\\\n1.563e$-$02 & 2.282e$-$07 & 4.94 & 2.646e$-$09 & 7.01 \\\\\n7.813e$-$03 & 7.164e$-$09 & 4.99 & 2.123e$-$11 & 6.96 \\\\\n3.906e$-$03 & 2.240e$-$10 & 5.00 & 1.706e$-$13 & 6.96 \\\\\n\\hline\n\\end{tabular}\n\\caption{Local error and deviation of the symmetrized defect-based error estimator for the fourth order integrator\nfrom~\\cite[\\texttt{Emb 4\/3 AK s}]{splithp} applied to~\\eqref{eq:cnls}.\\label{tab:conv2}}\n\\end{center}\n\\end{table}\n\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{||r||r|r||c|r||}\n\\hline\n\\multicolumn{1}{||c||}{$\\tau$} & \\multicolumn{1}{c|}{global error} & \\multicolumn{1}{|c||}{order} &\n\\multicolumn{1}{c|}{error of corrected solution} & \\multicolumn{1}{|c||}{order} \\\\ \\hline\n3.125e$-$02 & 7.894e$-$06 & 4.85 & 6.859e$-$07 & 5.97 \\\\\n1.563e$-$02 & 4.035e$-$07 & 4.29 & 2.771e$-$09 & 7.95 \\\\\n7.813e$-$03 & 2.471e$-$08 & 4.03 & 2.987e$-$11 & 6.54 \\\\\n3.906e$-$03 & 1.537e$-$09 & 4.01 & 4.622e$-$13 & 6.01 \\\\\n\\hline\n\\end{tabular}\n\\caption{\\rev{Global error and corrected solution for the fourth order integrator\nfrom~\\cite[\\texttt{Emb 4\/3 AK s}]{splithp} applied to~\\eqref{eq:cnls}}.\\label{tab:conv2b}}\n\\end{center}\n\\end{table}\n\n\\rev{\\paragraph{Adaptive time-stepping}\nThe error estimators introduced in this paper are intended to be used as the basis for an adaptive\ntime-stepping procedure to enhance the efficiency. To illustrate this aspect, we show step-sizes\ngenerated by the standard step-size selection strategy \\cite{haireretal87}. We solve problem\n(\\ref{eq:cnls}) with the initial condition\n$$ \\psi(x,0)= \\sum_{j=1}^2 \\frac{a_j {\\mathrm e}^{-{\\mathrm i} b_j x}}{\\cosh(a_j(x-c_j))}$$\nwith $a_1=a_2=2,\\ b_1=1,\\ b_2 =-3,\\ c_1=5,\\ c_2=-5,$ and a space discretization at\n512 points on the interval $[-16,16].$ Time integration is effected by the integrator from \\cite[\\texttt{Emb 4\/3 AK s}]{splithp}.\nThis example features two solitons which cross at $t\\approx2.3$,\nat which point the unsmooth solution demands smaller stepsizes.\nIf we prescribe a tolerance\nof $10^{-10}$ on the local error, we obtain the stepsizes shown in Figure~\\ref{fig:steps}.\nIt is found that the stepsizes indeed decrease in the region where the solitons cross,\nwhich corresponds with the behavior observed for adaptive time-stepping based on\nstandard error estimators in \\cite{koarl2paper}.\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{paper_v1.png}\n\\caption{\\rev{Step-sizes generated by an adaptive strategy based on the symmetric error estimator\nfor the integrator from \\cite[\\texttt{Emb 4\/3 AK s}]{splithp} for the problem (\\ref{eq:cnls})\nwith crossing solitons.}\\label{fig:steps}}\n\\end{center}\n\\end{figure}\n}\n\n\\subsection{Rosen--Zener model}\nAs a second example, we solve a Rosen-Zener model from~\\cite{blacastha17} by Magnus-type methods.\nThe associated Schr{\\\"o}dinger equation in the interaction\npicture is given by\n\\begin{equation}\\label{rosen}\n\\mathrm{i}\\dot\\psi(t) = H(t)\\psi(t)\n\\end{equation}\nwith\n\\begin{equation} \\label{zener}\n\\begin{aligned}\n& H(t) = f_1(t) \\sig_1 \\otimes I_{k\\times k} + f_2(t) \\sig_2 \\otimes R \\in\\mathbb{C}^{2k\\times 2k},\\quad k=50,\\\\\n& \\sig_1 = \\left( \\begin{array}{cc} 0 & 1 \\\\ 1 & 0 \\end{array} \\right),\\qquad\n \\sig_2 = \\left( \\begin{array}{rr} 0 & -{\\mathrm i} \\\\ {\\mathrm i} & 0 \\end{array} \\right),\\\\\n& R = \\mathrm{tridiag}(1,0,1) \\in \\mathbb{R}^{k\\times k},\\qquad f_1(t) = V_0 \\cos(\\omega t) \\left( \\cosh(t\/T_0) \\right)^{-1},\\\\\n& f_2(t) = V_0 \\sin(\\omega t) \\left( \\cosh(t\/T_0) \\right)^{-1},\n\\qquad \\omega=\\tfrac12,\\ T_0 =1,\\ V_0 = 1,\n\\end{aligned}\n\\end{equation}\nsubject to the initial condition $\\psi(0)=(1,\\dots,1)^T.$\n\nIn Tables~\\ref{tab:conv1rosen}--\\ref{tab:conv2rosene}, we give the local errors and\ndeviation of the symmetrized error estimators for the test problem~\\eqref{rosen}.\nTable~\\ref{tab:conv1rosen} gives the results for the exponential midpoint rule,\nwhere the symmetrized defect can be evaluated exactly. Tables~\\ref{tab:conv2rosenb}\nand~\\ref{tab:conv2rosen} give the empirical convergence orders for the commutator-free fourth order\nMagnus-type integrator~\\cite[\\texttt{CF4:2} in Table~2]{alvfeh11} in conjunction with the symmetrized\ndefect-based error estimator,\nevaluated by means of the Taylor variant\nin Table~\\ref{tab:conv2rosenb} and the Hermite variant\nin Table~\\ref{tab:conv2rosen}, respectively\n(see~Figure~\\ref{fig:algscfm}).\nFinally, Table~\\ref{tab:conv3rosen} gives the result\nfor the classical fourth order Magnus integrator, where the error estimator\nis evaluated by means of the Hermite variant.\n\\rev{Tables~\\ref{tab:conv1rosenb}, \\ref{tab:conv2rosenc}, \\ref{tab:conv2rosend} and\n\\ref{tab:conv2rosene} give the corresponding global errors on the interval $[0,1]$ of the basic solution\nand of the solution corrected by the symmetric error estimate.}\nIn all cases, the theoretical results are\nwell reflected in the numerical experiments.\n\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{||r||r|r||c|r||}\n\\hline\n\\multicolumn{1}{||c||}{$\\tau$} & \\multicolumn{1}{c|}{$\\|\\nL(\\tau,u_0)\\|_2$} & \\multicolumn{1}{|c||}{order} &\n\\multicolumn{1}{c|}{$\\|{\\nLLtildes}(\\tau,u_0) - \\nL(\\tau,u_0)\\|_2$} & \\multicolumn{1}{|c||}{order} \\\\ \\hline\n1.250e$-$01 & 3.343e$-$03 & 2.97 & 7.157e$-$06 & 4.96\\\\\n6.250e$-$02 & 4.198e$-$04 & 2.99 & 2.251e$-$07 & 4.99\\\\\n3.125e$-$02 & 5.254e$-$05 & 3.00 & 7.047e$-$09 & 5.00\\\\\n1.563e$-$02 & 6.569e$-$06 & 3.00 & 2.203e$-$10 & 5.00\\\\\n7.813e$-$03 & 8.212e$-$07 & 3.00 & 6.885e$-$12 & 5.00\\\\\n3.906e$-$03 & 1.026e$-$07 & 3.00 & 2.157e$-$13 & 5.00\\\\\n\\hline\n\\end{tabular}\n\\caption{Local error and deviation of the symmetrized defect-based error estimator for the second order exponential midpoint\nrule applied to~\\eqref{rosen}.\\label{tab:conv1rosen}}\n\\end{center}\n\\end{table}\n\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{||r||r|r||c|r||}\n\\hline\n\\multicolumn{1}{||c||}{$\\tau$} & \\multicolumn{1}{c|}{global error} & \\multicolumn{1}{|c||}{order} &\n\\multicolumn{1}{c|}{error of corrected solution} & \\multicolumn{1}{|c||}{order} \\\\ \\hline\n5.000e$-$01 & 2.713e-01 & & 7.652e-03 & \\\\\n2.500e$-$01 & 6.618e-02 & 2.04 & 4.638e-04 & 4.04\\\\\n1.250e$-$01 & 1.645e-02 & 2.01 & 2.880e-05 & 4.01\\\\\n6.250e$-$02 & 4.106e-03 & 2.00 & 1.797e-06 & 4.00\\\\\n3.125e$-$02 & 1.026e-03 & 2.00 & 1.123e-07 & 4.00\\\\\n1.563e$-$02 & 2.565e-04 & 2.00 & 7.018e-09 & 4.00\\\\\n\\hline\n\\end{tabular}\n\\caption{\\rev{Global error and corrected solution for the exponential midpoint rule applied to~\\eqref{rosen}}.\\label{tab:conv1rosenb}}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{||r||r|r||c|r||}\n\\hline\n\\multicolumn{1}{||c||}{$\\tau$} & \\multicolumn{1}{c|}{$\\|\\nL(\\tau,u_0)\\|_2$} & \\multicolumn{1}{|c||}{order} &\n\\multicolumn{1}{c|}{$\\|{\\nLLtildes}(\\tau,u_0) - \\nL(\\tau,u_0)\\|_2$} & \\multicolumn{1}{|c||}{order} \\\\ \\hline\n\\hline\n5.000e$-$01 & 1.884e$-$03 & 4.78 & 5.854e$-$05 & 6.61 \\\\\n2.500e$-$01 & 6.029e$-$05 & 4.97 & 4.875e$-$07 & 6.91 \\\\\n1.250e$-$01 & 1.892e$-$06 & 4.99 & 3.868e$-$09 & 6.98 \\\\\n6.250e$-$02 & 5.918e$-$08 & 5.00 & 3.033e$-$11 & 6.99 \\\\\n3.125e$-$02 & 1.850e$-$09 & 5.00 & 2.373e$-$13 & 7.00 \\\\\n\\hline\n\\end{tabular}\n\\caption{Local error and deviation of the symmetrized defect-based error estimator for the fourth order CFM integrator\n\\cite[\\texttt{CF4:2} in Table~2]{alvfeh11} applied to~\\eqref{rosen},\ndefect evaluation by Taylor variant.\n\\label{tab:conv2rosenb}}\n\\end{center}\n\\end{table}\n\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{||r||r|r||c|r||}\n\\hline\n\\multicolumn{1}{||c||}{$\\tau$} & \\multicolumn{1}{c|}{global error} & \\multicolumn{1}{|c||}{order} &\n\\multicolumn{1}{c|}{error of corrected solution} & \\multicolumn{1}{|c||}{order} \\\\ \\hline\n5.000e$-$01 & 2.098e-03 & & 5.330e-05 & \\\\\n2.500e$-$01 & 1.212e-04 & 4.11 & 7.419e-07 & 6.17\\\\\n1.250e$-$01 & 7.443e-06 & 4.03 & 1.126e-08 & 6.04\\\\\n6.250e$-$02 & 4.632e-07 & 4.01 & 1.745e-10 & 6.01\\\\\n3.125e$-$02 & 2.892e-08 & 4.00 & 2.768e-12 & 5.98\\\\\n1.563e$-$02 & 1.807e-09 & 4.00 & 1.175e-13 & 4.56\\\\\n\\hline\n\\end{tabular}\n\\caption{\\rev{Global error and corrected solution for the fourth order CFM integrator\n\\cite[\\texttt{CF4:2} in Table~2]{alvfeh11} applied to~\\eqref{rosen},\ndefect evaluation by Taylor variant.\\label{tab:conv2rosenc}}}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{||r||r|r||c|r||}\n\\hline\n\\multicolumn{1}{||c||}{$\\tau$} & \\multicolumn{1}{c|}{$\\|\\nL(\\tau,u_0)\\|_2$} & \\multicolumn{1}{|c||}{order} &\n\\multicolumn{1}{c|}{$\\|{\\nLLtildes}(\\tau,u_0) - \\nL(\\tau,u_0)\\|_2$} & \\multicolumn{1}{|c||}{order} \\\\ \\hline\n5.000e$-$01 & 1.884e$-$03 & 4.78 & 4.008e$-$05 & 6.64 \\\\\n2.500e$-$01 & 6.029e$-$05 & 4.97 & 3.277e$-$07 & 6.93 \\\\\n1.250e$-$01 & 1.892e$-$06 & 4.99 & 2.584e$-$09 & 6.99 \\\\\n6.250e$-$02 & 5.918e$-$08 & 5.00 & 2.023e$-$11 & 7.00 \\\\\n3.125e$-$02 & 1.850e$-$09 & 5.00 & 1.583e$-$13 & 7.00 \\\\\n\\hline\n\\end{tabular}\n\\caption{Local error and deviation of the symmetrized defect-based error estimator for the fourth order CFM integrator\n\\cite[\\texttt{CF4:2} in Table~2]{alvfeh11} applied to~\\eqref{rosen},\ndefect evaluation by Hermite variant.\n\\label{tab:conv2rosen}}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{||r||r|r||c|r||}\n\\hline\n\\multicolumn{1}{||c||}{$\\tau$} & \\multicolumn{1}{c|}{global error} & \\multicolumn{1}{|c||}{order} &\n\\multicolumn{1}{c|}{error of corrected solution} & \\multicolumn{1}{|c||}{order} \\\\ \\hline\n5.000e$-$01 & 2.098e-03 & & 3.203e-05 & \\\\\n2.500e$-$01 & 1.212e-04 & 4.11 & 4.402e-07 & 6.19\\\\\n1.250e$-$01 & 7.443e-06 & 4.03 & 6.702e-09 & 6.04\\\\\n6.250e$-$02 & 4.632e-07 & 4.01 & 1.041e-10 & 6.01\\\\\n3.125e$-$02 & 2.892e-08 & 4.00 & 1.676e-12 & 5.96\\\\\n1.563e$-$02 & 1.807e-09 & 4.00 & 1.052e-13 & 3.99\\\\\n\\hline\n\\end{tabular}\n\\caption{\\rev{Global error and corrected solution for the fourth order CFM integrator\n\\cite[\\texttt{CF4:2} in Table~2]{alvfeh11} applied to~\\eqref{rosen},\ndefect evaluation by Hermite variant.\\label{tab:conv2rosend}}}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{||r||r|r||c|r||}\n\\hline\n\\multicolumn{1}{||c||}{$\\tau$} & \\multicolumn{1}{c|}{$\\|\\nL(\\tau,u_0)\\|_2$} & \\multicolumn{1}{|c||}{order} &\n\\multicolumn{1}{c|}{$\\|{\\nLLtildes}(\\tau,u_0) - \\nL(\\tau,u_0)\\|_2$} & \\multicolumn{1}{|c||}{order} \\\\ \\hline\n5.000e$-$01 & 4.788e$-$03 & 4.56 & 1.214e$-$04 & 6.13 \\\\\n2.500e$-$01 & 1.618e$-$04 & 4.89 & 1.126e$-$06 & 6.75 \\\\\n1.250e$-$01 & 5.154e$-$06 & 4.97 & 9.201e$-$09 & 6.94 \\\\\n6.250e$-$02 & 1.618e$-$07 & 4.99 & 7.269e$-$11 & 6.98 \\\\\n3.125e$-$02 & 5.064e$-$09 & 5.00 & 5.693e$-$13 & 7.00 \\\\\n\\hline\n\\end{tabular}\n\\caption{Local error and deviation of the symmetrized defect-based error estimator\nfor the fourth order classical Magnus integrator~\\eqref{MC-4}\napplied to~\\eqref{rosen}, defect evaluation by Hermite variant.\n\\label{tab:conv3rosen}}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[!ht]\n\\begin{center}\n\\begin{tabular}{||r||r|r||c|r||}\n\\hline\n\\multicolumn{1}{||c||}{$\\tau$} & \\multicolumn{1}{c|}{global error} & \\multicolumn{1}{|c||}{order} &\n\\multicolumn{1}{c|}{error of corrected solution} & \\multicolumn{1}{|c||}{order} \\\\ \\hline\n5.000e-01 & 6.957e-03 & & 1.536e-04 & \\\\\n2.500e-01 & 4.362e-04 & 4.00 & 2.452e-06 & 5.97\\\\\n1.250e-01 & 2.728e-05 & 4.00 & 3.853e-08 & 5.99\\\\\n6.250e-02 & 1.705e-06 & 4.00 & 6.029e-10 & 6.00\\\\\n3.125e-02 & 1.066e-07 & 4.00 & 9.419e-12 & 6.00\\\\\n1.563e-02 & 6.662e-09 & 4.00 & 1.688e-13 & 5.80\\\\\\hline\n\\end{tabular}\n\\caption{\\rev{Global error and corrected solution for the\nfourth order classical Magnus integrator~\\eqref{MC-4} applied to~\\eqref{rosen},\ndefect evaluation by Hermite variant.\\label{tab:conv2rosene}}}\n\\end{center}\n\\end{table}\n\n\n\\section{Conclusion} \\label{sec:concl}\n\nWe have discussed a symmetrized defect-based estimator for self-adjoint\ntime discretizations of nonlinear evolution equations. We have introduced\nthe general construction principle extending the ideas from~\\cite{auzingeretal18a},\nand have elaborated the algorithms for an implicit Runge-Kutta method,\nfor splitting methods and for exponential Magnus-type integrators\nfor time-dependent linear problems. We have proven\nthat the deviation of the estimated error from the true error is two\norders in the step-size smaller than the basic integrator, and illustrated\nthe theoretical result for two examples solved by either splitting methods\nor exponential Magnus-type integrators of different orders.\n\nIt can be expected that in adaptive simulations, where choice of the step-size\nis delicate, the improved accuracy of the error estimator may add to\nthe reliability and efficiency of the integrator.\nHowever, this topic exceeds\nthe scope of the present work and will be explored elsewhere.\n\\rev{Here, we have confined ourselves to a numerical illustration that\nour error estimators induce adaptive step-sizes commensurate with the\nsolution behavior.}\nNote, moreover, that the numerical approximation\nbased on a scheme of order $ p $ and corrected\nby our error estimator (see~\\eqref{Sdach-nonlin})\nis very close to self-adjoint\nand has improved convergence order $ p+2 $\n(see Theorem~\\ref{Sdach-results}),\nthus providing a nearly self-adjoint higher order\napproximation at moderate computational cost.\nSince the additive correction is of high order,\nno stability problems will arise for\nthe corrected scheme~\\eqref{Sdach-nonlin}.\n\n\\section*{Acknowledgements}\nThis work was supported in part by the Vienna Science and Technology Fund (WWTF) [grant number MA14-002]\nand the Austrian Science Fund (FWF) [grant number P 30819-N32]. \\rev{We thank D. Haberlik, student at\nTU Wien, for contributing some of the numerical results, and}\nM.~Brunner, student at TU Wien, for contributing Figure~\\ref{fig:GALS}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe development of analog to digital converters (ADC) is advancing rapidly. The cutting-edge ADCs available commercially have a bandwidth of 20 GHz and can digitize signals at astonishing speed of about 65 GS\/s (giga samples per second). This rapid development in ADCs have enabled analysis of the signals from DC to microwave frequencies digitally. \n\nTraditionally resonator circuits (LC circuits) and cavity resonators have been used to analyze high frequency signals. The resonance frequency of an LC circuit can be tuned, which makes them useful in applications where the receiver needs the flexibility to analyze signals coming at different frequencies. However, due to the inherent power losses in the circuit the quality factor (Q-factor) that can be achieved in an LC circuit is only about $10^2$. Radio frequency cavities or microwave cavities, made up of a closed metal structure that confine electromagnetic field can have extremely high Q-factor (upto 10$^6$). Each cavity has a set of resonant modes (normal modes) that are determined by the physical parameters of the cavity. These are not easy to tune without compromising the performance of the cavity. Digital cavities overcome the limitation of these resonators, i.e. they are tunable, and at the same time, can have extremely high Q-factor. \n\nIn the simple case of a linear cavity formed by two reflecting surfaces, the mode spacing or the free spectral range $fsr$ is given by\n\\begin{equation}\\label{EQ1}\nfsr = \\frac{ c}{2 L},\n\\end{equation} \nwhere $c$ is the speed of light and $L$ is the distance between the surfaces. The frequencies of the normal modes are given by $N\\cdot fsr$ with $N\\in \\{1,2,3,..\\}$. A sinusoidal signal at the resonance frequency is amplified by the cavity due to constructive interference in the successive round trips in the cavity while the other signals are suppressed due to destructive interference. A time-periodic boundary condition applied to a sinusoidal signal leads to similar interference effects mimicking the functionality of a cavity. Here, we use the term time-periodic boundary condition to emphasize that the condition applied is similar to the periodic boundary condition (in spatial dimensions) used for example in molecular dynamics simulations ~\\cite{LEACH, ALLEN, KARKI_2011B} to simulate properties of a large system by tessellation of a smaller simulation box. The periodic-boundary condition in time is implemented by translating the signal $x(t)$ at time $t$ to $x(t+k\\cdot T)$, where $T$ is the period and $k\\in \\{0,1,2,...\\}$. Implementation of such a boundary condition during the digitization of a signal by an ADC leads to the formation of a digital cavity.\n\nFig.\\ref{FIG1} shows a pictorial representation of a digital cavity implemented using discrete values of digitized signals. In the figure the signal is discretized at a fixed interval of $\\Delta t$. A period of the digital cavity contains $n=10$ data points with period $T= n\\cdot \\Delta t$. Sinusoidal signals that are in resonance and off-resonance with the cavity are shown in panel (a) and (b), respectively. \n\n\n\\section{Theory}\n\nA digital cavity of length $n$ is defined as \n\n\\begin{equation}\\label{EQ2}\ny(j\\cdot \\Delta t;n):=\\sum_{k=0}^{N_c} x(j\\cdot \\Delta t+k\\cdot T); j=1,2,...n,\n\\end{equation}\nwhere the waveform $y$ is the response of the cavity to the signal $x$ and $N_c+1$ is the number of times the cavity function is applied to the signal. The cavity defined by Equ.\\eqref{EQ2} amplifies the resonant frequencies while suppressing (or filtering out) the non-resonant frequencies. \n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{Bothwaves.eps}\n \\caption{\\label{FIG1} Digital cavity with $n=10$ and cavity length $T=10 \\Delta t$. In (a) the signal is resonant with the cavity while it is non-resonant in (b). }%\n \\end{figure}\n\n The fundamental mode of the cavity is given by $f_0=1\/(n\\cdot \\Delta t)$, which can be tuned either by changing $n$ or $\\Delta t$. In an analogous way a physical cavity can be tuned either by changing the cavity length or the dielectric constant of the medium in the cavity. The free spectral range of a digital cavity is the same as the fundamental mode ($fsr = f_0$), hence the cavity amplifies a frequency comb~\\cite{CUNDIFF2005} with frequencies that are integer multiple of the fundamental mode, i.e. $f_0, 2f_0, 3f_0,...$ including the DC signal as shown in Fig.\\ref{FIG2}. We denote the resonant frequencies by $f_{res}$. Note that a DC signal is also amplified by an analog cavity. The DC offset of a signal in a digital cavity can be filtered out using a high band pass filter or an AC coupling prior to digitization. Another simple approach to filter out the DC offset is to use a generalized digital cavity as defined later. \n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{Freqdetune_comb.eps}\n \\caption{\\label{FIG2} A frequency comb that is amplified by a digital cavity. The $x$-axis is presented as frequency coordinate relative to the fundamental mode. The amplification as well as the line width of the resonant modes depend on $N_c$. }%\n \\end{figure}\n\nThe upper limit of the response of the cavity to a non-resonant frequency $f$ is given by\n\\begin{equation}\\label{EQ3}\n||y(j\\cdot \\Delta t)||\\leq \\frac{\\sqrt{5}}{2\\pi}\\frac{f_{res}}{f-f_{res}},\n\\end{equation}\nwhere $f_{res}$ is the resonant frequency closest to $f$ (see Appendix for the derivation), and the norm in the left side of the equation is taken to be the absolute value of the waveform. As shown by Equ.\\eqref{EQ3} there is no upper bound for resonant frequencies. In this case the waveform gets amplified linearly with $N_c$, i.e. $||y(j\\cdot \\Delta t)|| = (N_c+1) ||x(j\\cdot \\Delta t)||$, and the cavity response can be made arbitrarily large by choosing $N_c$ appropriately. If we define the amplitude of the response of the cavity to a signal as $A(f)=max(||y(j\\cdot \\Delta t)||)$ then the ratio of the amplitude for a resonant frequency to a non-resonant frequency, $A(f_{res})\/A(f)$, can be arbitrarily large. This, in other words, means that the line-width of the cavity can be arbitrarily small. We take full-width at half the maximum (FWHM) as the measure of the line-width. Let $\\xi_{f_0}$ denote the half-width at half the maximum. Then for $f = f_0+ \\xi_{f_0}$ we have $A(f) = (N_c+1)\/2$, and using Equ.\\eqref{EQ3} we get\n\\begin{equation}\\label{EQ4}\n2\\xi_{f_0} = \\frac{2 \\sqrt{5} }{\\pi(N_c+1)}.\n\\end{equation} \nAs the waveform generated after the cavity, $y(j\\cdot \\Delta t)$, is a set of numbers, the definition of the amplitude can be generalized. If the amplitude is defined as $A(f) = max(||(y(j\\cdot \\Delta))^r||)$, where $r \\geq 1$, then the generalized FWHM is given by \n\\begin{equation}\\label{EQ5}\n2\\xi_{f_0} = \\frac{2^{1\/r}\\sqrt{5} }{\\pi (N_c+1)}.\n\\end{equation}\nAs shown by Equ.\\eqref{EQ5} the line-width of a digital cavity can be made arbitrarily narrow by choosing large $N_c$, and in addition it can also be digitally narrowed by simply choosing larger $r$. Fig.\\ref{FIG3} shows line-widths for a digital cavity with different values of $N_c$ and $r=2$.\n\n\\begin{figure}\n\\includegraphics[width=7.5cm]{Freqdetune_Cycles_FWHM_bw.eps}\n \\caption{\\label{FIG3} Line-width of a digital cavity using $r=2$ and different values of $N_c$. The line-width narrows progressively with increasing $N_c$. The figure also shows that at certain non-resonant frequencies, indicated as nodes, the cavity completely suppresses the signal. }%\n \\end{figure} \n\nFig.\\ref{FIG3} also shows nodes at certain frequencies at which a digital cavity completely suppresses the signal even for low values of $N_c$. The frequencies at which these nodes occur depend on $N_c$, and they occur at $f=f_{res}+ p f_0\/(N_c+1)$ for $p\\in \\{1,2,...,N_c\\}$. As pointed out later these nodes can be useful in demultiplexing a signal that is multiplexed over different frequencies.\n\n\n\\section{Discussion on the Applications and Scope}\nOne of the scientific applications of a digital cavity is in phase-sensitive signal detection. We developed the concept for real time analysis of phase and amplitude of different frequencies present in the fluorescence signal from a sample that is excited with phase modulated laser pulses as in the fluorescence detected wave-packet interferometry~\\cite{RICE1991,MARCUS2006} and the two-dimensional electronic coherence spectroscopy.~\\cite{MARCUS2007} In these experiments, the fluorescence from the sample is multiplexed over many frequencies due to the contribution of the linear and the non-linear response of the sample. The signals at different frequencies provide a plethora of information about the system being investigated. A data-acquisition system that is capable of de-multiplexing the different contributions simultaneously would make this technique promising for chemical imaging. This in conjunction with near-field scanning techniques, can provide detailed information of chemical composition of a system with sub-wavelength resolution.~\\cite{KARKI_2012B} A digital cavity can be used to de-multiplex the signal and analyze the different contributions simultaneously. Thus, it could enable multiplexed chemical imaging using the fluorescence detected interferometric techniques. Fig.\\ref{FIG4} shows two signals at 4 and 6 kHz (panels (b) and (c), respectively) recovered from a signal that is synthesized by digitally mixing four frequencies at 4, 5, 6 and 7 kHz. The signal is digitized at the rate of 96 kS\/s (kilo samples per second). Two digital cavities with $n=24$ and 16 are implemented in real time to recover the two different contributions to the signal. Other contributions can also be recovered at the same time simply by implementing other cavities with different cavity lengths. \n\n\\begin{figure}\n\\includegraphics[width=7.5cm]{Sines_Filtered_bw_p.eps}\n \\caption{\\label{FIG4} Signal de-multiplexing using a digital cavity. Panel (a) shows the signal that is generated using direct digital synthesis by summing signals at frequencies 4, 5, 6 and 7 kHz. All the frequencies have same amplitude and zero phase offset. The signal in (a) is digitized using an ADC at a rate of 96 kS\/s. A digital cavity with $n=24$ selects the signal at 4 kHz (panel (b)) and another cavity with $n=16$ selects the signal at 6 kHz (panel(c)). $N_c$ is set to 3 and 5 in (b) and (c), respectively. All the waveforms have been normalized by the maximum value. }%\n \\end{figure} \n\nDe-multiplexing technique as shown in Fig.\\ref{FIG4} can also be used in data transmission employing multiple frequencies. When the frequencies are well defined, even small values of $N_c$ can select the signal contribution from the desired frequency. The 4 kHz signal in Fig.\\ref{FIG4} is selected by using $N_c=3$ and the 6 kHz signal is selected by using $N_c=5$. Note that for the 4 kHz signal all the other frequencies, 5,6 7 kHz, lie on the nodes of the cavity response when $N_c=3$, i.e. they are completely suppressed, and the same is true for the 6 kHz contribution when $N_c=5$. Multiplexed data transmission can dramatically improve the data transfer rate compared to the transmission using single frequency as is commonly done in wireless communication. Clever implementation of digital cavities can have significant impact in this field. \n\n\nA digital cavity described by Equ.\\eqref{EQ2} is generalization of coherent sampling technique~\\cite{NEUVO1984,SVANBERG2007} in which a common reference clock is used for signal generation as well as for digitization. Coherent sampling has been used for phase-sensitive detection, similar to that shown in Fig.\\ref{FIG4}, of signals mainly at $f_0$ and $2f_0$ to investigate rare gases using high-resolution tunable diode laser spectroscopy.~\\cite{SVANBERG2007} A digital cavity is generalization of the previously implemented techniques in that it does not need the same reference clock for signal generation and digitization. Furthermore the definition of the digital cavity can be generalized as \n\\begin{align}\\label{EQ6}\ny(j\\cdot \\Delta t):=&\\sum_{k=0}^{N_c} x(j\\cdot\\Delta t+n\\cdot \\Delta t\\cdot k) \\cos(k\\cdot\\theta)+\\nonumber\\\\\n&\\sum_{k=0}^{N_c} x((j+l)\\cdot\\Delta t+n\\cdot \\Delta t\\cdot k) \\sin(k\\cdot\\theta),\n\\end{align} \nwhere $l\\in \\{n\/4,n\/2,1,2,3,..\\}$ is used to offset the fundamental frequency $f_0$ to $f_g = nf_0\/(4l)$, and $\\theta = 2 \\pi k(4l-n)\/(4l)$ (see Appendix for the derivation). Equ.\\eqref{EQ2} is a special case of the generalized cavity in which $4l=n$ . For $4l=2n$ the generalized cavity functions as a comb-pass filter amplifying signals at $f_0\/2, 3f_0\/2, 5f_0\/2,...$, and for other $l$ it acts as a comb-pass filter amplifying signals at $f_g, f_g+f_0, f_g+2f_0,...$. Note that a DC offset of a signal in the generalized cavity, except for the case when $4l=n$, becomes non-resonant with the cavity. The generalized digital cavity can be tuned by changing either of $\\Delta t$, $n$ or $l$. Such tunability significantly increases the scope of a digital cavity compared to previously known techniques like the coherent sampling.\n\nAs a digital cavity is freely tunable to signal source(s) at a particular frequency it can be used in applications like software defined radio in which the tuning of the radio to a certain frequency is done by the software. When a signal needs to be transmitted over a noisy channel then the cavity can be used to suppress the random (white) noise. Note that the cavity averages a cycle of the resonant signal $N_c$ times, which improves the signal to noise ratio by a factor of $\\sqrt{N_c}$ for the random noise. Fig.\\ref{FIG5} shows two sinusoidal signals retrieved from their corresponding noisy parent signals. Some examples where suppression of white noise becomes important is in signal transmission over long distance (satellite communication, interplanetary communication, etc.), detection of faint radio\/microwave frequencies from astronomical bodies, detection of cosmic microwave background, and perhaps even signals from extra-terrestrial intelligent life forms as sought after by the SETI project. In some cases deliberate adulteration of signal with white noise can be useful to avoid eavesdropping. Digital cavities can significantly improve the instrumentation and reduce the cost related to high-end data acquisition\/analysis systems needed in the works mentioned above. \n\n\\begin{figure}[htbp]\n\\includegraphics[width=8cm]{NNoise_reduction_bw.eps}\n \\caption{\\label{FIG5} Reduction of random (white) noise in a digital cavity due to waveform averaging. The signals are generated by mixing random noise five and ten times the amplitude of a sine function at 960 Hz in (a) and (b), respectively. Digitization is done at the rate of 96 kS\/s. $x-$ axis shows the index $j$ used to denote the digitization position - one cycle is completed for $j=100$; two cycles are shown in the figures. The signal to noise ratio improves in both the cases ((a) and (b)) with larger $N_c$. The dotted lines are fits to the cavity response using sine functions. }%\n \\end{figure} \n\nAnother potentially important application of the digital cavity can be in high resolution spectroscopy. As shown in Equ.\\eqref{EQ5} the line-width of the cavity becomes narrower with increasing $N_c$, i.e. data acquisition time. This is the consequence of the fact that the precision of any interferometer depends on the number of the cycles of the sinusoidal signal that interfere.~\\cite{VAUGHAN1977} In analog cavities the multi-pass within the cavity increases the number of cycles that interfere, however, the power losses in each pass sets an upper limit. Power loss in this case equates to information loss. A digital cavity is not prone to this limitation as information present in each cycle is stored digitally. Consequently, a digital cavity can have arbitrarily high precision, depending on the time over which the data is acquired. Data acquired for 1s gives a precision of about 1 Hz in the measurement of a frequency, and higher measurement time results in better precision regardless of the frequency being measured. Every digitizer, however, has the problem of electronic jitter, which introduces uncertainty in time at which digitizations happen. Jitter in digital cavity is analogous to various noise sources in an analog cavity like the thermal noise and zero point fluctuation (vacuum fluctuations). In the high end digitizers, jitter can be less than 100 fs. The jitter does not set an upper limit but increases the data acquisition time to reach the desired precision. With the currently available digitizers it would be possible to analyze the hyperfine transitions in atoms and molecules in general, and atoms used for atomic clocks in particular, with extremely high precision. As the digitization is controlled by a clock, precision measurements with a digital cavity can also be used to synchronize clocks. When a microwave of 20 GHz generated by a source controlled by a master clock is used for synchronization then the frequency measurement done by a digital cavity can be used to synchronize the clock of the digitizer to $\\approx 10^{-14}$s, and a day of measurement enables synchronization better than $10^{-15}$s , which rivals the synchronization that can be achieved using analog techniques based on radio\/microwave freqeuncies.~\\cite{CUNDIFF2005} However,implementing a digital cavity with stable operation for a day or longer could be challenging.\n\n\\section{Conclusion}\nWe have presented the theoretical description of the digital cavity and derived some of its properties. Digital cavities, in general, implement the functionality of the analog cavities in time domain and eliminate problems related to power losses in the analog cavities. We have also shown examples of how digital cavities can be used in signal analysis and discussed other possible uses in signal transmission, high precision spectroscopy and precise synchronization of clocks.\n\n\n\\textbf{Acknowledgments}\n\nKK thanks Werner-Gren Foundation for the generous post-doctoral fellowship awarded to him.\nFinancial support from the Knut and Alice Wallenberg Foundation is greatfully acknowledged.\n\n\\vspace{1cm}\n\\section{Appendix A:} \\textbf{ Bounded and unboundedness of the cavity response}\n\nTo simplify the derivation of the properties of a digital cavity we represent the signal $x ( i \\cdot \\Delta t)$, $i \\in \\mathbbm{Z}$ as a Fourier series. Without any loss of generality, we consider a component of the series, a sine function, $\\sin ( \\omega t + \\phi)$, digitized at time intervals $\\Delta t$. We use the following equation to to define a digital cavity of length $n$, $n \\in \\mathbbm{N}^+$.\n\\begin{equation}\n y ( j \\cdot \\Delta t ; n) : = \\sum^{N_c}_{k = 0} \\sin ( \\omega \\cdot j\n \\cdot \\Delta t + \\omega \\cdot k \\cdot n \\cdot \\Delta t + \\phi) . \n \\label{AEQ3}\n\\end{equation}\nIn (\\ref{AEQ3}) $n \\cdot \\Delta t = T$ gives the period of the cavity, thus for each $j \\in ( 0, 1, \\ldots, n - 1)$ the equation sums up the values at equivalent points in the successive periods for $N_c$ times. Instead of using frequency $f$ as done in the main text, we use angular frequency $\\omega$ in the equation to reduce notations in the equations. We define $\\omega_c = 2 \\pi \/ T$ as the fundamental frequency of the cavity. Using this definition, Equ.(\\ref{AEQ3}) can be written as:\n\\begin{equation}\n y ( j \\cdot \\Delta t) = \\sum_{k = 0}^{N_c} \\sin \\left( \\omega \\cdot j\n \\cdot \\Delta t + \\frac{2 \\pi \\omega}{\\omega_c} k + \\phi \\right) . \n \\label{AEQ4}\n\\end{equation}\n\n\nWe further analyse Equ.(\\ref{AEQ4}) in the following two cases:\n\nFirst, when $\\omega = \\omega_c$, i.e. when the cavity is resonant with the\nfunction, the cavity simply amplifies the sine function $N_c + 1$ times: \\ \\\n\\begin{equation}\n y ( j \\cdot \\Delta t) = ( N_c + 1) \\sin ( \\omega_c \\cdot j \\cdot \\Delta\n t + \\phi) . \\label{AEQ5}\n\\end{equation}\n\n\nEqu.(\\ref{AEQ5}) is true for any $\\omega = m \\omega_c$ , where $m \\in\n\\mathbbm{Z}$. Thus for all the frequencies that are integer multiple of the cavity frequency, application of the digital cavity amplifies the signal and at the same time increases the signal to noise ratio. As there are no any physical constrains that limit the amplification, it does not saturate. In other words, the amplification factor is not bounded.\n\n\nNow we analyze the case when $\\omega \\neq m \\omega_c$, i.e. when the cavity is not resonant with the function. For simplicity we analyse the case $\\omega_c \\leqslant \\omega \\leqslant 1.5 \\omega_c$. By symmetry, the following argument also applies for the other cases. We can rewrite Equ.(\\ref{AEQ4}) as\n\\begin{equation}\n y ( j \\cdot \\Delta t) = \\sum_{k = 0}^{N_c} \\sin \\left( \\frac{2 \\pi\n \\omega}{\\omega_c} k + \\phi_2 \\right), \\label{AEQ6}\n\\end{equation}\nwhere $\\phi_2 = \\omega \\cdot j \\cdot \\Delta t + \\phi$. Let $\\omega \/ \\omega_c\n= 1 + x$ with $0 \\leqslant x \\leqslant 0.5$. Equ.(\\ref{AEQ6}) can be written as\n\\begin{equation*}\n y ( j \\cdot \\Delta t) = \\cos \\phi_2 \\sum_{k = 0}^{N_c} \\sin ( \\Omega k) + \\sin \\phi_2 \\sum_{k = 0}^{N_c} \\cos ( \\Omega k), \\;\\; \\textrm{with}\\;\\; \\Omega = 2 \\pi x.\n\\end{equation*}\nThe sums on the right hand side have an upper bound, which can be calculated as follows. Assume $N_c$ is very large. Take $r$ such that $\\Omega r \\leqslant\n\\pi \\leqslant \\Omega ( r + 1)$. Then\n\\begin{equation}\n \\left\\| \\sum_{k = 0}^{N_c} \\sin ( \\Omega k) \\right\\| \\leqslant \\sum_{k = 0}^r \\sin ( \\Omega k) \\frac{\\Delta k}{\\Delta k} .\n\\end{equation}\nLet $\\Omega k = y$, then $\\Delta k = \\frac{\\Delta y}{\\Omega}$. Using $\\Delta\nk$ in the above\n\\begin{eqnarray*}\n \\left\\| \\sum_{k = 0}^{N_c} \\sin ( \\Omega k) \\right\\| & \\leqslant &\n \\frac{\\sum_{y = 0}^{\\pi} \\sin y \\Delta y}{\\Omega \\Delta k}\\\\\n & \\leqslant & \\frac{\\int_{y = 0}^{\\pi} \\sin y d y}{\\Omega} = \\frac{\\omega}{\\pi ( \\omega - \\omega_c)}\n .\n\\end{eqnarray*}\n\nSimilarly\n\\begin{equation*}\n \\left\\| \\sum_{k = 0}^{N_c} \\cos ( \\Omega k) \\right\\| \\leqslant \n \\frac{\\omega_c}{2 \\pi ( \\omega - \\omega_c)} .\n\\end{equation*}\nThus for $\\omega \\neq \\omega_c$, $\\mathcal{F} ( j \\cdot \\Delta t)$ is bounded by\n\\begin{equation}\n \\| \\mathcal{F} ( j \\cdot \\Delta t) \\| \\leqslant \\frac{\\omega_c}{2 \\pi (\n \\omega - \\omega_c)} ( 2 \\cos \\phi_2 + \\sin \\phi_2) . \n \\label{AEQ7}\n\\end{equation}\nNote that the singularity at $\\omega_c$ indicates the divergence of digital cavity map at $\\omega_c$, which is also clear in Eq.(\\ref{AEQ5}).\n\nThe maximum value \\ $( 2 \\cos \\phi_2 + \\sin \\phi_2)$ can have is $\\sqrt{5}$, which gives\n\\begin{equation}\n \\| \\mathcal{F} ( j \\cdot \\Delta t) \\| \\leqslant \\frac{\\sqrt{5}}{2 \\pi} \n \\frac{\\omega_c}{\\omega - \\omega_c} . \\label{AEQ8}\n\\end{equation}\nEqu.(\\ref{AEQ8}) shows that for the case when $\\omega \\neq \\omega_c$, or in general when $\\omega \\neq m \\omega_c$, the cavity response has an upper bound.\n\n\n{\\textbf{Determination of frequencies completely suppressed by a digital\ncavity.}}\n\nIf we take $\\omega = \\omega_c \/ N_c$ in Equ.(\\ref{AEQ6}) then\n\\begin{eqnarray}\n y ( j \\cdot \\Delta t) & = & \\sum_{k = 0}^{N_c} \\sin \\left( \\frac{2 \\pi\n k}{N_c} + \\phi_2 \\right) = 0 \\label{ANSUP}\n\\end{eqnarray}\nfor all $j$ as the summation is over a period of the sine function. Thus for a\ndigital cavity in which the successive periods are summed $N_c + 1$ times\nthere are $N_c$ frequencies between 0 and $\\omega_c$ given by $\\omega = p\n\\omega_c \/ ( N_c + 1)$ where $p \\in \\mathbbm{Z} \\textrm{and} p \\leqslant N_c$\nthat are completely filtered out. We define $\\omega_c \/ N_c$ as the node\ndistance. \\ The nodes are present at $\\omega = i \\cdot \\omega_c \\pm p \\omega_c\n\/ ( N_c + 1)$ for all $i \\in \\mathbbm{N} \\wedge p \\in \\{ 1, 2, \\ldots, N_c\n\\}$.\n\n\n\\section{Appendix B:}\n\n{\\textbf{Frequencies selected by digital cavity and tuning the cavity.}}\n\nAccording to Equ.(\\ref{AEQ4}) for all $\\omega = i \\omega_c$ where $i$ is an\ninteger we have\n\\begin{equation}\n y ( j \\cdot \\Delta t) = ( N_c + 1) \\sin ( \\omega \\cdot j \\cdot \\Delta t + \\phi) . \n \\label{AEQ12}\n\\end{equation}\nThus the cavity selects all the frequencies that are integer multiple of the\nfundamental frequency, in other words it selects a frequency comb with carrier\noffset frequency and comb tooth spacing both given by $\\omega_c$. The\nfundamental frequency, and thereby the comb, selected by the cavity can be\nchanged either by changing $n$ (number of samples digitized per unit time) or\n$\\Delta t$ (digitization interval); remember $\\omega_c = 2 \\pi \/ ( n \\cdot\n\\Delta t)$. However, the definition of the digital cavity can be generalized\nsuch that carrier offset frequency and comb tooth spacing are independent. For\ne.g. if we define the cavity as\n\\begin{eqnarray}\n y ( j \\cdot \\Delta t) & := & \\sum_{k = 0}^{N_c} ( - 1)^k \\sin \\left(\n \\omega \\cdot j \\cdot \\Delta t + \\frac{2 \\pi \\omega}{\\omega_c} k + \\phi\n \\right) \\label{AEQ13}\n\\end{eqnarray}\nthen the carrier offset frequency is given by $\\omega_c \/ 2$ while the tooth\nspacing is given by $\\omega_c$. Equ (\\ref{AEQ13}) is a special case of the\ngeneral digital cavity defined as\n\\begin{equation}\n y ( j \\cdot \\Delta t ; \\theta) := \\sum_{k = 0}^{N_c} \\sin \\left( \\omega \\cdot j \\cdot \\Delta t + 2 \\pi\n \\left( \\frac{\\omega + \\frac{\\theta \\omega_c}{2 \\pi}}{\\omega_c} \\right) k +\n \\phi \\right) . \\label{AEQ14}\n\\end{equation}\nFor $\\omega$ to be resonant with the cavity we need\n\\begin{eqnarray}\n \\frac{\\omega + \\frac{\\theta \\omega_c}{2 \\pi}}{\\omega_c} & = & 1 \\nonumber\\\\\n \\Rightarrow \\theta & = & 2 \\pi \\left( 1 - \\frac{\\omega}{\\omega_c} \\right) \\label{AEQ16}\n\\end{eqnarray}\nThe values $\\theta$ can have is contrained due to the fact that data is\naccumulated at discrete time points and $\\phi$ is not known in general. The\ngeneral expression for the allowed values of $\\theta$ in a digital cavity can\nbe found as follows:\n\nEqu(\\ref{AEQ14}) can be written as\n\\begin{equation*}\n y ( j \\cdot \\Delta t) = \\sum_{k = 0}^{N_c} \\sin \\left( \\omega \\cdot j \\cdot \\Delta t +\n \\frac{2 \\pi \\omega}{\\omega_c} k + \\phi \\right) \\cos ( \\theta k) + \\sum_{k =\n 0}^{N_c} \\sin \\left( \\omega \\cdot j \\cdot \\Delta t + \\frac{2 \\pi\n \\omega}{\\omega_c} k + \\phi + \\frac{\\pi}{2} \\right) \\sin ( \\theta k).\n\\end{equation*}\nUsing the relations\n\\begin{equation*}\n \\omega_c = \\frac{2 \\pi}{n \\cdot \\Delta t}, \n\\end{equation*}\nand\n\\begin{equation*}\n \\frac{\\pi}{2} = \\frac{\\omega_c \\cdot n \\cdot \\Delta t}{4}\n\\end{equation*}\nwe analyze the second term\n\\begin{equation*}\n 2 \\textrm{nd}\\; \\textrm{term} = \\sum_{k = 0}^{N_c} \\sin \\left( \\omega \\cdot j\n \\cdot \\Delta t + \\omega \\cdot n \\cdot \\Delta t \\cdot k + \\frac{n}{4} \\cdot\n \\Delta t \\cdot \\omega_c + \\phi \\right) .\n\\end{equation*}\nLet $\\omega_c = r \\omega$, then we have\n\\begin{equation*}\n 2 \\textrm{nd}\\; \\textrm{term} = \\sum_{k = 0}^{N_c} \\sin \\left( \\omega \\cdot j\n \\cdot \\Delta t + \\omega \\cdot n \\cdot \\Delta t \\cdot k + \\frac{n}{4} \\cdot\n \\Delta t \\cdot r \\cdot \\omega + \\phi \\right)\n\\end{equation*}\nAccording to Equ.(\\ref{AEQ16}) we also have\n\\begin{equation}\n \\theta = 2 \\pi \\left( 1 - \\frac{1}{r} \\right) \\label{AEQ17}\n\\end{equation}\nLet\n\\begin{eqnarray*}\n \\frac{n \\cdot r}{4} & = & l, \\textrm{with}\\; l \\in \\mathbbm{Z} \\;\n \\textrm{then}\\\\\n r & = & \\frac{4 l}{n},\\; \\textrm{and}\\\\\n \\theta & = & 2 \\pi \\left( \\frac{4 l - n}{4 l} \\right) .\n\\end{eqnarray*}\nThe frequency that resonates with the cavity is then given by\n\\begin{equation}\n \\omega = \\frac{n \\omega_c}{4 l} . \\label{AEQ18}\n\\end{equation}\nFinally the general expression for the cavity can be written as\n\\begin{eqnarray}\n y ( j \\cdot \\Delta t) \n & = & \\sum_{k = 0}^{N_c} \\sin ( \\omega \\cdot j \\cdot \\Delta t + \\omega\n \\cdot n \\cdot \\Delta t \\cdot k + \\phi) \\cos \\left( 2 \\pi \\left( \\frac{4 l -\n n}{4 l} \\right) k \\right) + \\nonumber\\\\\n & & \\sum_{k = 0}^{N_c} \\sin ( \\omega \\cdot ( j + l) \\cdot \\Delta t +\n \\omega \\cdot n \\cdot \\Delta t \\cdot k + \\phi) \\sin \\left( 2 \\pi \\left(\n \\frac{4 l - n}{4 l} \\right) k \\right) . \n\\end{eqnarray}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nModels of innovation diffusion seek to understand how new ideas, products,\nor practices spread within a society through various channels~\\cite{R03}.\nInnovation may refer to new technologies or deviations from existing social\nnorms. Rather than a single theory, innovation diffusion represents a\ntheoretical framework that encompasses a range of social models in which the\nterm ``diffusion\" can mean contagion, imitation, and social\nlearning~\\cite{Kincaid04,R&G43,CKM}.\n\nMany of the traditional approaches~\\cite{1960s} to innovation diffusion\nmodeling are based on a mean-field approximation and are referred to as\naggregate models. An influential example is the seminal Bass\nmodel~\\cite{B69,B80,M90,B04,CEJOR2012,Hopp04,BMapplications}, where\ninnovation spreads as the result of either an adopter converting a\nsusceptible (contagion), or through external influences on susceptibles\n(advertising and mass media). The basic outcome of the Bass model is that the\ntime dependence of the fraction of adopters exhibits a sigmoidal\nshape~\\cite{B69,B80,M90,B04,R03,PY09}. Thus significant adoption arises\nonly after some latency period, after which complete adoption is quickly\nachieved.\n\nWhile the Bass and related models have been successful in fitting historic\ndata~\\cite{Sultan90}, there are several limitations of these approaches:\n\\begin{itemize}\n\\item The predictive power of the Bass\n model is uncertain~\\cite{BMparameters,Hohnisch08}.\n\n\\item Aggregate models are based on infinitely large, homogeneous\n populations~\\cite{CEJOR2012,PY09} and cannot account for\n sample-specific differences and related fluctuation phenomena.\n\n\\item Bass-like models do not account for behavioral patterns that result from\n social reinforcement and ``bandwagon'' pressure~\\cite{C10,Cth,bandWagon,reinf}.\n\n\\item Aggregate models assume a ``pro-innovation bias'' and thus cannot\n reproduce phenomena such as incomplete\n adoption~\\cite{G00,R03,C10,bandWagon}.\n\\end{itemize}\n\nWe are particularly interested in situations where innovation can be\naccompanied by controversy, suspicion, or rejection within some social\ncircles, potentially leading to incomplete adoption. As an example, mobile\nphones are owned by 90\\% of Americans~\\cite{mobiles1} as of 2014, but their\nuse is accompanied by continued health and safety concerns~\\cite{mobiles2}.\nSimilarly, the coverage of the measles, mumps and rubella vaccine in the\nUnited Kingdom reached 92.7\\% in 2013--14, below the target level of 95\\%\ncoverage for herd immunity. This incomplete adoption level may result from\ndoubts about vaccine effectiveness and safety concerns promulgated by\nanti-vaccination movements~\\cite{vaccine1,vaccine2}. Such doubts seem to\npersist even in the face of their apparently negative consequences, such as\nthe measles epidemic that seemed to have its inception in Disneyland at the\nstart of 2015.\n\nMotivated by these facts, we introduce a model for the diffusion of an\ninnovation, using a statistical physics approach~\\cite{DOInets}, in which we\naccount for the competing role of ``Luddites'' in hindering the spread of the\ninnovation. Agents may either be Luddites (opposed to innovation),\n``Ignorants'' (no knowledge of the innovation), ``Susceptibles'' (receptive\nto innovation), or ``Adopters'' of the innovation. We dub this the LISA\n(Luddites\/Ignorants\/Susceptibles\/Adopters) model. The main new feature of\nthe LISA model is the existence of agents that reject the innovation in\nresponse to the spread of adoption. \nPrevious work \\cite{resistance} has considered the introduction of `resistance leaders' who spread a negative response to the innovation, akin to the spread of a competing innovation. The LISA model differs from this approach by considering agents who respond in particular to the rate of uptake of the innovation.\nWe use the term ``Luddites'' in\nreference to the 19th-century Luddism movement in which English textile\nartisans protested against newly developed labor-saving\nmachinery~\\cite{Luddites_hist}. We are interested in determining how Luddism\nlimits the final level of adoption and how the presence of Luddites leads to\na trade-off between adoption levels and adoption times scales. \n\n\nThe LISA model is defined in the next section, while the behavior of the\nmodel in the mean-field limit and on complete graphs is investigated in\nSection~\\ref{sec:MF}. Section~\\ref{sec:graphs} focuses on the model dynamics\non random graphs and on a one-dimensional regular lattice. For all these\nsubstrates, we investigate how Luddism affects the final level of adoption\nand the time scale of adoption. We also elucidate a dichotomy between the\ncases of slow but relatively universal adoption for low values of an\nintrinsic innovation rate, and the rapid but limited spread of innovation\nthat occurs in the opposite limit. Our conclusions are presented in\nSection~\\ref{sec:conc}.\n\n\\section{The LISA model}\n\n\nAs a helpful preliminary, let us review the simpler two-state Bass model of\ninnovation diffusion. Here a population consists of two types of agents:\nsusceptibles $\\mathcal{S}$ or adopters $\\mathcal{A}$. In the Bass model,\nsusceptibles can become adopters via either of two processes:\n\\begin{itemize}\n\\item[(a)] Contagion-driven conversion: a susceptible converts to an adopter\n by interacting with another adopter, as represented by the process\n $\\mathcal{S}+\\mathcal{A} \\to \\mathcal{A}+\\mathcal{A}$.\n\\item[(b)] Spontaneous adoption: a susceptible converts to an adopter,\n $\\mathcal{S} \\to \\mathcal{A}$.\n\\end{itemize}\nThe characteristic feature of the Bass model is that the adopter density exhibits a \nsigmoidal time dependence, in\nwhich the time derivative of this density has a sharp peak (corresponding to\nan inflection point in the time dependence of the density itself), before\ncomplete adoption eventually occurs~\\cite{B69,B80,M90,B04,R03,PY09}.\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{model}\n\\caption{(\\textit{Color online}) Schematic depiction of the LISA model. An\n ignorant $\\mathcal{I}$ can become a Luddite $\\mathcal{L}$ with rate\n $r\\dot{A}$ (in a mean-field setting); an ignorant can also become a\n susceptible $\\mathcal{S}$ by contagion with rate proportional to the\n susceptible density. A susceptible spontaneously becomes an adopter at\n rate $\\gamma$.}\n\\label{fig:model}\n\\end{figure}\n\nOur LISA model is a four-state system that consists of a population of $N$\nindividuals that can each be in the states of Luddite ($\\mathcal{L}$),\nignorant ($\\mathcal{I}$), susceptible ($\\mathcal{S}$), or adopter\n($\\mathcal{A}$). Ignorant agents may either be persuaded to become\nsusceptible, and thence reach the adopter state, or they may become a Luddite\nand permanently oppose the spread of the innovation. Specifically, the\nelemental steps of our LISA model are the following (see\nFig.~\\ref{fig:model}):\n\\begin{enumerate}\n\\item[(a)] Contagion-driven conversion: An ignorant agent becomes susceptible\n by interacting with another susceptible agent. That is,\n $\\mathcal{I}+\\mathcal{S} \\to \\mathcal{S}+\\mathcal{S}$ with rate 1.\n\n\\item[(b)] Spontaneous adoption: A susceptible agent spontaneously becomes an\n adopter, $\\mathcal{S} \\to \\mathcal{A}$ with rate $\\gamma$\n\\footnote{Adoption could also occur by contagion, according to\n $\\mathcal{S} + \\mathcal{A} \\to \\mathcal{A} + \\mathcal{A}$. Yet, \nthis two-body process would yield similar features as our LISA model, \nbut would be technically more tedious to handle.}.\n\n\\item[(c)] Luddism: Ignorants may permanently reject the innovation\n and become Luddites, $\\mathcal{I}\\to \\mathcal{L}$, with a rate\n proportional to the change in the density of adopters in its\n neighborhood.\n\\end{enumerate}\n\nThe Luddism mechanism outlined above incorporates two aspects of\nnegative behavior towards innovation. The first represents a fear of\ninnovation or its consequences, as in the case of the historical\nLuddism movement, where the introduction of labor-saving machinery\ncaused fear over job security~\\cite{Luddites_hist}. The second is\nthat of non-conformity; agents may oppose the innovation simply due to\nits rapid increase in popularity~\\cite{bandWagon}. We model this\nfeature by defining the rate at which the Luddite density increases to\nbe \\emph{proportional} to the adoption rate, with constant of\nproportionality denoted by $r$, the Luddism parameter.\n\nThe multi-stage progression $\\mathcal{I} \\to \\mathcal{S} \\to \\mathcal{A}$ may\nalso be viewed as a type of social reinforcement mechanism in which\nadoption follows from a succession of prompts from\nneighbors~\\cite{Cth,reinf}. \nThe equivalent 3-state model with only Luddites, ignorants and adopters creates a polarized community creates a polarized community where the ratio of adopters to Luddites is dependent only on the Luddism parameter, $r$. Other relevant models \\cite{resistance} have found that high levels of advertising can prompt a negative response to innovation which cannot be replicated with only three states.\nThe combination of a multi-stage progression\nto adoption, together with the Luddite mechanism, arguably represents the simplest\ngeneralization of the Bass model that gives rise to non-trivial long-time\nstate with incomplete adoption of an innovation. \n\n\\section{Mean-field descriptions}\n\\label{sec:MF}\n\nWe first consider the LISA model in the mean-field limit, where agents\nare perfectly mixed. The densities of each type of agent are given by\n$(L,I,S,A)=(N_{\\mathcal{L}}, N_{\\mathcal{I}}, N_{\\mathcal{S}},\nN_{\\mathcal{A}})\/N$, where $N_{X}$ is the number of agents\nof type $X \\in \\{\\mathcal{L}, \\mathcal{I}, \\mathcal{S},\n\\mathcal{A}\\}$, and $N$ is the total number of agents. We consider\nthe limit $N\\to\\infty$, so that all densities are continuous\nvariables and all fluctuations are negligible. In this setting, the\nevolution of the agent densities is described by the rate equations:\n\\begin{align}\n\\begin{split}\n\\label{eqn:MF}\n\\dot{L}&= r \\dot{A} I \\equiv (\\alpha-1) SI,\\\\\n\\dot{I}&= -(1+\\gamma r )SI \\equiv -\\alpha SI ,\\\\\n\\dot{S}&= S(I-\\gamma),\\\\\n\\dot{A}&= \\gamma S,\n\\end{split}\n\\end{align}\nwhere the dot denotes the time derivative and we define\n$\\alpha\\equiv 1+\\gamma r$. Since the total density is conserved, i.e.,\n$L+I+S+A=1$, the sum of these rate equations equals zero. A natural initial\ncondition is a population that consists of a small density of susceptible\nagents that initiate the dynamics, while all other agents are ignorant; that\nis, $I(0)=1-S(0)=I_0$ and $L(0)=A(0)=0$.\n\nTo solve these rate equations, it is useful to introduce the modified time\nvariable $d\\tau= S(t)\\,dt$, which linearize the rate equations to\n\\begin{align}\n\\begin{split}\n\\label{eqn:RE}\nL'&= (\\alpha-1) I,\\\\\nI'&=-\\alpha I,\\\\\nS'&= I-\\gamma,\\\\\nA'&= \\gamma,\n\\end{split}\n\\end{align}\nwith solution\n\\begin{align}\n\\begin{split}\n\\label{eqn:sol_tau}\nL&= \\frac{\\alpha-1}{\\alpha}I_0(1-e^{-\\alpha\\tau}),\\\\\nI&= I_0 e^{-\\alpha\\tau},\\\\\nS&=\\frac{I_0}{\\alpha}(1-e^{-\\alpha\\tau})+1-I_0-\\gamma\\tau,\\\\\nA&= \\gamma \\tau.\n\\end{split}\n\\end{align}\n\nThere are two basic regimes of behavior that are controlled by the adoption\nrate $\\gamma$, as illustrated in Fig.~\\ref{fig:single}:\n\\begin{enumerate}\n\\item[(a)] \\emph{Extensive adoption}. When $\\gammaI_0$, the\n susceptibles quickly become adopters, leaving behind a substantial static\n population of ignorants and a small fraction of adopters, as well as\n Luddites.\n\\end{enumerate}\nNumerical simulations of the LISA model on a large complete graph \\emph{and}\nnumerical integration of the rate equations \\eqref{eqn:MF}, illustrated in\nFig.~\\ref{fig:single}, give results that are virtually indistinguishable.\n\n\\begin{figure}[ht]\n\\centering\n\\subfigure[]{\\includegraphics[width=0.9\\linewidth]{single_gamlti0}}\n\\subfigure[]{\\includegraphics[width=0.9\\linewidth]{single_gamgti0}}\n\\caption{(\\textit{Color online}) Evolution of a realization of the LISA model\n on a complete graph of $10^6$ nodes with $I_0=0.8$ and Luddism parameter\n $r=0.9$. (a) $\\gamma = 0.3$ (extensive adoption) (b) $\\gamma=1$\n (sparse adoption). Evenly distributed samples of the stochastic simulation ($\\Box$) are\n indistinguishable from the solution of Eq.~\\eqref{eqn:MF} (solid line). The\n completion times for (a) and (b) are $60$ and $17$ respectively.}\n\\label{fig:single}\n\\end{figure}\n\nWe can express the densities in terms of the physical time $t$ by inverting\n$d\\tau=S(t)dt$ to give $t=\\int_0^{\\tau}d\\tau'\/S(\\tau')$. Substituting\n$S(\\tau)$ from the third of Eqs.~\\eqref{eqn:sol_tau} and taking the limits of\nlow adoption, $\\gamma \\ll 1$ and $\\alpha\\approx1$,\nwe have~\\footnote{Here the term $-\\gamma \\tau'$ has been neglected.\n This approximation is legitimate since $\\tau'$ is integrated from\n $0$ to $\\tau\\ll \\tau_{\\infty} \\approx 1\/\\gamma$ and therefore\n $\\gamma \\tau' \\ll 1$ in the regime being considered. A similar\n reasoning, with $\\gamma \\tau \\leq \\gamma \\ln(I_0\/\\gamma) \\ll 1$,\n leads to (\\ref{eqn:t*_approx}) when $\\gamma \\ll 1$.}\n\\begin{equation}\nt=\\int_0^{\\tau}\\!\\!\\frac{ d\\tau'}{1-I_0 e^{-\\tau'}}\\approx \\tau +\n\\ln\\left[1-I_0 e^{-\\tau}\\right]\\,.\n\\end{equation}\nIn particular, \nthe physical inception time $t_{\\rm inc}$ is,\n\\begin{eqnarray}\n\\label{eqn:t*_approx}\nt_{\\rm inc}\\approx \\int_0^{\\ln(I_0\/\\gamma)}\\!\\!\\frac{ d\\tau'}{1-I_0 e^{-\\tau'}}\n\\approx \\ln\\left[\\frac{I_0}{(1-I_0)\\gamma}\\right]\n\\end{eqnarray}\nand therefore grows as $\\ln(1\/\\gamma)$.\n\nThe stationary state is reached when all susceptibles disappear, so that no\nfurther reactions can occur. This gives the criterion $S(\\tau_{\\infty})=0$\nwhich defines the value of $\\tau_{\\infty}$. By solving the third line of\nEq.~\\eqref{eqn:sol_tau}, we obtain\n\\begin{equation}\n\\label{eqn:Xinf}\n\\tau_{\\infty}=\\frac{1}{\\gamma}\\!-\\!\\frac{I_0r}{\\alpha}\\!+\\!\\frac{1}{\\alpha} \nW_0\\left(-\\frac{I_0}{\\gamma} e^{I_0r-\\alpha\/\\gamma}\\right)\\,,\n\\end{equation}\nwhere $W_0(z)$ is the principal branch of the Lambert function $W(z)$, which\nis defined as the solution of $z=We^{W}$. Here $\\tau_{\\infty}$ is a\ndecreasing function of the adoption rate $\\gamma$, with $\\tau_{\\infty} \\sim 1\/ \\gamma$ in the high and low adoption rate regimes.\n\nWe now determine the final densities by substituting $\\tau_\\infty$ into\nEqs.~\\eqref{eqn:sol_tau}. For small adoption rate ($\\gamma \\ll 1$), this\ngives \n\\begin{align}\nA_\\infty&=1\\!-\\!\\mathcal{O} (\\gamma),\\nonumber\\\\\nI_\\infty&\\to 0,\\nonumber\\\\\nL_\\infty &\\approx (\\alpha-1) I_0=\\mathcal{O} (\\gamma).\\nonumber\n\\end{align} \nSimilarly, the densities at the inception time are obtained by substituting $\\tau_{\\rm inc}$\ninto Eqs.~\\eqref{eqn:sol_tau}. This yields $A(\\tau_{\\rm inc}) + S(\\tau_{\\rm inc}) =\n1-[(\\alpha-1)I_0+\\gamma]\/\\alpha$. Since $(\\alpha-1)I_0 \\sim {\\cal O}(\\gamma)$, when $\\gamma \\ll 1$ and $r$ is finite, here the stationary density of adopters approximately equals the sum of the adopter and susceptible densities at the inception time, $A_\\infty \\approx A(\\tau_{\\rm inc}) + S(\\tau_{\\rm inc})$. Hence, in the low adoption rate regime (when $r$ is finite), we can infer the final level of adoption from the adopter and susceptible densities at the inception time, i.e., well before the stationary state.\n \n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{parameters_gam}\n\\includegraphics[width=0.8\\linewidth]{parameters_r}\n\\caption{(\\textit{Color online}) Dependences of the final-state densities\n $L_\\infty, I_\\infty$ and $A_\\infty$ for a complete graph of $10^4$ nodes\n and $I_0=0.9$. In the top panel $r=0.9$ while $\\gamma$ varies, whereas in\n the bottom panel $\\gamma=0.3$ while $r$ varies. Simulations ($\\Box$) in complete agreement with \\eqref{eqn:sol_tau} with substitution \\eqref{eqn:Xinf} (solid line). }\n\\label{fig:parameters}\n\\end{figure}\n\nThe dependence of\nthe final densities for different parameter ranges is shown in\nFig.~\\ref{fig:parameters}. Again simulation results for the complete graph\nare indistinguishable from numerical integration of the rate equations.\nInterestingly, $L_\\infty$ varies non-monotonically on $\\gamma$ when the\ninitial state consists mostly of ignorants and the fixed rate of Luddism $r$\nis not too high, as in Fig.~\\ref{fig:parameters}~(top).\nThis non-monotonic dependence on $\\gamma$ can be understood by noting \nthat $dL_\\infty\/d\\gamma \\sim r(1 - e^{-1\/\\gamma}) > 0$ for $\\gamma \\ll 1$ and $dL_\\infty\/d\\gamma \\sim -e^{-1\/\\gamma}\/\\gamma^2 < 0$\nfor $\\gamma \\gg 1$.\nWe therefore expect that $L_\\infty$ is peaked for an intermediate value of $\\gamma$ on a range between the slow and quick adoption regimes. It is also worth\nnoting that in the absence of Luddites, complete adoption is almost, but not\ncompletely achieved, since the final densities of adopters and ignorants are\n$A_\\infty\\approx 1- I_\\infty$ and $I_\\infty \\approx e^{-1\/\\gamma}$, see Fig.3 (bottom).\n\nTo assess the role of finite-$N$ fluctuations on the dynamics, we simulate\nthe LISA model on complete graphs of $N$ nodes using the Gillespie\nalgorithm~\\cite{Gillespie}. At long times we find that the densities of each\nspecies, $N_{X}\/N$, fluctuates around the corresponding mean-field density,\nwith a root-mean-square fluctuation of amplitude $\\sim N^{-1\/2}$, as\nexpected from general properties of this class of reaction\nprocesses~\\cite{noise}. We also find that the probability distribution of\n$N_{X}\/N$ is a Gaussian of width of order $N^{-1\/2}$ that is centered on the\nmean-field density. We also estimate the completion time $T_C$ for the\nsystem to reach its final state by the physical criterion that\n$S(t\\!=\\!T_C)=1\/N$. That is, completion is defined by the presence of a\nsingle susceptible remaining in the population~\\cite{reinf}. By linearizing\nthe rate equations \\eqref{eqn:MF} around $S_\\infty=0$, the density of\nsusceptibles asymptotically vanishes as $S(t)\\sim e^{-(\\gamma-I_\\infty)t}$.\nHence, we estimate the mean completion time to be\n$T_C \\approx (\\ln~N)\/(\\gamma-I_\\infty)$. This prediction is confirmed by our\nsimulations.\n\n\\section{LISA model on random graphs and lattices}\n\\label{sec:graphs}\n\nWe now consider the behavior of the LISA model on Erd\\H os-R\\'enyi random\ngraphs and one-dimensional lattices. We are particularly interested in\nuncovering dynamics that are characterized by genuine non mean-field effects.\n\nA graph with $N$ nodes can be represented by its $N\\times N$ adjacency matrix\n${\\bf A}=[A_{ij}]$, where $A_{ij}=1$ if nodes $i$ and $j$ are connected and\n$0$ otherwise. We implement the LISA model on such a graph using the\nGillespie algorithm~\\cite{Gillespie}. The propensity for a susceptible to\nbecome an adopter is $\\gamma$, independent of the local environment. The\npropensity for an ignorant node $i$ to become susceptible if it has $s_i$\nsusceptible neighbors is $s_i\/N$. The propensity of an ignorant node $i$\nto become a Luddite is $r\\gamma s_i\/k_i$, where $k_i=\\sum_j A_{ij}$ is the\ndegree (number of neighbors) of node $i$, and $s_i\/k_i$ is the fraction of\nnodes in the neighborhood of $i$ that are in the susceptible state. Thus the\npropensity of $i$ to become a Luddite is proportional to the sum of its\nsusceptible neighbors' propensities to adopt. This rate encodes node $i$'s\nlocal knowledge of the rate of adoption. These reaction rates approach those\nof the complete graph, described in Section~\\ref{sec:MF}, as the average\ndegree of the graph increases.\n\n\\subsection{Erd\\H os-R\\'enyi random graphs}\n\\label{sec:ER}\nWe first study the LISA model on the class of Erd\\H os-R\\'enyi (ER) random\ngraphs in which an edge between any two nodes occurs with a fixed probability\n$p$. This construction leads to a binomial degree distribution for the ER\ngraph in which each node has, on average, $k = p(N-1)$\nneighbors~\\cite{Newman2010}. Under the assumption of no correlations between\nthe degrees of neighboring nodes, the adjacency matrix may be written as\n$A_{ij}\\approx k_i k_j \/(N k) \\approx k\/N$. The LISA dynamics on ER graphs\ncan now be approximately described by a natural generalization of the\nmean-field theory in which there are suitably defined reaction rates. In\nparticular,\nif $S_i$ is the probability that a node $i$ is susceptible and $I_j$\nis the probability that a node $j$ is ignorant, then the density of\nsusceptibles $S$ evolves as\n\\begin{align*}\n\\dot{S_i}=S_i\\Big[\\sum_j (A_{ij}\/N)I_j -\\gamma \\Big]\\approx S\\big[(k\/N)I-\\gamma\\big],\n\\end{align*}\nsince each susceptible interacts with $k$ of its $N$ neighbors on average.\nThus on the ER graph there is a rescaling of the rate of the two-body\ncontagion process $\\mathcal{I}+ \\mathcal{S} \\to \\mathcal{S}+\\mathcal{S}$,\nwhereas the rates of the remaining one-body processes remain unaltered.\nHence we obtain the effective rate equations\n\\begin{align}\n\\begin{split}\n\\label{eqn:MF_ERG}\n\\dot{L}&= \\gamma r S I \\equiv \\left(\\beta - \\frac{k}{N}\\right) SI,\\\\\n\\dot{I}&= -\\left(\\gamma r +\\frac{k}{N}\\right)S I \\equiv -\\beta SI,\\\\\n\\dot{S}&= S\\left(\\frac{k}{N}~I-\\gamma\\right),\\\\\n\\dot{A}&= \\gamma S,\\\\\n\\end{split}\n\\end{align}\nwhere, for later convenience, we define $\\beta\\equiv \\gamma r + (k\/N)$.\n\nAs in the case of the mean-field dynamics, the above equations predict two\nregimes of behavior (see Fig.~\\ref{fig:er_single}):\n\\begin{figure}[ht]\n\\centering\n\\subfigure[]{\\includegraphics[width=0.8\\linewidth]{ER_ensemble_gamlti0}}\n\\subfigure[]{\\includegraphics[width=0.8\\linewidth]{ER_ensemble_gamgti0}}\n\\caption{(\\textit{Color online}) The evolution, averaged over 100\n realizations, of the LISA model on an ER graph with $N=10^3$ nodes, $k=10$, and $I_0=0.8$. (a) $\\gamma = 0.002$, such that $\\gamma < (k\/N) I_0$\n and (b) $\\gamma = 0.1$ such that $\\gamma > (k\/N) I_0$. Shown are the\n evenly distributed samples of the stochastic simulation ($\\Box$) and the solution of Eq.~\\eqref{eqn:MF_ERG} (solid line).\n The Luddism parameter $r=0.9$.}\n\\label{fig:er_single}\n\\end{figure}\n\\begin{enumerate}\n\\item[(a)] {\\it Slow but extensive adoption} ($\\gammakI_0\/N$). The density of\n $\\mathcal{S}$'s vanishes quickly so that the density of adopters and\n Luddites quickly reach their steady-state values.\n\\end{enumerate} \n\nThe simulation results presented in Fig.~\\ref{fig:er_single} indicate\nthat the mean-field approximation \\eqref{eqn:MF_ERG} correctly\ncaptures the main qualitative features of the dynamics on large ER\ngraphs. \nWhen $\\gamma (2\/N)I_0$. \nFrom simulations, illustrated in Fig.7, we observe the following three regimes:\n\n\\begin{enumerate}\n\\item[(A)] When $\\gamma\\ll 2I_0\/N$, there is slow adoption as well as a\n time-scale separation. First, almost all $\\mathcal{I}$'s are converted to\n $\\mathcal{S}$'s~\\cite{1D} in a time of the order of $N^2$. When the\n lattice consists almost entirely of $\\mathcal{S}$'s, these become adopters\n after a mean time of the order of $\\gamma^{-1}$. As a consequence, when\n $\\gamma\\ll N^{-1}$ the size of the adopter domains grows abruptly after a\n time of order $\\sim N^2 +\\gamma^{-1}$, when all ignorants have disappeared\n and the entire lattice is covered with adopters.\n\n\\item[(B)] When $\\gamma \\sim 2I_0\/N$, the domains of adopters grow initially\n nearly linearly in time, whereas the average size of ${\\cal I}$ clusters\n remains approximately constant and of a comparable size to ${\\cal A}$\n domains.\n\n\\item[(C)] When $\\gamma\\gg (2\/N)I_0$, adoption occurs quickly and the final\n state is reached in a time of order ${\\cal O}(1\/\\gamma)$. The final\n adopter density is limited by the formation of Luddites at the ends of ignorant domains which prevent further conversion within each domain.\n\\end{enumerate}\n\\begin{figure}[ht]\n\\includegraphics[width=0.8\\linewidth]{1dlattice_gamma_pop}\n\\includegraphics[width=0.9\\linewidth]{1dlattice_singleA}\n\\includegraphics[width=0.9\\linewidth]{1dlattice_singleB}\n\\includegraphics[width=0.9\\linewidth]{1dlattice_singleC}\n\\caption{(\\textit{Color online}) Final simulated average proportions of\n adopters (red\/gray $\\Box$), ignorants (green\/dark gray $\\Box$) and Luddites (blue\/black $\\Box$) for\n varying values of $\\gamma$, averaged over 100 simulations. Theoretical\n predictions using ignorant domain length (see Appendix~\\ref{ap:1D} for\n details) are overlaid (solid line). Parameters are $N=1000$, $r=0.5$. Initially\n ignorants and susceptibles are randomly distributed, with densities\n $I_0 = 0.8$ and $S_0 = 0.2$. The three regimes discussed in the text are\n separated by dashed lines corresponding to regions where\n $(2\/N)I_0\\ll \\gamma$ and $(2\/N)I_0\\gg \\gamma$. Typical realizations of the\n model for $N=100$ in each of the three regimes are given (bottom). On the\n vertical axis the iteration corresponds to a single step of the Gillespie\n algorithm, with one reaction taking place per iteration.}\n\\label{fig:1dlattice_gamma}\n\\end{figure}\n\nWhile the mean-field approximation \\eqref{eqn:MF_ERG} predicts the correct\nregimes of behavior, the agreement is only qualitative. In\nFig.~\\ref{fig:lattice_single} we compare typical simulations of the LISA\nmodel on a one-dimensional lattice with the mean-field predictions of\n\\eqref{eqn:MF_ERG} for the case of $k=2$.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{1D_ensemble}\n\\caption{(\\textit{Color online}) Time dependence of the densities in each\n state for a one-dimensional lattice of size $N=10^5$ averaged over 100\n realizations. The corresponding mean-field predictions from\n Eq.~\\eqref{eqn:MF_ERG} with $k=2$ (solid line) deviate dramatically from the simulation samples ($\\Box$). The parameters are\n $\\gamma = 0.005, r = 0.9$, and $I_0 = 0.8$.}\n\\label{fig:lattice_single}\n\\end{figure}\nThe simulations and mean-field predictions \\eqref{eqn:MF_ERG} systematically\ndeviate; the latter always overestimates the final density of adopters and\nunderestimates the final density of ignorants. This can be attributed to the\ntopological constraints on one-dimensional lattices. Initially the lattice\ncomprises of contiguous domains of ignorants that are separated by domains of\none or more neighboring susceptibles. Since ignorants can only become\nsusceptible if a neighbor is susceptible, domains of ignorants shrink at\ntheir interfaces with susceptibles. Crucially, the evolution of an\nignorant-susceptible interface ceases if either the susceptible at the\ninterface adopts or the ignorant at the interface becomes a Luddite. Thus in\none dimension both Luddites and adopters act as barriers to the spread of\nadoption, an effect that is not captured by the mean-field description.\n\nSince domains of ignorants decrease in size and evolve independently, we can\ndetermine analytically the expected final length of ignorant domains\n$\\langle x\\rangle$ and hence the final fractions of each type of agent. The\ndetails of these calculations are given in Appendix~\\ref{ap:1D}. Briefly, we\nfirst determine the probability $P_n(m)$ that a domain of ignorants of\ninitial length $n$ shrinks by $m$. We then use $P_n(m)$ to calculate the\nexpected final length of ignorant domains $\\langle x\\rangle$ and the final\nfraction of ignorants. Since Luddites only form at the boundaries of\nignorant domains, we are able also to determine the expected final fraction\nof Luddites and hence, using the conservation relation $L+I+S+A=1$, the final\nfraction of adopters. The resulting final fractions of each type of agent\nare plotted in Fig.~\\ref{fig:1dlattice_gamma} and agree extremely well with\nthe corresponding numerical simulations. In principle, this method allows us\nto derive explicit formulas for the final fractions of each agent; however,\nin practice these formulas prove cumbersome.\n\n\\section{Discussion \\& conclusion}\n\\label{sec:conc}\n\nInnovations are often accompanied by societal debates and controversies that\nmay lead to divisions between adopters of an innovation and those who\npermanently reject that innovation. Consequently, innovations are rarely\nadopted by the whole population, as various examples, ranging from technology\nto medicine, demonstrate. Classical models of innovation diffusion, such as\nthat proposed by Bass, assume a ``pro-innovation bias'' and predict the\ncomplete adoption of innovations.\n\nMotivated by these considerations we have introduced a multi-stage\ngeneralization of the Bass model, the LISA model, that does not unavoidably\nlead to complete adoption. The main new feature of our model is the\nintroduction of Luddites that permanently oppose the spread of innovation in\ntheir neighborhood. In the LISA model, ignorant individuals can successively\nbecome susceptibles and then adopters, or turn to Luddism in response to a\nhigh rate of adoption and permanently reject the innovation.\n\nWe carried out a detailed analysis of the properties of the LISA model on\ncomplete graphs and Erd\\H os-R\\'enyi random graphs, as well as on\none-dimensional lattices. In particular, we focused on the steady states and\ncompletion time (time to reach stationarity). We showed that significant\ninsights can be gained from a simple mean-field analysis that aptly captures\nthe qualitative aspects of the two basic regimes of the LISA dynamics. When\nthe rate of adoption is low, the population slowly converges to a final state\nthat consists of a high concentration of adopters. In the converse case, the\nstationary state is reached much more quickly, but the final fraction of\nadopters is much lower and is severely limited by the significant densities of\nLuddites and ignorants.\n\nSince most models of innovation diffusion are formulated at mean-field level,\nan important aspect of this work has also been to reveal the limitations of\nthe mean-field approximation. In particular, for Erd\\H os-R\\'enyi random\ngraphs with low mean degree and one-dimensional lattices, the mean-field\napproximation proves inaccurate. This is due to the formation of Luddites \nwhich isolate domains of ignorants from the innovation, an effect particularly\napparently in one dimension.\nIt would be worthwhile to investigate the LISA model on\nmodular networks, where Luddism has the potential to block the spread of\ninnovation to entire communities. In\naddition to the work described in this paper, we also found that the\nmean-field approximation proves better on two-dimensional lattices than on\none-dimensional lattices. \n\nIn summary, the LISA model is a simple, but non-trivial, innovation diffusion\nmodel that accounts for the possibility that the promotion of an innovation\nmay be tempered by the alienation of some individuals. These in turn affect\nthe spread of the innovation. Interestingly, our model outlines two possible\nmarketing scenarios: If one is interested in reaching a high level of\nadoption then this can only be achieved over long time scales, since the rate\nof adoption must be low. However, if the priority is to attain a finite\nlevel of adoption as quickly as possible regardless of the alienation that\nthis may cause, then a high rate of adoption is preferable.\n\n\\section{Acknowledgements}\n\nThis work is supported by an EPSRC Industrial Case Studentship Grant\nNo. EP\/L50550X\/1. SR is supported in part by NSF Grant No.\\ DMR-1205797.\nPartial funding from Bloom Agency in Leeds U.K. is also gratefully\nacknowledged.\n\n\n\\begin{appendix}\n\\section{Analysis of one-dimensional dynamics}\n\\label{ap:1D}\n\nIn this appendix we describe the calculation of the final fractions of each\ntype of agent on one-dimensional lattices. These results are compared with\nsimulations in Fig.~\\ref{fig:1dlattice_gamma} of Section~\\ref{sec:1D}.\n\n\\subsection{Analysis of ignorant domains}\n\nInitially, the nodes on the one-dimensional lattice are either ignorant, with\nprobability $I_0$, or susceptible, with probability $S_0=1-I_0$. Thus the\ninitial configuration consists of connected domains of ignorant nodes\nbordered by susceptibles. Moreover, since ignorants can only become\nsusceptible if a neighbor is susceptible, domains of ignorants only evolve at\ntheir ignorant-susceptible interfaces. We will refer to these as ``active\ninterfaces''. At an active interface one of three events can occur:\n\\begin{itemize}\n\\item The ignorant node becomes susceptible, thus reducing the domain length\n by one, with probability\n\\begin{equation*}\np_S= \\frac{1\/N}{1\/N+r\\gamma\/2+\\gamma}.\n\\end{equation*}\n\n\\item The ignorant node becomes a Luddite, thus reducing the length of the\n domain by one and causing the interface to become inactive, with\n probability\n\\begin{equation*}\np_L= \\frac{r\\gamma\/2}{1\/N+r\\gamma\/2+\\gamma}.\n\\end{equation*}\n\n\\item The susceptible node becomes an adopter, thereby terminating the\n interface evolution, with probability\n\\begin{equation*}\np_A= \\frac{\\gamma}{1\/N+r\\gamma\/2+\\gamma}.\n\\end{equation*}\n\\end{itemize}\n\nFor an isolated ignorant node with two susceptible neighbors, these\nprobabilities respectively become\n\\begin{align*}\n\\hat{p}_S= &\\frac{2\/N}{2\/N+r\\gamma+\\gamma},\\\\\n\\hat{p}_L= &\\frac{r\\gamma}{2\/N+r\\gamma+\\gamma}, \\\\\n\\hat{p}_A= &\\frac{\\gamma}{2\/N+r\\gamma+\\gamma}.\n\\end{align*}\n\nLet $Q_n(m)$ be the probability that a domain of ignorants of initial\nlength $n$ with a \\emph{single} ignorant-susceptible interface has a\n\\emph{final} length $n-m$, with $0\\le m\\le n$. \nWe can determine $Q_n(m)$ as follows: If the final length of ignorants is\n$n-m$, with $01\n\\end{array}\n\\right. .\n\\label{eq:Pmm}\n\\end{equation*}\nAlso, the probability that a single ignorant node that initially has\ntwo susceptible neighbors becomes a susceptible or Luddite is given\nby\n\\begin{equation*}\nP_1(1)=\\hat{p}_A(p_L+p_S)+\\hat{p}_L+\\hat{p}_S.\n\\label{eq:P01}\n\\end{equation*}\nThus the solution to the recursion relation \\eqref{eq:recur2} for\n$00$ is given by\n$p_0(n)=I_0^{n-1}S_0$ for large $N$. Thus we find that\n\\begin{equation*}\n\\langle x\\rangle\n=\\sum_{n=0}^Nnp_0(n)-\\sum_{n=0}^Np_0(n)\\sum_{l=0}^nlP_n(l).\n\\label{eq:meanx}\n\\end{equation*}\nIn principle, we may use the above to obtain an explicit expression for\n$\\langle x \\rangle$. In practice, however, we use the solutions to \\eqref{eq:recur2} to\ncalculate $\\langle x \\rangle$ numerically.\n\n\\subsection{Calculation of population densities}\nInitially, the mean number of ignorants is given by $I_0 N$ and so dividing\nby the mean length of ignorant domains, $1\/(1-I_0)$, yields the expected\nnumber of ignorant domains, $(1-I_0)I_0N$.\nThus the final density of ignorants is $$I_\\infty=(1-I_0)I_0\n\\langle x \\rangle.$$\n\nThe probability that an ignorant domain survives is\n$$q= 1 - \\sum_{n=0}^{\\infty}p_0(n)P_n(n).$$\nSurviving ignorant domains have two interfaces, which are either\nignorant-adopter or ignorant-Luddite, with probabilities $p_A\/(p_L + p_A)$\nand $p_L\/(p_L + p_A)$, respectively. Thus the expected number of Luddites at\nthe interfaces of non-vanishing ignorant domains is given by\n\\begin{equation}\n\\eta_{+}=\\frac{2 p_L}{p_L + p_A} q(1-I_0)I_0 N.\n\\label{eq:edgeLuddites}\n\\end{equation}\n\nIt is also possible for Luddites to arise when a domain vanishes. By\nidentifying the terms in $P_n(n)$ that result in Luddites, it is possible to\ndetermine that the expected number of Luddites that arise when a domain of\ninitial size $n>1$ vanishes is given by\n$$l_n=\\left(\\hat{p}_Ap_L+\\hat{p}_L\\right)p_S^{n-1}\n+(n-1)\\left(2p_Lp_S^{n-1}+p_L^2p_S^{n-2}\\right)$$ and\n$l_1=\\hat{p}_Ap_L+\\hat{p}_L$. Thus the expected number of Luddites\nthat arise from domains of ignorants that vanish is\n\\begin{equation}\n\\eta_0=(1-I_0)I_0N\\sum_{n=0}^{\\infty}p_0(n)l_n.\n\\label{eq:vanishLuddites}\n\\end{equation}\nSumming Eqs.~\\eqref{eq:edgeLuddites} and \\eqref{eq:vanishLuddites} and\ndividing by $N$ we arrive at the final density of Luddites\n$$L_\\infty = I_0(1-I_0) \\left( \\frac{2 p_L}{p_L + p_A} q + \\sum_{n=0}^{\\infty}p_0(n)l_n \\right).$$\nSince the dynamics cease when $S=0$, the number of adopters can be\nfound using the conservation law $A_\\infty= 1 - L_\\infty - I_\\infty$.\n\n\\end{appendix}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe galactic nano-quasar GRO J1655-40 is an interesting low mass X-ray binary (LMXB) with a primary mass \n$M = 7.02\\pm0.22~M_\\odot$ (Orosz \\& Bailyn 1997) and the companion star mass \n= $2.3~M_\\odot$ located at a distance of $D = 3.2 \\pm 0.2$~kpc (Hjellming \\& \nRupen 1995). The disk has an approximate inclination angle of $\\theta = 69.5^\\circ\\pm0.1^\\circ$ \n(Orosz \\& Bailyn 1997) to the line of sight. \nIn the last week of February 2005 it became X-ray active \n(see Shaposhnikov et al. 2007 and references therein) and remained so for \nthe next 260 days before returning to the hard state. \nIn this Letter, we thoroughly analyse the data of the first two weeks of \nthe very initial stage (rising phase) and the last three weeks of the final stage \n(decline phase) of the 2005 outburst. We clearly observe very smooth day to day variation of the\nQPO frequency in these two phases. We propose that a satisfactory explanation of this\nbehavior can be obtained if we assume that an oscillating shock which is sweeping inward\nthrough the disk in the rising phase and outward in declining phase is responsible for the QPO. In the next\nSection, we briefly present an overview of QPOs observed in the black hole candidates. \nIn \\S 3, we present the shock oscillation solution for the generation of QPOs.\nIn \\S 4, we present the observational results in detail and show our model fit of the\nQPO frequencies from day to day. We interpret the results and extract the shock parameters. \nIn \\S 5, we give a coherent description of the rising and the declining phases of the outburst.\n\n\\section{Low and intermediate frequency QPOs in black hole candidates}\n\nObservations of low and intermediate frequency quasi-periodic oscillations (QPOs) \nin black hole candidates have been reported quite extensively in the literature. One\nsatisfactory model shows that the oscillation of X-ray intensity is actually due to the\noscillation of the post-shock (Comptonizing) region\n(Molteni, Sponholz \\& Chakrabarti 1996 [hereafter MSC96]; Chakrabarti \\& Manickam 2000 [hereafter CM00]). \nPerturbations inside a Keplerian disk also have been assumed to be the cause of low-frequency QPO also \n(e.g., Trudolyubov, Churazov and Gilfanov 1999; see, Swank 2001 for a review). The numerical simulations\nof low-angular momentum accretion flows including the thermal cooling effects (MSC96; Chakrabarti, \nAcharyya \\& Molteni 2004, hereafter CAM04) or dynamical cooling (through outflows, e.g., Ryu, \nChakrabarti \\& Molteni 1997) show clearly that the shocks oscillated with frequencies similar to \nthe observed QPO frequencies. Not only were the shock locations found to be a function of the cooling \nrate (MSC96), they were found to propagate when viscous effects are turned on (Chakrabarti \\& Molteni 1995). \n\n\\section{The properties of low and intermediate frequency QPOs from shock oscillations}\n\nIt has been argued in the past that steady, propagating and oscillating shocks can form in a low angular momentum \nflow (e.g., CAM04 and references therein). In the shock oscillation solution (MSC96, CM00; CAM04) of QPOs, \nthe oscillations take place at a frequency inverse to the infall time in the post-shock region (i.e.,\nthe region between the shock at $r=r_s$ and the horizon). In a shock-free low angular momentum flow, \nthis infall time is $t_{infall} \\sim r_s\/v = r_s(r_s-1)^{1\/2}$, where $v=1\/(r_s-1)^{1\/2}$ is the free-fall\nvelocity in a pseudo-Newtonian potential (Paczynsk\\'i \\& Wiita, 1980) $\\phi_{PN}=-1\/(r_s-1)$ Here,\ndistance, velocity and time are measured in units of the Schwarzschild radius $r_g=2GM\/c^2$, the\nvelocity of light $c$ and $r_g\/c$ respectively and where, $G$ and $M$ are the universal constant\nand the mass of the black hole. However, in the presence of a significant angular momentum \ncapable of producing centrifugal pressure supported shocks around a black hole, the \nvelocity is reduced by a factor of $R$, the compression ratio $R=\\rho_-\/\\rho_+$, where, \n$\\rho_-$ and $\\rho_+$ are the densities in the pre-shock and the post-shock flows,\nbecause of the continuity equation $\\rho_- v_-= \\rho_+ v_+$ across a thin shock.\n\nIn the presence of a shock, the infall time in the post-shock region is therefore given by\n$$\nt_{infall}\\sim r_s\/v_+ \\sim R r_s(r_s-1)^{1\/2} \n\\eqno{(1)}\n$$ \n(CM00; Chakrabarti et al. 2005). Of course, to trigger the oscillation, the accretion rate \nshould be such that the cooling time scale roughly match the infall time scale (MSC96).\nThus, the instantaneous QPO frequency $\\nu_{QPO}$ (in $s^{-1}$) is expected to be\n$$\n\\nu_{QPO} = \\nu_{s0}\/t_{infall}= \\nu_{s0}\/[R r_s (r_s-1)^{1\/2}]. \n\\eqno{(2)}\n$$\nHere, $\\nu_{s0}= c\/r_g=c^3\/2GM$ is the inverse of the light crossing time of the black hole \nof mass $M$ in $s^{-1}$ and $c$ is the velocity of light. In a drifting shock scenario, \n$r_s=r_s(t)$ is the time-dependent shock location given by\n$$\nr_s(t)=r_{s0} \\pm v_0 t\/r_g.\n\\eqno{(3)}\n$$\nHere, $r_{s0}$ is the shock location when $t$ is zero and $v_0$ is the shock velocity \n(in c.g.s. units) in the laboratory frame. The positive sign in the second term is to be used for an outgoing \nshock in the declining phase and the negative sign is to be used for the in-falling shock \nin the rising phase. Here, $t$ is measured in seconds from the first detection of the QPO. \n\nThe physical reason for the oscillation of shocks appears to be\na 'not-so-sharp' resonance between the cooling time scale in the post-shock region and the infall time \nscale (MSC96) or the absence of a steady state solution\n(Ryu, Chakrabarti \\& Molteni, 1997). In both the cases, the QPO \nfrequency directly gives an estimate of the shock location (Eq. 1). \nThe observed rise of the QPO frequencies with luminosity (e.g., Shaposhnikov \\& Titarchuk 2006, \nhereafter ST06) is explained easily in this model since an enhancement \nof the accretion rate increases the local density and thus the cooling rate. \nThe resulting drop of the post-shock pressure reduces the shock location\nand increases the oscillation frequency. In CM00 and Rao et al. (2000) it was shown that QPOs \nfrom the higher energy Comptonized photons, thought to be from\nthe post-shock region, (Chakrabarti \\& Titarchuk, 1995), have a higher $Q$ value. \nThe latter model requires two components, one\nKeplerian and the other having an angular momentum lower than the Keplerian (referred to hereafter\nas sub-Keplerian), and explains a wide variety of observations of\nblack hole candidates (Smith, Heindl, \\& Swank 2002; Smith, Heindl, Markwardt \\& Swank 2001; \nSmith, Dawson \\& Swank 2007). As the shocks are the natural solutions to this sub-Keplerian component,\nexplanation of QPOs from shocks is justified.\nWhen the Rankine-Hugoniot relation is not exactly satisfied at the shock\nor the viscous transport rate of the angular momentum is different on both sides, \nthe mean shock location would drift slowly due to a difference in pressure\non both sides. In the rising phase of the outburst, a combination of the ram pressure of the incoming \nflow and rapid cooling in the post-shock region (which lowers the \nthermal pressure) pushes the oscillating shock inward. \nIn the decline phase, the Keplerian disk itself recedes outward creating a lower\nthermal pressure in the post-shock region. In this case, the shock drifts outward.\n\n\\section {Observational results and analysis}\n\nWe concentrate on the publicly available data of $51$ observational IDs (corresponding to observations of\na total of $36$ days) of GRO J1655-40 acquired with the RXTE\nProportional Counter Array (PCA; Jahoda et al., 1996). Out of these IDs, $27$ are of the rising phase \n(from MJD 53426 to MJD 53441) and $24$ are of declining phase (from MJD 53628 \nto MJD 53648). We extracted the light curves (LC), the PDS and the energy spectra from the\ngood detector unit PCU2 which also happens to be the best-calibrated. We used the\nFTOOLS software package Version 6.1.1 and the XSPEC version 12.3.0. \nFor the timing analysis (LC \\& PDS), we used the Science Data of the Normal mode \n($B\\_8ms\\_16A\\_0\\_35\\_H$) and the Event mode ($E\\_125us\\_64M\\_0\\_1s$, $E\\_62us\\_32M\\_36\\_1s$). \nTo extract LC from Event mode data files, we used ``sefilter\" task and for the\nnormal mode data files, we used ``saextrct\" task. For the energy spectral analysis, \nthe {\\bf ``Standard2f\"} Science Data of PCA was used. For PCA background estimation \npurpose the ``pcabackest\" task was used while to generate the response files \nthe ``pcarsp\" task was utilized. For the rebinning of the `pha' files \ncreated by the ``saextrct\" task, we used the ``rbnpha\" task. For all the analysis,\nwe kept the hydrogen column density ($N_{H}$) fixed at 7.5$\\times$ 10$^{21}$ \natoms cm$^{-2}$ and the systematics at $0.01$.\n\n\\begin {figure}[t]\n\\vskip 0.8 cm\n\\centering{\n\\includegraphics[width=8.0cm]{fig1ab.eps}}\n\\caption{Variation of QPO frequency with time (in day)\n(a) of the rising phase and (b) of the declining phase.\nError bars are FWHM of fitted Lorentzian curves in the\npower density spectrum. The dotted curves are the solutions from oscillating\nand propagating shocks. While in (a) the shock appears to be drifting at a\nconstant speed towards the black hole, in (b) the shock initially moves very\nslowly and then extends at a roughly constant acceleration. According to the\nfitted solution, the shock wave goes behind the horizon on the $16.14$th day, about\n$15$ hours after the last observed QPO.}\n\\end {figure}\n\nFigures 1(a-b) show the variation of the QPO frequencies in (a) the rising and (b) the declining phases\nof the outburst. The full widths at half maxima of the fitted QPOs have been used as the error bars.\nIn the rising phase (a), the $0^{th}$ day starts on MJD=53426. The fitted curve represents our fit\nwith Eqs. (2-3) which requires that the shock is launched at $r_s=1270$ which \ndrifts slowly at $v_0=1970$cm s$^{-1}$. On the \n$15^{th}$ day after the outburst starts, the noise was high, but we could clearly observe two\ndifferent QPO frequencies with a very short time interval. \nAt the time of the last QPO detection ($15.41^{th}$ day) \nat $\\nu=17.78$Hz, the shock was found to be located at $r \\approx 59$. \nThe strength of the shock $R$, which may be strong at the beginning with $R=R_0\\sim 4$\nshould become weaker and ideally $R\\sim 1$ at the horizon $r=1$, as it is impossible to maintain\ndensity gradient on the horizon. If for simplicity we assume the variation of the shock strength \nas $1\/R\\rightarrow 1\/R_0 + \\alpha t_d^2$, where $\\alpha$ is a very small number limited by the \ntime in which the shock disappears (here $t_{ds} \\sim 15.5$days).\nThus, the upper limit of $\\alpha \\sim (1-1\/R_0)\/t_{ds}^2 =0.75\/t_{ds}^2 = 0.003$. \nWe find that for a best fit, $\\alpha \\sim 0.001$ and the reduced $\\chi^2=0.96$. However,\nthe fit remains generally good ($\\chi^2= 1.71$ for $xs0=1245$ and $v0=1960$cm\/s) even \nwith a shock of constant strength ($R=R_0$). Hence \nfor a generally good fit the number free parameters could be assumed to be three: the shock strength, initial shock\nlocation and the shock velocity.\n\nIn the declining phase (Fig. 1b), the QPO frequency on the first day\n($MJD=53631$) corresponds to launching the shock at $ \\sim r_s=40$. It evolves\nas $ \\nu_{QPO} \\sim t_d^{-0.2}$. Since $ \\nu_{QPO} \\sim r_s^{-2\/3}$ (Eq. 1), \nthe shock was found to drift very slowly with time ($r_s \\sim t_d^{0.13}$) until \nabout $t_d=3.5$ day where the shock location was $\\sim r=59$. There is a discontinuity in the\nbehavior at this point whose possible origin is obtained from spectral studies presented below.\nAfter that, it moves out roughly at a constant acceleration ($r_s \\sim t_d^{2.3}$) \nand the QPO frequency decreases as $\\nu_{QPO}\\sim r_s^{-2\/3} \\sim t_d^{-3.5}$.\nFinally, when the QPO was last detected, on $t_d=19.92$th day ($MJD=53648$), \nthe shock went as far as $r_s=3100$ and the oscillation could not\n be detected any longer. The strength of the shock was kept at $R=4$. The reduced $\\chi^2$ for the\nfit is given by $0.236$.\nIn Figs. 2(a-b), we present the dynamic PDS where the vertical direction indicates the QPO frequency. \nResults of five dwells are given in both the rising and the declining phases. \nThe grayscale has been suitably normalized so as to identify the QPO features prominently.\n\n\\begin {figure}[h]\n\\centering{\n\\vskip -1.0 cm\n\\includegraphics[width=6.5cm,angle=270]{fig2a_bw.eps}\\\\\n\\vskip -1.2 cm\n\\includegraphics[width=6.5cm,angle=270]{fig2b_bw.eps}}\n\\vskip -0.8 cm\n\\caption {(a) Dynamic power density spectra over five\ndays in the rising phase. \n(1) Obs. ID=91404-01-01-01, QPO=0.382 Hz,\n(2) Obs. ID=91702-01-01-03, QPO=0.886 Hz,\n(3) Obs. ID=90704-04-01-00, QPO=2.3130 Hz,\n(4) Obs. ID=91702-01-02-00, QPO=3.45 \\& 6.522 Hz with a break frequency\nat 0.78 Hz and (5) Obs. ID=91702-01-02-01, QPO=14.54 \\& 17.78 Hz.\n(b) Dynamic power density spectra over five days in the decline phase. (1) Obs. ID=91702-01-76-00, \nQPO=13.14 Hz, (2) Obs. ID=91702-01-79-01, QPO=9.77 Hz,\n(3) Obs. ID=91702-01-80-00, QPO=7.823 \\& 15.2 Hz with a break\nfrequency at 1.32 Hz, (4) Obs. ID=91702-01-80-01, QPO=4.732 Hz \nwith a break frequency=0.86 Hz, (5) Obs. ID=91702-01-82-00, QPO=0.423 Hz.}\n\\end {figure}\n\n\n\\begin {figure}[b]\n\\vskip -1.0 cm\n\\centering{\n\\includegraphics[width=8cm]{fig3.eps}}\n\\caption{Power density spectrum (left panel) and energy spectral index (right panel)\nof the rising phase (i and ii) and the declining phase (iii-v). \nThe QPO frequencies are (i) $0.513$ Hz (0.632), (ii) $3.45$ \\& $6.522$ Hz (0.441), (iii) $13.14$ Hz \n(0.848), (iv) $7.823$ Hz (1.14) and (v) $1.347$ Hz (1.29) respectively. Numbers next to frequencies \nare the reduced $\\chi^2$ values. \nThe components used for fitting the energy spectra are marked: `DATA' for total fit, \n`BB' for black body, `G' for Gaussian, `CST' for Comptonization using the Sunyaev-Titarchuk (1980) model \nwith an exponential cutoff, `PL' for a power-law component without a cut-off.}\n\\end {figure}\n\n\nIn order to examine how the disk was re-adjusting itself in these two phases, \nwe studied both the timing and the spectral properties and plot the results in\nFig. 3 (panels i to v) we give an idea of this by providing\nthe PDS (left panels) and the fitted energy spectrum (right panels) of\nfive observations. The observation IDs are: 91702-01-01-01 ($11.0315^{th}$ day) and 91702-01-02-00G \n($14.6957^{th}$ day) of the rising phase and 91702-01-76-00 ($0^{th}$ day), 91702-01-80-00 \n($3.27^{th}$ day) and 91702-01-81-01 ($6.118^{th}$ day) of the declining phase respectively. \nIn the rising phase, the spectrum clearly becomes softer as the shock moves in and the QPO frequency increases. \nThe softening of the spectrum with the increase in the QPO frequency \nhas been reported by various authors (e.g., Chakrabarti et al. 2005;\nST06; Shaposhnikov et al., 2007). From a theoretical point of view,\nMSC96 explicitly showed that increased cooling reduces the shock location \nand increases the QPO frequency. This was also reported by ST06\nusing the data of Cyg X-1. { We observe that the black body (BB) component from the\nKeplerian disk and the Gaussian (G) components are also strengthened. The Gaussian component\ncould be from iron lines, but due to poor resolution of RXTE this cannot be said with certainty. \nIn the declining phase, as the QPO frequency \ndecreases, all three of the black body (BB), the power-law (PL) component\nand an additional Comptonization component (using Comptonization through the Sunyaev-Titarchuk \nor CST model) from the region decreases. Interestingly, this additional cooler CST component with \na cut-off was required only before the discontinuity observed on $3.5$th day in the \noutgoing phase, signifying that perhaps there were two sources of X-rays: one (PL) from\nthe post-shock region and the other (CST) from the outflow region.} Ultimately, after the\ndiscontinuity, only a weak power-law component remains which is emitted from\na hot tenuous sub-Keplerian flow. This is all that remains after \nthe outburst is over. We interpret this observation to be associated \nwith a possible change in the flow behavior at $r\\sim 59$: either \nthe shock is propagating from the disk region to the outflow\nregion or the shock is moving away from the black hole due to the low \npressure in the emptied disk. We also note that the disk component \nwhich was becoming stronger in the rising phase is absent in the \nlate declining phase, indicating that the Keplerian component disappears \nsoon after the outburst is over. \n\n\\section{Discussions and concluding remarks}\n\nIn this Letter, we show that during the rising phase of the outburst of GRO J1655-40, \nthe QPO frequency increases very slowly in the first few days and then rapidly \nincreases to about $18$Hz before it disappears altogether. Our slowly drifting shock oscillation solution\nexplains this variation very accurately. Our estimation suggests that the shock was at \n$1270$ Schwarzschild radii when the first observation of QPO was made and it went to \n$r=59$ Schwarzschild radii when it was last observed. Within $15$ hours of this,\nthe shock front went inside the horizon and thus when the observation was made on the\nnext day, the QPO was absent. The inward drift velocity of the shock was slow, only about $20$\nm s$^{-1}$. As the shock proceeds close to the black hole,\nthe Keplerian disk follows the shock and the energy spectrum gradually became \nsofter with the black body component becoming stronger each day. Thus the whole scenario is \nconsistent with our theoretical understanding (MSC96) that this drift \nis due to the reduction of the post-shock pressure by the increased cooling\neffects in addition to the higher upstream ram pressure. Both MSC96 and CAM04 computed the cooling timescale\nusing bremsstrahlung and Comptonization respectively and showed that oscillations occur\nwhen the infall time scale is comparable to these coolings in super-massive and\nstaller mass black holes respectively. As a consistency check, one could also use the \nfitted spectrum to compute the electron temperature and calculate the cooling time scale\nfrom $E_{th}\/{\\dot E}$ where, $E_{th}$ is the thermal energy content of the electrons\nand ${\\dot E}$ is the rate of cooling due to Comptonization. However, $E_{th}$ depends on the \naccretion rate and electron temperature and ${\\dot E}$ depends on the enhancement factor (Dermar\net al. 1991). However, the PCA data from 3-25keV does not allow one to compute the these unknowns\nunambiguously.\n\nIn the decline phase, the QPO frequency was found to be decreasing \nmonotonically. This means that an oscillating shock is propagating outwards. The nature\nof the reduction of QPO frequency is very curious. For the first $\\sim 3.5$ days, the shock \nlocation did not change very much, from $\\sim 40$ to $\\sim 59$ or so, as if it was stalling. \nIn this phase an additional cooler Comptonized component was required.\nAfter this, the shock suddenly moved away with almost constant acceleration for another\n$16$ days before the QPO disappeared completely. The spectrum remained \nhard and the intensity decreased monotonically. Towards the end, the spectrum\nis dominated only by the power-law component. We conjecture that in the first $\\sim 3.5$ days the\nshock remained inside the disk drifting slowly outward, perhaps due to interaction of the receding shock\nwith the still incoming Keplerian flow. After that, the receding Keplerian disk \ncreated a vacuum in the system and accelerated the shock wave\noutward. This is indicated by almost square-law behavior of the shock location. \n\nIn the two component advective flow model that we are using, the shock forms in the\nsub-Keplerian component since a Keplerian flow is subsonic and cannot have a shock. At the\nonset of the rising phase of the outburst, only the sub-Keplerian flow rate increases rapidly\nsince it is almost freely falling. The shocks we consider\nmay have actually formed farther out (than $1270r_g$, for example)\nwe could detect them only when the rms value of the QPO is high enough. Similarly, when the \nshock comes closer to the black hole horizon, the noise also rises and QPOs may not be detectable\nas well.\n\nThe observational result we described here is unique in the sense that we are able to connect the \nQPO frequency of one observation with that of the next by a simple analytical formalism. To our knowledge\nno competing model exists which uses a true solution of the flow, such as shocks in our case,\nto explain such a behavior. In any case, even if the QPOs were generated by certain non-axisymmetric\nfeature (such as a blob), it is impossible that it will survive beyond a few \norbital time-scales. Furthermore, the smooth decrease of QPO frequency requires that the blob moves outwards\nthrough a differentially rotating disk for some weeks, odds of which is insignificant.\nIf our explanation of the propagation of shocks is correct, then \nthis outburst shows convincingly how a shock wave smoothly disappears behind the \nhorizon after $\\sim 16$ days of its initial detection. Thus, such an observation \nwould be unlikely in neutron star candidates. Our interpretation is generic and other \noutburst sources should also show such systematic drifts. A discussion of other similar sources will be dealt \nwith in future.\n\n\\section*{Acknowledgments}\n\nD. Debnath acknowledges the support of a CSIR scholarship \nand P.S. Pal acknowledges the support of an ISRO RESPOND project. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \n\nMassive stars are the principal source of heavy elements and ultraviolet radiation, and a \nmajor supplier of wind and supernova energy, within the Universe. Individually, they dominate cluster formation and,\ncollectively, they influence the evolution of galaxies. \nHowever, our knowledge of massive stars remains limited \\citep{2007ARA&A..45..481Z}.\nWe know they are born in massive and dense clumps embedded within \ngiant molecular clouds \\citep{1991psfe.conf....3B}. We also think we understand fairly well how physical \nmechanisms conspire to describe the emergence of low- and intermediate-mass stars. But the events which conspire \nto produce the high-mass counterparts are still controversial \\citep{2007ARA&A..45..565M}. This is due to a combination \nof factors including their rarity, large distances, rapid evolution, high extinction and their confusion with associated clusters. \n\nA formation model requires integration of evolutionary models for the internal star, the environment\nand the feedback. Together, these constitute a `protostellar system' which consists of several distinct components including a hydrostatic core,\ndisk, envelope, wind, jets and bipolar outflow. Recently, with Herschel, BLAST and Planck following Spitzer \n\\citep[e.g.][]{2010A&A...518L..97E,2009ApJ...707.1824N,2011A&A...536A..22P}, we have acquired the quality and \nquantity of data to explore the early formation stages. \nProcesses that can now be addressed include clump dispersal, cluster formation, mass inflow, stellar accretion, stellar flux and mass outflow.\n\nThe aim here is to link the system components and underlying processes together through a model \nframework that predicts resulting observable correlations.\nWe thus develop a scheme which explores plausible paths from clump to exposed star. It is set up to track how mass is passed between\nthe components, while accounting for direct ejection from the supplying clump and indirect ejection through jets.\n\nThe interaction of a gradually emerging star with a massive clump is described in terms of a time sequence\ngoing from compact and hot molecular cores to extended H\\,{\\small II} regions \\citep{2002ARA&A..40...27C} as expansion occurs. This has emphasized an issue concerning the time scales with too many bound hypercompact or the later ultracompact \n H\\,{\\small II} regions being observed. This may be resolved by taking less ideal approaches to accretion and dispersal from \nand to the clump \\citep{2010ApJ...719..831P}, allowing for choking of the cavity through the infall of dense filaments and fragments of the clump.\n\nThe very early stages of protostar formation within cold clumps have now been identified in large numbers as Infrared\nDark Clouds (IRDCs) \\citep{2006ApJ...653.1325S,2006ApJ...641..389R} .\nThese could be the star-less objects from which massive stars will form. However,\nthe earliest signs of accretion have also been signposted by directed outflows of cold molecular gas and, it turns \nout, most of the IRDCs studied so far host weak 24$\\mu$m emission sources and already drive molecular outflows,\nboth strong indicators for active star formation \\citep{2010ApJ...715..310R}. In addition, the Herschel Space Telescope has helped reveal \nthe full population of early evolutionary stages at the very onset of massive star formation \\citep{2010A&A...518L..78B,2012A&A...547A..49R}.\n\nIt has been proposed that the large-scale evolution can be split into two phases: an accretion phase and a clean-up phase.\nInitially, as the protostar gains in mass, its sphere of influence grows, leading to\naccelerated accretion \\citep{2003ApJ...585..850M}. There is observational support for this as discussed by \\citet{2011MNRAS.416..972D} which favours either turbulent core \n\\citep{2003ApJ...585..850M} or competitive accretion \\citep{2001MNRAS.323..785B} models. Subsequently, in a distinct second phase, termed the clean-up phase, the accretion has stopped abruptly and the remaining clump material is partly dispersed or integrated into a surrounding cluster \\citep{2008A&A...481..345M}. \n\nIn another scenario, the massive star does not fully form early. Low mass star formation dominates until the clump has considerably \nevolved. These stars would remain difficult to detect. It has been remarked that the current Initial Mass Function implies two peaks \nof star formation with the majority of low mass stars forming first and high mass stars forming later \\citep{2001A&A...373..190B}.\nIn competitive accretion, gas is funnelling down to the cloud centre where\nstars, initially accreting gas with low relative velocity, already have large mass before accreting the late-arriving higher velocity gas \\citep{2006MNRAS.370..488B}.\n\nStellar collisions and mergers could supplement gas accretion \\citep{2002MNRAS.336..659B}. \nMassive stars could form via coalescence of intermediate mass stars within very dense systems. Although not considered here, along with other scenarios \ninvolving fragmentation, these should lead to alternative predictions.\n\n\nDo high-mass stars form in a scaled-up analogue of the low-mass formation scenario? \\citet{2008A&A...481..345M} find a consistent \ninterpretation in favour of an analogy. They analysed wide Spectral Energy Distributions (SEDs) to derive bolometric luminosity \n($L_{bol}$) and clump mass ($M_{clump}$). This leads to the diagnostic diagram of $L_{bol}$ versus $M_{clump}$ (we have replaced the term \nenvelope with clump here, employing envelope to refer to the inner part of the clump which accretes directly on to the protostar). \nThe high-mass objects were then shown to occupy a sequence of regions \non this diagram in a similar manner to those evolving Classes which populate the low-mass regions. \nHerschel data have recently extended this to include protostars \\citep[e.g.][]{2010A&A...518L..97E}. \n\nThe SED parameters now available include fluxes derived from the Red MSX Source survey, Spitzer IRAC and MIPS, \nSOFIA FORCAST, Herschel PACS, Herschel SPIRE, and BLAST.\nFurther data are available across the radio, sub-millimetre and infrared. From these, we can derive clump mass, temperature, \nluminosity, ultraviolet flux, outflow mass and outflow power. So we can now construct several diagrams to employ as diagnostic tools \nto estimate the evolutionary stages. \n\nThe bolometric luminosity and temperature, $T_{bol}$, have been used in isolation to test accretion models for a version of the model tuned to low-mass protostars \\citep{2006MNRAS.368..435F}. A similar approach but just using the luminosity function was adopted by \n\\citet{2011MNRAS.416..972D} for high-mass stars using MSX data and radio identifications to distinguish protostars from \n later stages. In place of an SED derivation of luminosity, the 21\\,$\\mu$m flux was utilised with the assumption that the bolometric luminosity\n can be taken as a reliable proxy for the set of protostars being investigated. Both these approaches led to constraints on the time scale of \n young stars and the general form of the accretion process.\n\nA potential test would use the radio luminosity, $L_r$, produced by free-free emission after extreme ultraviolet excitation. The ratio of \n$L_r$\/$L_{bol}$ should then provide a measure of the development while $L_{bol}$\/ $M_{clump}$ provides a distinct measure. Plotted together,\nsuch a diagram provides a {\\em distance independent} distribution of evolutionary phases. \n\nOutflow parameters have also been used to distinguish the phases of low-mass stars where accretion is known to decelerate \nrather than accelerate with time \\citep{2006A&A...449.1077C}. This leads to a decrease \nin the force of the outflow as the source ages \\citep{2010MNRAS.408.1516C}.\n \\citet{2002A&A...383..892B} showed that bipolar outflows are\nindeed ubiquitous phenomena in the formation process of massive stars, suggesting similar flow-formation scenarios for all masses,\nconsistent again with scaled-up, but otherwise similar, physical processes - mainly accretion - to their low-mass counterparts. Going further,\nit was shown that the measured molecular hydrogen outflow luminosity is tightly related to the source bolometric luminosity for \nlow mass stars \\citep{2006A&A...449.1077C}, and this relationship extends to massive objects \\citep{2008A&A...485..137C}. \nIt is clear that this only applies to the youngest protostars in which the bolometric luminosity is dominated by the \nrelease of energy through accretion. Those sources associated with jets are very young (well before the Main Sequence turn-on), while \nthose without detectable jets possess ultracompact H\\,{\\small II} regions \\citep{2005ccsf.conf..105B,2011BSRSL..80..235R}.\n\nThe objectives of this first paper is to set up the model framework and consider the effects of radiation feedback. We impose simple accretion\nrates that slowly vary. Subsequent works will tackle accretion outbursts, the outflow properties and the strength of feedback in self-regulation.\n\nThe accelerated-accretion model generates a particular $L_{bol}$ versus $M_{clump}$ relation \\citep{2008A&A...481..345M}. We begin here by exploring how general this is. A constant accretion rate is a common working assumption while there is evidence for both a declining rate as well as a sporadic\/episodic rate. These models involve less dramatic \ncut-offs in the accretion and should generate models with different statistical properties. \n\nA major issue to address later is the existence of two distinct phases.\nIf accelerated accretion is followed by a clean-up phase, we expect the outflow feedback phase to precede the radiative feedback phase.\nIt may prove difficult for accretion phenomena to still dominate once the rapid rise \nin ultraviolet flux has started at late times. Quite remarkably, nature has no problem: all of the sources with infall signatures onto\nUltracompact H\\,{\\small II} regions have corresponding outflow signatures as well \\citep{2011arXiv1112.0928K}. This observation\nsuggests that accretion may continue, consistent with the gravo-turbulent model \\citep{2004A&A...419..405S}. However,\nboth accretion and collimated outflows are probably weak when the star has advanced to its Ultracompact H\\,{\\small II} \nstage \\citep{2010MNRAS.404..661V}. An ultimate aim of this study will therefore be to determine the conditions under which the accretion-outflow phase can significantly\noverlap with the UV phase, associated with the final contraction of a massive star.\n\n\\section{Method} \n\\label{method}\n\n\\subsection{ Model construction: mass movement}\n\\label{model}\n\nThe model is constructed upon (1) the extension of the Unification Scheme for low-mass stars, \n(2) the strategy and model invoked for high-mass stars, (3) the detailed evolutionary tracks of an accreting massive protostar, \n(4) the results for a range of potential accretion rates and (5) predicted outcomes for radiative and outflow feedback. All algorithms\nand graphics are written and processed in IDL.\n\nThe first task is the construction of a model for the environment in which a given clump mass, $M_{clump}$, is redistributed\nin time according to a prescribed formula, constrained by mass conservation. The clump directly supplies three entities: an \ninner envelope. $M_{env}$, a surrounding cluster $M_{stars}$ and dispersal into the ambient cloud $M_{gas}$ (with \nsome additional help from the jet-driven outflow). The inner envelope is here assumed to supply the accretion disk at a rate \n$\\dot M_{acc}(t)$ which, feeds instantaneously both the star, $M_*$, and the jets, $M_{jets}$. The jet material accumulates \nin an inner outflow, $M_{out}$, which also pushes out a fraction of the clump. Hence:\n\\begin{equation}\n \\dot M_{clump}(t) + \\dot M_{env}(t) + \\dot M_{stars}(t) + \\dot M_{gas}(t) = 0,\n\\end{equation}\n\\begin{equation}\n - \\dot M_{env}(t) = \\dot M_{acc}(t) = \\dot M_{*}(t) + \\dot M_{jets}(t) .\n\\end{equation}\n\nWe will consider two important free parameters. The most critical is the fraction, $\\xi$, of the initial clump mass which ends \nup as part of the star. Even for low mass stars, it is well known that there must be a much larger obscuring mass than necessary \nto form the star \\citep{1998ApJ...492..703M}. This mass prolongs the embedded phase and extends the late Class 0 and early Class 1 stages. The best estimate for the excess mass was found to be a factor of two in the low-mass version of the present scheme on using accretion rates derived from gravo-turbulent models \\citep{2006MNRAS.368..435F}. \n\nAlso for high-mass protostars, the surrounding bound clumps are estimated to exceed the star's mass by a factor which can exceed 30 \\citep{2008A&A...481..345M}. We thus anticipate quite low values for $\\xi$. Hence, in this work, we assume that the clump mass is sufficient to form the protostar and the associated stellar cluster (see Sub-section~\\ref{clumps}) in addition to gas expelled directly from the clump. We assume both mass loss rates to be constant with the same time scale. \n\nThe second parameter is the fractional efficiency $\\epsilon$ of mass diversion from inflow to outflow, from the disk to the jets. The extended magneto-centrifugal \nmodel is expected to be quite efficient and the X-wind model is expected to reach the thirty per cent level \\citep{1988ApJ...328L..19S,1994ApJ...429..781S}. In doing so, \nsuch magneto-centrifugal mechanisms can carry away the total angular momentum and kinetic energy of the accreting disk material. \n\nPreviously, we found that a constant efficiency $\\epsilon$ was inconsistent with the observations of low-mass stars \\citep{2000IrAJ...27...25S}. Hence we took\n\\begin{equation}\n \\epsilon(t) = \\eta \\left[\\frac{\\dot M_{acc}(t)}{\\dot M_o}\\right]^\\zeta,\n\\label{eqn-hm}\n\\end{equation}\nwhere $\\dot M_o$ is the peak accretion rate and $\\zeta$ is a constant. \n\nThe variable jet efficiency was introduced in order to account for evolving outflow properties of low-mass protostars. The Class 0 outflows appear to have \na mechanical luminosity of order of the bolometric luminosity of the protostellar core. On the other hand, Class 1 outflows have mechanical \nluminosities and momentum flow rates up to a factor of 10 lower \\citep{1996A&A...311..858B,2006A&A...449.1077C}. \nWith the same mass outflow efficiency, $\\zeta = 0$, this would then require that the {\\em jet speed} is higher in Class 0 \noutflows. This is, however, contrary to the observations which associate lower velocities ($\\sim 100$\\,km\\,s$^{-1}$) to \nthe Class 0 outflows. Here, we shall assume a constant outflow mass fraction, taking the case $\\epsilon = 0.3$ throughout this work.\n\nIn general, the mass left over which accretes on to the core to form the star is\n\\begin{equation}\nM_*(t) = \\int_0^t (1\\,-\\,\\epsilon(t))\\,\\dot M_{acc}(t) dt.\n\\end{equation}\n\n\n\n\n\\subsection{ Model construction: accretion rates}\n\nThe mass accretion rate $\\dot M_{acc}(t)$ is the main prescribed parameter. We choose the four forms as shown in Fig.~\\ref{rateversustime} \nand set out below. Note that the power-law and exponential models both include a significant phase of accelerated accretion prior to \nthe prolonged decline.\n\n\\begin{figure\n\\includegraphics[width=8.7cm]{d-6-40-c.eps}\n\\caption{Accretion rates as a function of time (log-log plot). The rate of loss of mass from the envelope is displayed for the five models discussed in the text. In each case a star of mass 100~M$_\\odot$ is formed on assuming 30\\% of the envelope mass is ejected in jets. The models are (i) constant fast accretion for 10$^5$~yr\n(solid line), (ii) constant slow accretion for $\\times$~10$^6$~yr (long-dashed), (iii) power law with maximum $\\dot M_{acc}$ = 4.01 ~$\\times$~10$^{-3}$~M$_\\odot$~yr$^{-1}$ (dashed), (iv) exponential with maximum $\\dot M_{acc}$ = 2.77 ~$\\times$~10$^{-3}$~M$_\\odot$~yr$^{-1}$ (dot-dashed), and (v) accelerated accretion for 10$^5$~yr\n(dotted line). \\label{rateversustime}}\n\\end{figure}\n\n\\subsubsection{Constant rate models}\n\nConstant accretion models were favoured following the work of \\citet{1977ApJ...214..488S} on singular isothermal spheres. It is clear,\nhowever, that the rate must eventually fall (by the pre-main sequence stage for low-mass stars) as the reservoir becomes\nexhausted. Here we assume a constant rate until a cut-off time, $t_o$, at which the accretion is abruptly halted.\n\nMore generally, the collapse of isothermal isotropic cores supported by thermal pressure yields infall rates of \nthe form $\\dot M_{acc} = f(t)\\left[c_s^3\/G\\right]$ where $c_s$ is the sound speed \\citep{1993ApJ...416..303F}. \nThe function $f(t)$ can be anything from gradually decreasing over a time $\\sim\\,200\\tau$ \nto a sharply peaked function at time $\\sim\\,0.1\\tau$ where the time and mass flow scales are\n\\begin{equation}\n \\tau = 1\/{\\surd}(4{\\pi}G\\rho_c) = 1.3\\,10^5 \n \\left[\\frac{7.6\\,10^{-20}\\, {\\rm g\\,cm}^{-3}}\n {\\rho_c}\\right]^{1\/2} {\\rm yr}\n\\end{equation}\nand\n\\begin{equation}\n \\frac{c_s^3}{G} = 1.6\\,10^{-6} \\left[\\frac{c_s}\n {0.19\\,{\\rm km\\,s}^{-1}}\\right]^3 {\\rm M}_{\\odot}\\,{\\rm yr}^{-1},\n\\end{equation}\nwhere $\\rho_c$ is the initial central density. Inclusion of envelope spin, turbulence, magnetic field and fragmentation \nwould introduce further initial parameters (see review by \\citet{2003RPPh...66.1651L}).\n\n\\subsubsection{Exponential models}\n\n One type of model assumes the envelope loses mass in proportional to its mass,\n$\\dot M_{acc}\\,=\\,-\\dot M_{env}\\,\\propto\\,M_{env}$ \\citep{1996A&A...311..858B,1998ApJ...492..703M}. This\nleads to an exponentially decreasing envelope mass \n\\begin{equation}\n M_{env}(t) = M_o\\,e^{-t\/\\tau_f},\n\\end{equation}\nand an accretion rate\n\\begin{equation} \n \\dot M_{acc}(t) = \\frac{M_o}{t_f}\\,e^{-t\/\\tau_f},\n\\end{equation}\nwhich thus begins with an established high accretion rate.\nA modified exponential model allows for a rapid rise and fall by modelling\n\\begin{equation}\n \\dot M_{acc}(t) = \n \\dot M_o\\,e^{2\\left(\\tau_r\/\\tau_f\\right)^{1\/2}}\\,\n e^{-\\tau_r\/t}\\,e^{-t\/\\tau_f},\n\\label{eqnexp}\n\\end{equation}\nand we can put $\\tau_r = 0$ if desired. The maximum accretion rate $\\dot M_o$ \noccurs at time $t_m = {\\surd}(\\tau_r\\tau_f)$.\n\n\\subsubsection{Power law models}\n\nThe favoured model for low-mass evolution involves a sharp exponential rise followed by a prolonged\n power law decrease in time \\citep{1999ASPC..188..117S,2000IrAJ...27...25S}. The power-law has substantial observational\nsupport \\citep{2000prpl.conf..377C}. The early peak may reach \n $\\dot M_{acc} = 10^{-4}M_{\\odot}\\,{\\rm yr}^{-1}$ for 10$^4$ years, and eventually fall to \n $\\dot M_{acc} = 10^{-7}M_{\\odot}\\,{\\rm yr}^{-1}$ for 10$^6$\\,years, corresponding to Class 0 and Class 2 or \n Classical T~Tauri stars, respectively. \n\nWe choose the accretion rate to take the form\n\\begin{equation}\n \\dot M_{acc}(t) = \n \\dot M_o \\left(\\frac{e}{\\alpha}\\right)^{\\alpha} \\left(\\frac{t}{t_o}\\right)^{-\\alpha} e^{-t_o\/t}.\n\\label{eqnpower}\n\\end{equation}\nNote that we can choose $\\alpha$ to simulate various models: an asymptotic constant-accretion model corresponds to $\\alpha \\sim 0$ and $t_o$ small,\ngradual-accretion corresponds to $\\alpha \\sim 0.5$ and abrupt accretion to $\\alpha \\sim 2-3$. In this work, we take $\\alpha = 1.75$ as a default value \nThe envelope evolution can be written analytically in terms of an incomplete\nGamma function on integrating Eqn.\\,\\ref{eqnpower}:\n\\begin{equation}\n M_{env}(t) = \n \\dot M_ot_o (e\/{\\alpha})^{\\alpha}\\left[1\\,-\\,{\\Gamma}\n (\\alpha-1,t_o\/t)\\right].\n\\end{equation}\n\n\\subsubsection{ Accelerated accretion}\n\nA power-law form for the accretion rate, in which the star's growth accelerates, takes a simple form: \n\\begin{equation}\n \\dot M_{acc}(t) = \\dot M_o \\left(\\frac{t}{t_o}\\right)^{n},\n\\label{eqnaccel}\n\\end{equation}\nfor $t < t_o$ with the final, maximum rate $\\dot M_o$. The accretion in this case is assumed to be\n $\\dot M_{acc} = 0$ for $t > t_o$. This yields a final mass for the star of\n \\begin{equation}\n M_f = \\frac{\\dot M_o t_o (1-\\epsilon)}{1+n},\n\\label{eqnmasspower}\n\\end{equation} \nassuming $\\epsilon$ is constant.\n\nThe sudden drop in the accretion rate at $t = t_o$ appears as \na sharp spike in $L-M$ plots since the contribution to the bolometric luminosity from the accretion luminosity\ndisappears. This can be seen in the tracks calculated by \\citet{2008A&A...481..345M}. Here,\nwe include a brief period of linear decline down to a minimum accretion rate of $10^{-4}$~$\\dot M_o$: \n\\begin{equation}\n \\dot M_{acc}(t) = 20 \\times \\dot M_o ( 1.05 - t\/t_o) \\label{eqnmassadjust}\n\\end{equation} \nfor $t_o < t < 1.05t_o$. \n\nThe turbulent core model is reproduced with $n = 1$ which leads to\n$\\dot M_{*} \\propto M_{*}^{1\/2}$. This model has been shown to lead to some predictions which are \nconsistent with different sets of data \\citep{2008A&A...481..345M,2011MNRAS.416..972D}.\n\nNote that observed correlations between accretion rates and mass such as\n$\\dot M_{acc} \\propto M_{*}^{1.8 \\pm 0.2}$ \\citep{2006A&A...452..245N} apply to a\nrelatively mature stage of young stars. In contrast, $\\dot M_{acc} \\propto M_{*}^{1}$ was\n uncovered \\citep{2011MNRAS.415..103B,2012MNRAS.421...78S}. However, it is no surprise that\nthe results differ according to the precise sample selection criteria \n\n\\subsection{ Growth of the star}\n\nThe structure of a protostar while it accretes at a constant rate has been calculated by \\citet{2009ApJ...691..823H} in the\n`Hot Accretion' scenario corresponding to spherical free-fall. If, instead of free-fall on to the surface, the gas settles via an \naccretion disk, the `Cold Accretion' structure is appropriate \\citep{2010ApJ...721..478H}.\nIn the Hot Accretion case, the stellar radius swells up to over 100~$R_\\odot$ for $\\dot M_{*} > 10^{-3}$~M$_\\odot$~yr$^{-1}$.\nThe accretion may continue until after the arrival on the Main Sequence, arriving at higher masses for higher accretion rates.\n\nFor this work, we have fitted analytical functions to the template figures provided in the above two studies for the radius, $R_*$, and luminosity, $L_*$.\nThe four functions correspond to the four main stages with \nsmooth interpolation between these stages. The stages are (1) the adiabatic accretion, (2) the swelling (or bloating), (3) the Kelvin-Helmholtz contraction and, finally, (4) main-sequence accretion. The resulting functions are shown in Figs. \\ref{radius} and \\ref{luminosity}. Figure \\ref{luminosity} displays both the accretion and stellar luminosities, and demonstrates that the accretion luminosity dominates until the radial swelling stage which is apparent as a dip in the bolometric luminosity.\n\nThe stellar structure depends on the stellar mass, the initial interior state and the accretion rate history. For the case of\n a constant accretion rate, the above published works provide accurate templates for fiducial cases. However, to employ these figures for time-varying accretion,\n \\citet{2011MNRAS.416..972D} took the current accretion rate to look up the radius and luminosity from \\citet{2009ApJ...691..823H}. This method\nmay be a reasonable approximation when the accretion rate continues to increase such that most of the mass has been accumulated \nwithin a factor of two of the present accretion rate. \nMore accurately for the adiabatic phases, we here calculate how the star has accumulated the mass and entropy over its entire evolution. Thus a mass-averaged accretion rate, proportional to the accumulated entropy, is employed. \n However, this method would still lead to very large errors when the accretion rate varies by large amounts, decreases considerably or varies rapidly in the post-adiabatic stages.\n \n\nWhen the above method yields a Kelvin-Helmholtz time that is comparable or shorter than the accretion time scale, the adiabatic approximation is invalid. \nA compromise solution would employ both the current and the mass-avaeraged accretion rates so as to deliver the correct stellar parameters for the two limiting cases of adiabatic accretion and rapid loss of entropy. This is parameterised with\nthe use of the factor $f = t_{acc}\/t_{KH}$ where, as usually defined, $t_{acc} = M\/\\dot M$ and $t_{KH} = G M^2\/(R L_{int})$, as determined by the mass-averaged accretion rate. We then take this factor to determine the stellar radius and luminosity relative to the mass averaged values, $R_o$, $L_o$, by the additive factors\n $(1 - exp(-f)) \\times (R(t) - R_o)$ where $R(t)$ is the stellar radius calculated from the current rate. While this compromise is more accurate, this would still lead to spurious\n properties if accretion variations are very rapid.\n \nIn this paper, therefore, we will use a second method, where the structure at any time is determined by the mass-averaged accretion rate over the history of the star limited to the past Kelvin-Helmholtz time. We implement an iterative process to determine the radius, luminosity and mass-average accretion rate since these are themselves functions of $t_{KH}$. Upon testing, we find that the maximum difference between the two methods is typically a few per cent and occurs in the Kelvin-Helmholtz phase. In comparison, simply assuming the current accretion rate and reading the stellar structure from the constant-rate simulations, generated errors of order of 10\\% in radius throughout the first three evolutionary stages for the slowly varying rates displayed in Fig.~\\ref{rateversustime}. Although not ideal, it could still be implemented for most present purposes.\n\n\n\\begin{figure\n\\includegraphics[width=8.7cm]{d-0-13-h.eps}\n\\includegraphics[width=8.7cm]{d-0-13-c.eps}\n\\caption{Stellar radius evolution for Hot Accretion (upper panel) and Cold Accretion (lower panel).. The mass is accreted for four constant accretion rates, $\\dot M_*$, as indicated. These were calculated from analytical approximations to the data presented by \\citet{2009ApJ...691..823H} for the case of hot accretion and\n\\citet{2010ApJ...721..478H} for the case of cold accretion. \\label{radius}}\n\\end{figure}\n\\begin{figure\n\\includegraphics[width=8.7cm]{d-0-14-h.eps}\n\\includegraphics[width=8.7cm]{d-0-14-c.eps}\n\\caption{Luminosity evolution for Hot Accretion (upper panel) and Cold Accretion (lower panel). The total (solid), accretion (dotted) and stellar (dashed) luminosities are displayed for four constant accretion rates, $\\dot M_*$, creating stars of final mass $M_f$ with $\\dot M_* = 10^{-3} $M$_\\odot $yr$^{-1}$ and $M_f = 100 $M$_\\odot $ (black lines), $\\dot M_* = 10^{-4} $M$_\\odot $yr$^{-1}$ and $M_f = 50 $M$_\\odot $ (red lines), $\\dot M_* = 10^{-5} $M$_\\odot $yr$^{-1}$ and $M_f = 30 $M$_\\odot $ (green lines),\nand $\\dot M_* = 10^{-6} $M$_\\odot $yr$^{-1}$ and $M_f = 17 $M$_\\odot $ (blue lines),\ncalculated from analytical approximations to the data presented by \\citet{2009ApJ...691..823H} (hot accretion)\nand \\citet{2010ApJ...721..478H} (cold accretion). \\label{luminosity}}\n\\end{figure}\n\n\\subsection{Radiation feedback modelling}\n\\label{radiation}\n\nThe bolometric luminosity, $L_{bol}$, and the jets' mechanical power, $P_{jets}$, are equated to the \ngravitational energy released through accretion and the stellar luminosity:\n\\begin{equation}\n L_{bol} + P_{jets} = L_{acc} + L_* + P_{jets} = \\frac{M \\dot M_{acc}}{R_*} + L_* .\n\\end{equation} \n\nThe jets' power is related to the jet mass outflow and accretion rates by \n\\begin{equation}\n P_{jets} = \\frac{1}{2}\\dot M_{jets} v_{jet}^2 = \\frac{1}{2}{\\epsilon}\\dot M_{acc} v_{jet}^2.\n \\end{equation} \n We take\n \\begin{equation} \n v_{jet}^2 = \\chi^2 \\frac{G M_*}{R_*},\n \\end{equation} \n with $\\chi = 1$ as the default constant value, corresponding to a jet speed proportional to the escape speed from the stellar surface.\n\nThe accretion luminosity is thus\n\\begin{equation}\n L_{acc} = (1- \\frac{1}{2}\\chi^2{\\epsilon}) \\frac{M \\dot M_{acc}}{R_*} .\n\\end{equation} \nIt should be noted that this remains an approximation since various factors, such as stellar rotation,\nare not accounted for.\n\nThe radiation feedback into the environment consists of an ionizing effect and a heating effect on the surrounding envelope.\n The ultraviolet Lyman flux from the protostar and the accreting material ionizes the environment, generating an H\\,{\\small II} region. \n This region can be observed in the radio continuum through free-free emission. \n \n The Lyman flux was tabulated in the work of \\citet{2011MNRAS.416..972D} for hot main sequence stars. This provides an excellent up-to-date blueprint\n for stars evolving on to the Main Sequence although expected uncertainties remain very high. \n Since they considered that significant Lyman flux in their sample would only occur when the star was as good as on the Main Sequence, they were able \n to directly use their tabulated values given the stellar mass. In the present study, we have calculated approximations to the Lyman flux based on the work of\n \\citet{1973AJ.....78..929P} and \\citet{2011MNRAS.416..972D} to relate surface temperature to the Lyman flux, and then employed the stellar temperature and radius to determine the flux of Lyman photons. We find that the two works are in close agreement and power-law approximations appropriately\nrepresent them In detail, the fit used is $L_{Ly} = 4 \\pi R_*^2 F_{Ly}$ with\n \\begin{equation}\n log( F_{Ly}) = 23.14 + 15.8(log(T_*) - 4.5 ) ~~~~~~log(T_*) < 4.55\n \\end{equation} \n \\begin{equation}\n log( F_{Ly}) = 24.7 + 5.1(log(T_*) - 4.7 ), ~~~~~~log(T_*) > 4.55. \n \\end{equation} \n \nTo convert a predicted ultraviolet Lyman flux into a high-frequency radio flux density is straightforward. Provided the observed region is \nionisation bounded and the frequency is sufficiently high to ensure that self-absorption is negligible, then the two are approximately proportional\n \\citep{1968ApJ...154..391R,2013A&A...550A..21S}:\n \\begin{equation}\n \\frac{N_{Ly}}{ counts~~s^{-1}} = 43.6 \\left(\\frac{S_{5GHz}}{mJy}\\right) \\left( \\frac{D}{ kpc }\\right)^2 ,\n \\end{equation} \nwhich yields\n \\begin{equation}\n \\frac{L_{Ly}}{ L_\\odot} = -0.63 \\left(\\frac{S_{5GHz}}{mJy}\\right) \\left( \\frac{D}{ kpc }\\right)^2 .\n \\end{equation} \n \n\\subsection{Envelope \\& bolometric temperature}\n\\label{envelope}\n\nThe observed spectral energy distributions of protostars are often complex with multiple peaks. In this work, we restrict the analysis to \npredicting the bolometric temperature of the optically-thick core of the clump given a spherically-symmetric model. \nThis provides an indication of how the peak wavelength will change with age. However, this should be only considered indicative since, even in this \nhomogeneous spherically symmetric approximation, we require several other major assertions concerning the gas and dust distribution.\n \n We consider two methods corresponding to different clump detection and measurement scenarios. In Method 1, we first set up a large clump of mass \n$M_{clump}$ with a mass related to the most massive star (see Equation~\\ref{clumpmass} below) with a radial power-law density distribution.\nThis clump model was explored by \\citet{2006MNRAS.368..435F}, and the sensitivity to parameter ranges was tested. Following those results, we take \na small fixed inner envelope radius, $R_{in}$, of 30\\,AU, in order to sustain a high accretion rate. We calculate an optical depth of the clump $\\tau_c$ \nassuming a spherical density structure with $\\rho \\propto R^{-\\beta}$. The outer clump radius, $R_{out}$, is taken to be located where the clump merges into\nthe ambient cloud, i.e. where the temperature has fallen to that of the ambient molecular cloud, taken here as 12\\,K. The inner clump density is then \n\\begin{equation}\n \\rho_{in} = \\frac{ (3 - \\beta )M_{clump}}{4{\\pi}R_{in}^3\\zeta }\n\\label{eqnrhoe}\n\\end{equation}\nwhere $\\zeta = (R_{out}\/R_{in})^{(3-\\beta)} - 1$, and the clump optical depth is\n\\begin{equation}\n \\tau_c = \\kappa\\,\\rho_{in}R_{in} \\frac{ 1-\\zeta^{1-\\beta} } {\\beta - 1}.\n\\label{eqntaue}\n\\end{equation}\nWe follow Myers et al (1998) and take the emissivity at 12$\\mu$m as $\\kappa = 4\\,{\\rm cm}^2\\,{\\rm g}^{-1}$. \n\nThe inner temperature of the clump is\n\\begin{equation}\n T_{\\rm in}(t) = \\left[\\frac{L_{bol}(t)}{4 \\pi \\cdot \\sigma \\cdot R_{in}^2}\\right]^{1\/4}. \n\\end{equation}\nThis yields an optical depth through the envelope proportional to the emissivity: \n\\begin{equation}\n \\tau_{c}(t) = \\kappa \\cdot \\rho_{in}(t) \\cdot R_{in} \\cdot \\frac{1 - \\zeta(t)^{1 - \\beta}}{\\beta - 1}. \n\\end{equation} \nFollowing \\citet{1998ApJ...492..703M}, we take$A = 1.59~10^{-13}$~cm$^2$\\,g$^{-1}$\\,Hz$^{-1}$, $h$\nas the Planck constant, $k$ the Boltzmann constant, and calculate the bolometric temperature from\n\\begin{equation}\n T_{bol}(t) = \\frac{\\Gamma(9\/2)\\cdot \\zeta(9\/2)}{\\Gamma(5)\\cdot \\zeta(5)}\n \\cdot \\left[\\frac{h \\cdot \\kappa \\cdot T_{in}(t)}{k \\cdot A \\cdot\n \\tau_{e}(t)}\\right]^{1\/2}. \n\\label{tbolometric}\n\\end{equation}\n\nThe above Method 1 was devised for the low-mass protostellar case in which the clump is described as a core, and its mass may correspond to that of \nthe enhanced density and temperature which distinguishes it from the embedding molecular cloud. This case leads to linear isotherms on the diagnostic \nL$_{bol}$--M$_{clump}$ logaritmic diagrams with the bolometric temperature simply proportional to M$_{clump}$ \/ L$_{bol}$ for the case $\\beta = 1.5$. \nAs shown in the top panel of Fig.~\\ref{isotherms}, the predicted isotherms are quite close together thus requiring a wide range of bolometric temperatures to cover the data \nfor the displayed Herschel clumps.\n\n\\begin{figure\n\\includegraphics[width=8.7cm]{d-6-32-c-m5a.eps}\n\\includegraphics[width=8.7cm]{d-6-32-c-m7a.eps}\n\\caption{Isotherms for the bolometric temperature of clumps plotted on the L$_{bol}$--M$_{clump}$ diagram. The upper panel employs Method 1 with a fixed inner radius of 30\\,AU\nand an outer radius of temperature 12\\,K while Method 2 is displayed in the lower panel where the inner radius is determined by dust sublimation and the outer radius is a constant of \n50,000\\,AU (i.e. full size of 0.5\\.pc). The isotherms are for temperatures of 15\\,K (full red), 30\\,K (dashed blue) and 45\\,K (dot-dash black) with $\\beta$ = 1.2\n(lower thick), 1.5 and 1.8 (upper thin). \nThe data are from the Herschel $l = 30^\\circ$ field as analysed by \\citet{2013A&A...549A.130V} with protostellar clumps (diamonds) and pre-stellar clumps (crosses,\ntaken as clumps with no detected stars but which are gravitationally bound if the temperature is used to derive the relevant internal measure of pressure).\\label{isotherms} }\n\\end{figure}\n\nIn the high-mass case, we associate the clump with a size that is only mildly dependent on the clump mass. The observed median half-size is $ 5 \\times 10^4$~AU \n\\citep{2008A&A...481..345M,2013A&A...549A.130V}. Here, we will assume this value as a constant outer radius with larger clumps having the extra mass `squeezed in'. as concluded by \n\\citet{2013A&A...553A.115B}. To complete the model for Method 2, the inner radius, R$_{in}$, of the clump is taken to be the sublimation radius i.e. T$_{in} = 1,400$\\,K. \nWith these two boundary conditions replacing those of Method 1 (the distribution otherwise the same as described above), this Method 2 yields\nwell-separated isotherms as shown in the lower panel of Fig.~\\ref{isotherms}. We will display isotherms derived from Method 2 in the following.\n\n\\subsection{Disk accretion}\n\nAt the inner radius, the envelope feeds a circumstellar disk. We assume here, and will test in a following study, the working assumption that the disk is `viscous'\nand steady, and that the gas spends relatively little time in the disk and so reaches an inner accretion disk at the same rate as with which it is supplied by the envelope. \nThe inner accretion may be non-steady, the gas either being expelled in the jets or accumulated onto the surface of the protostar, later to become the star itself. The disk mass is thus proportional to the accretion rate and the accretion time scale.\n\nStandard turbulent viscosity is efficient at separating the flux of angular momentum from the mass out to radii of about 100\\,AU for the initial rapid \naccretion rates from the envelope. Hence massive outer disks could build up until the viscous mechanisms associated with \nself gravity are effective. This could lead to the formation of secondary objects (stars, brown dwarfs), and so cut off both the star and jet supply\nline. Perhaps more likely is that high accretion rates lead to simultaneous binary formation and powerful molecular jets.\n\nWe assume here that the inner disk processes the material fast and so remains steady. The outer part of the disk will lag behind. The outer radius of the\nsteady state disk can be found by requiring the disk accretion time scale $t_{\\nu}(R) = R^2\/\\nu$ to be less than the time scale for changes in the \naccretion rate $\\dot M_{acc}\/\\ddot M_{acc}$. This yields a steady disk extent $R_s$. \n\nThe disk temperature $T_d$ and sound speed $c_d$ are given by standard expressions for an optically thick and isothermal structure \\citep{1981ARA&A..19..137P}.\nThe accretion energy is radiated locally:\n\\begin{equation}\n T_d^4(R,t) = \\frac{3GM(R,t)\\dot M_a(t)}{8{\\pi}{\\sigma}R^3} \\left[1\\,-\\,\\left(\\frac{R_*}{R}\\right)^\\frac{1}{2}\\right],\n\\end{equation} \nwhere $M(R,t) = M_*(t)\\,+\\,M_d(R,t)$ is the sum of the protostellar and disk mass internal to $R$.\nThe sound speed for a molecular gas disk with a mean molecular weight of 2.3 is $c_d = 6.01\\,10^3\\,T_d^{1\/2}$\\,cm\\,s$^{-1}$.\n\nThe disk mass $M_d$ is not expressible analytically since the accretion is driven by two viscosity components with differing functional forms. \nFirst, we take the usual turbulent viscosity $\\nu_t = {\\alpha_d}c_dH$ where H is the disk thickness and $\\alpha_d$ is a dimensionless parameter which \nwe set to 0.1 unless otherwise stated \\citep{1973A&A....24..337S}. This yields\n\\begin{equation}\n \\nu_t = \\frac{2\\,\\alpha_d\\,c_d^2}{3\\,\\Omega},\n\\end{equation}\nwhere the angular rotation speed is $\\Omega\\,=\\,{\\surd}(GM\/R^3)$. The component of viscosity related to self-gravitational forces is \nparameterised, as suggested by \\citet{1987MNRAS.225..607L}:\n\\begin{equation}\n \\nu_g = \\frac{2{\\mu_d}c_d^2}{3\\Omega}\\left(\\frac{Q_c^2}{Q_t^2} - 1\\right)\n\\end{equation}\nfor $Q_t < Q_c$ and $\\nu_g = 0$ otherwise. Here, the efficiency parameter $\\mu_d$ and the instability parameter $Q_c$ will also be set to unity\n(see \\citet{1990ApJ...358..515L}). Hence, the parameter\n\\begin{equation}\n Q_t = \\frac{c_d\\,\\Omega}{\\pi\\,G\\,\\Sigma}\n\\end{equation} \ndetermines the importance of viscosity through the disk's own gravity. Note that the viscosity can be determined by self-gravity even when the \nprotostellar mass is large when the disk column density is large. This may well arise where the turbulent viscosity is inefficient, in the outer \ndisk regions, leading to a build up of mass until self-gravity takes effect. \n\nThe disk column density is given by \n\\begin{equation}\n \\Sigma = \\frac{1}{2\\pi\\,R}\\frac{dM}{dR} \n\\end{equation}\nand the cumulative mass distribution\n\\begin{equation}\n \\frac{dM}{dR} = \\frac{R\\,\\dot M_{acc}}{\\nu_t\\,+\\,\\nu_g}\n\\end{equation}\nis given by the viscosity. From the disk column it is straightforward to calculate the disk radius at which the optical depth is unity, $R_{{\\tau}=1}$. \n This completes the set of equations which is integrated from an inner radius to yield the full disk structure. The results for disks associated \nlow-mass protostars were presented by \\citet{2000IrAJ...27...25S} and will be extended in a following work. \n\n\n\n\\section{Results} \n\\label{results}\n\n\\subsection{Clump Mass v. Bolometric Luminosity}\n\\label{clumps}\n\n\\begin{figure\n\\includegraphics[width=8.7cm]{d-0-12-h.eps}\n\\includegraphics[width=8.7cm]{d-6-12-h.eps}\n\\caption{ L$_{bol}$--M$_{clump}$ diagrams for Constant Slow Accretion (simultaneous star and clump evolution, upper panel) and \nConstant Fast Accretion (early formation of the massive star, lower panel), both forming via hot accretion.. \nThe five solid lines correspond to systematically increasing final stellar masses of 10, 15, 30, 50 and 100\\,M$_\\odot$. \nInitial clump masses, final stellar mass, accretion timescale and accretion rates are provided in Table~\\ref{maintable}.\nThe contributing luminosity components derived from accretion (dashed lines) and interior (dotted lines, where visible) are indicated. The filled arrowheads are laid at equal time intervals of one tenth the total time, and the ten unfilled arrowheads are similarly placed in the first of these intervals.\nFor comparison, we overlay (1) the data set presented by \\citet{2008A&A...481..345M} where the triangles denote the infrared sample while the squares denote the millimetre sample. \nThe three straight dot-dashed lines correspond to bolometric temperatures of 40\\,K (blue), 60\\,K (green) and 80\\,K (red) as calculated by Method~2 with an envelope density radial power-law index of 1.5. \n \\label{lbol-menv0}}\n\\end{figure}\n \n\\begin{table*}\n \\caption[table1]\n {Model data employed in Figs.~\\ref{lbol-menv0} and \\ref{lbol-menv1} in forming a 100~M$_\\odot$ star from an initial clump of mass 6,561~M$_\\odot$ (left) and a 10~M$_\\odot$ star from an initial clump of mass 247~M$_\\odot$ (right), all with 30\\% of the mass ejected through jets or outflow.}\n \\label{maintable}\n \\begin{tabular}{l|lrr|lrr}\n \\hline \\noalign{\\smallskip}\n \\sf{Accretion Model} & \\sf{Max. accretion} & \\sf{Accretion} & \\sf{Clump}\n & \\sf{Max. accretion } & \\sf{Accretion} & \\sf{Clump } \\\\\n & \\sf{rate} & \\sf{timescale} & \\sf{timescale}\n & \\sf{rate} & \\sf{timescale} & \\sf{timescale} \\\\\n \n & \\sf{10$^{-3}$~M$_\\odot$~yr$^{-1}$ } & \\sf{10$^{5}$~yr} & \\sf{10$^{5}$~yr}\n & \\sf{10$^{-3}$~M$_\\odot$~yr$^{-1}$ } & \\sf{10$^{5}$~yr} & \\sf{10$^{5}$~yr} \\\\\n \\noalign{\\smallskip} \\hline\n \\noalign{\\smallskip}\n {Constant - Slow } & 0.141 & 10.0 & 10 & 0.0014 & 100 & 100\\\\\n {Constant - Fast } & 1.413 & 1.0 & 10 & 0.0141 & 10 & 100 \\\\\n {Accelerated} & 2.717 & 1.0 & 10 & 0.0907 & 3 & 10 \\\\\n {Power Law} & 2.861 & 0.2 & 10 & 0.2861 & 0.2 & 10 \\\\\n {Exponential} & 1.980 & 1.0 & 10 & 0.1980 & 1.0 & 10\\\\\n & & \\\\\n \\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n \\end{tabular}\n\\end{table*}\n\n\\begin{figure\n\\includegraphics[width=8.7cm]{d-5-12-h.eps}\n\\includegraphics[width=8.7cm]{d-1-12-h.eps}\n\\includegraphics[width=8.7cm]{d-2-12-h.eps}\n\\caption{ L$_{bol}$--M$_{clump}$ diagrams for Accelerated (upper), Power Law (middle) and Exponential (lower panel) accretion models assuming hot accretion.\nThe model parameters are provided in Table~\\ref{maintable}. The data are the set of Herschel objects from the $l = 30^\\circ$ field from the Herschel Infrared GALactic plane survey (Hi-Gal) \\citep{2010A&A...518L..97E} as revised by \\citet{2013A&A...549A.130V}. The three straight lines correspond to bolometric temperatures of 10\\,K (blue), 20\\,K (green) and 40\\,K (red) as calculated by Method~2 with an envelope density radial power-law index of 1.5. See the caption of Fig.~\\ref{lbol-menv0} for other details. \\label{lbol-menv1}}\n\\end{figure}\n \n Cloud mass, bolometric luminosity and bolometric temperature are quantities derived from observations. The bolometric temperature is sensitive to the geometry, orientation \n and uniformity of the clump. Therefore, the relationship between cloud mass and bolometric luminosity is most often employed as a diagnostic tool\n for the formation of high-mass stars \\citep{2008A&A...481..345M,2013A&A...552A.123B} as well as lower-mass stars \\citep{1993A&A...273..221R,1996A&A...309..827S,2000IrAJ...27...25S}.\n More recently, L$_{bol}$--M$_{clump}$ or L$_{bol}$\/M$_{clump}$--M$_{clump}$ \n diagrams have been utilised to analyse Herschel Space Telescope data \\citep{2010A&A...518L..97E,2012A&A...547A..49R}.\n \n To interpret these data points as a time sequence, it is assumed that the clump mass decreases in time as the luminosity increases \\citep{1993ApJ...406..122A}. \nModel evolutionary tracks are then easily calculated based on the principles discussed in Section~\\ref{method} upon choosing an accretion type and rate.\n\nThe initial clump mass is assumed to be directly related to the final mass of the most massive star, $M_{*f}$, by the relation\n\\begin{equation}\n \\log M_{clump}(0) = A + 1.41 \\log M_{*f}.\n \\label{clumpmass}\n\\end{equation}\nHere, $A = 0.55$ could be taken \\citep{2008A&A...481..345M} on the assumption that the clump generates the bound cluster \\citep{2009A&A...503..801F}. However, accounting for the global gas escape as well as that from protostellar outflows, we shall take $A = 0.85$ corresponding to a star formation efficiency of 50\\%. As will be seen below,\nsuch high efficiencies are inconsistent and much larger values of $A$ need to be considered. \n\n The clump gas is taken to be reduced at a constant rate (excluding the inner envelope which supplies the massive star). However, other relationships between the stellar cluster \nand most-massive star have been considered and, above all, there is a wide spread in the measured values of $A$ \\citep{2010MNRAS.401..275W} corresponding to an order \nof magnitude in mass. \n\nDiagrams for models with constant hot accretion are shown in Fig.~\\ref{lbol-menv0} and for variable rates in Fig.~\\ref{lbol-menv1}. Also shown are observational data \nfor far-infrared and infrared sources and the theoretical bolometric temperature isotherms. Note that these isotherms as calculated from Equation~\\ref{tbolometric} are in agreement with the bolometric temperatures directly derived in the literature.\n\nThe top panel of Fig.~\\ref{lbol-menv0} displays the tracks for a slow evolution \\citep[e.g.][]{2001A&A...373..190B} with a simultaneous cloud evolution. This yields a strong dependence between the two parameters with a wide distribution of sources predicted. In recent works, the clump mass is assumed to fall at a slow constant rate with time while the massive star forms abruptly from a compact envelope, which is consistent with both available data and theoretical expectations \\citep{2002Natur.416...59M}. We therefore discount the Slow Accretion scenario as a model for\nclump\/cluster evolution although it is relevant in following the possible evolution of an isolated core especially in the low-mass case.\n\nThe constant Fast Accretion case is shown in the lower panel of of Fig.~\\ref{lbol-menv0}. It is clear that massive infrared protostars would be rarer in this case (as indicated by the arrowheads placed at regular time intervals). In addition, a much narrower range in clump masses for a given luminosity interval is predicted in the latter case.\n\nThe Accelerated Accretion Model generates similar results (top panel of Fig.~\\ref{lbol-menv1}) with two distinct track stages. This was the model explored by \\citet{2008A&A...481..345M} but we note here that both the constant and power law fall-off models produce very similar tracks. To differentiate between models will require a detailed statistical analysis. \nThe Power Law Model generates tracks in which there is an extended transition stage between the accretion and clean-up stages (middle panel of Fig.~\\ref{lbol-menv1}). This transition stage\noccurs at bolometric temperatures between 60\\,K and 80\\,K. \n\nIt should also be remarked that the track direction changes from almost vertically up to down for a short period. This is caused by the swelling of the protostar\nwhich reduces the bolometric luminosity by reducing the accretion luminosity. While hardly visible in Fig.~\\ref{lbol-menv1}, this effect becomes prominent in the cold accretion scenario\nwhere the expansion phase is stronger. This is shown in Fig.~\\ref{lbol-menv2} for the Power Law Model in which it is most obvious. In general, however, we find that Cold Accretion does not significantly alter the tracks on the L$_{bol}$--M$_{clump}$ plots.\n\n\\begin{figure\n\\includegraphics[width=8.7cm]{d-5-12-c.eps}\n\\includegraphics[width=8.7cm]{d-0-12-c.eps}\\caption{ Cold Accretion: L$_{bol}$--M$_{clump}$ diagrams for the Accelerated Accretion Model and the Constant-Slow Accretion Model with Cold Accretion. The model parameters are as on the corresponding Hot Accretion case provided in Table~\\ref{maintable}. See the caption of Fig.~\\ref{lbol-menv1} for further details.\n \\label{lbol-menv2}}\n\\end{figure}\n \n\\subsection{Bolometric Temperature}\n\nThe bolometric isotherms calculated through Method 2 are shown on the L$_{bol}$--M$_{clump}$ plots. The temperatures are broadly consistent with the range of temperatures\nderived from the spectral energy distributions for the Herschel sources. We do not expect more than this given the known sensitivity to\ngeometry and orientation. Indeed, as shown in Fig.~\\ref{isotherms}, there is a strong dependence on the radial density distribution as given by the index $\\beta$.\n\nHowever, we can compare the accretion models statistically to determine if they would yield significantly different statistics in terms of numbers of source in any temperature interval. These numbers are provided in Table~\\ref{tabletime}. As also illustrated in Fig.~\\ref{tbolversustime}, there is a remarkable difference between the Accelerated Accretion model and the alternative\nevolutions. In the Accelerated Accretion Model, significantly more time elapses in the low temperature ($<$ 30\\,K) regime. For the 100\\,M$_\\odot$ case, all the other models investigated (with the exception of Slow Accretion) spend much less time at low temperatures, by a factor of two to three. This result is consistent with expectations: accelerated accretion takes time to get off the ground.\n \n\n\\begin{figure\n\\includegraphics[width=8.7cm]{d-6-42-h-1m-100.eps}\n\\includegraphics[width=8.7cm]{d-6-42-h-1m-30.eps}\n\\caption{The early evolution of the bolometric temperature for Hot Accretion models which yield 100~M$_{\\odot}$ (upper panel) and 30~M$_{\\odot}$ (lower panel) stars, with lines as denoted in Fig.~\\ref{rateversustime} and parameters from Table~\\ref{maintable}. Hot accretion is assumed here. The panels assume that the initial clump mass is twice that necessary to generate the associated cluster ( A = 0.85).\n \\label{tbolversustime}}\n\\end{figure} \n\\begin{figure\n\\includegraphics[width=8.7cm]{d-6-42-c-1m-100.eps}\n\\includegraphics[width=8.7cm]{d-6-42-c-3m-100.eps}\\caption{The early evolution of the bolometric temperature for Cold Accretion models with \n the initial clump mass twice (upper panel) and 9.4 times (lower panel) the fiducial value required to generate the stellar cluster.\neach model displayed forms a 100~M$_{\\odot}$ stars, with lines as denoted in Fig.~\\ref{rateversustime} and parameters from Table~\\ref{maintable}. \nAlthough cold accretion is assumed, hot accretion is found to closely follow this behaviour. \\label{tbolversustime2}}\n\\end{figure} \n\n \n\\begin{table*}\n \\caption[table1]\n {The fraction of the total time required to reach a bolometric temperature of 100\\,K in order to traverse the indicated temperature range for the \n models to form a star of mass 100~M$_\\odot$ through hot (upper section) and cold (lower section) accretion. The clump mass is given by Equation~\\ref{clumpmass}\n and the radius of the clump is 50,000~AU, consistent with the R$_{env}$ values derived by \\citet{2008A&A...481..345M}. Herschel Hi-Gal fractions are\n taken from the two fields analysed by \\citet{2010A&A...518L..97E} and \\citet{2013A&A...549A.130V}. }\n \\label{tabletime}\n \\begin{tabular}{l|rrrrrr}\n \\hline \\noalign{\\smallskip}\n \\sf{Accretion Model} & \\sf{$<20\\,$K} & \\sf{ $<30\\,$K } & \\sf{30\\,K -- 50\\,K}\n & \\sf{50\\,K -- 70\\,K} & ~~~~~\\sf{$< 100$\\,K time ($\\times$ 1000 yrs)} \\\\\n \\noalign{\\smallskip} \\hline\n \\noalign{\\smallskip}\n & \\\\\n HOT ACCRETION: & \\\\\n {Constant - Slow} & 0.250 & 0.442 & 0.279 & 0.158 & 866 \\\\ \n {Constant - Fast} & 0.027 & 0.057 & 0.397 & 0.356 & 845 \\\\\n {Accelerated} & 0.058 & 0.080 & 0.373 & 0.356 & 845\\\\ \n {Power Law} & 0.020 & 0.049 & 0.446 & 0.321 & 847\\\\ \n {Exponential} & 0.025 & 0.051 & 0.402 & 0.356 & 845 \\\\\n & & \\\\\n COLD ACCRETION: \\\\\n {Constant - Slow} & 0.250 & 0.442 & 0.279 & 0.158 & 866 \\\\ \n {Constant - Fast} & 0.022 & 0.057 & 0.397 & 0.356 & 845 \\\\\n {Accelerated} & 0.052 & 0.080 & 0.373 & 0.356 & 845\\\\ \n {Power Law} & 0.017 & 0.049 & 0.446 & 0.321 847\\\\ \n {Exponential} & 0.021 & 0.051 & 0.402 & 0.356 & 8459 \\\\ \n & & \\\\\n Hi-Gal Data: (Elia et al 2010) & & & & & number \\\\\n$l = 30^\\circ$ field & 0.463 & 0.832 & 0.128 & 0.022 & 311 \\\\ \n$l = 59^\\circ$ field & 0.417 & 0.846 & 0.142 & 0.011 & 91 \\\\\nH-Gal YSOs (Veneziani et al 2013) & 0.463 & 0.989 & 0.011 & 0 & 284 \\\\\n \\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n \\end{tabular}\n\\end{table*}\n\n \n\n\\begin{table*}\n \\caption[table2]\n {The fraction of the total time required to reach a bolometric temperature of 100\\,K in order to traverse the indicated temperature range for cold accretion \n with fixed clump mass as indicated but different final stellar masses. The radius of the clump is 30,000~AU \n consistent with the values derived by \\citet{2013A&A...549A.130V}. }\n \\label{tabletime2}\n \\begin{tabular}{l|rrrrrr}\n \\hline \\noalign{\\smallskip}\n \\sf{Final Stellar Mass} & \\sf{$<20\\,$K} & \\sf{ $<30\\,$K } & \\sf{30\\,K -- 50\\,K}\n & \\sf{50\\,K -- 70\\,K} & \\sf{$< 100$\\,K time (yrs)} \\\\\n \\noalign{\\smallskip} \\hline\n \\noalign{\\smallskip}\n & \\\\\n {Constant - Fast} & \\\\\n CLUMP MASS: 6,683~M$_\\odot$ & \\\\\n CLUMP RADIUS: 50,000~AU & \\\\\n 100 & 0.022 & 0.057 & 0.396 & 0.356 & 845\\\\ \n 50 & 0.042 & 0.233 & 0.538 & 0.148 & 929 \\\\\n 30 & 0.077 & 0.581 & 0.294 & 0.081 & 960\\\\\n 15 & 0.550 & 0.810 & 0.133 & 0.036 & 1472 \\\\\n 10 & 0.715 & 0.880 & 0.084 & 0.023 & 988\\\\ \n & & \\\\\n CLUMP MASS: ~5,740~M$_\\odot$ & \\\\\n CLUMP RADIUS: ~30,000~AU & \\\\\n STELLAR MASS:~30~M$_\\odot$ \\\\ \n {Constant - Slow} & 0.672 & 0.826 & 0.114 & 0.037 & 2,936 \\\\ \n {Constant - Fast} & 0.473 & 0.778 & 0.156 & 0.043 & 2,935 \\\\ \n {Accelerated} & 0.473 & 0.777 & 0.156 & 0.043 & 978 \\\\ \n {Power Law} & 0.497 & 0.781 & 0.153& 0.042 & 978\\\\ \n {Exponential} &0.472 & 0.777 & 0.156 & 0.043 & 978 \\\\ \n & \\\\ \n Hi-Gal Data: & & & & & number \\\\\n $l = 30^\\circ$ field & 0.463 & 0.832 & 0.128 & 0.022 & 311 \\\\ \n $l = 59^\\circ$ field & 0.417 & 0.846 & 0.142 & 0.011 & 91 \\\\ \nH-Gal YSOs (Veneziani et al 2013) & 0.463 & 0.989 & 0.011 & 0 & 284 \\\\\n \\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n \\end{tabular}\n\\end{table*}\n\n\n \n\\subsection{Herschel Hi-Gal data}\n \\label{herschel}\n \n Data directly comparable to that predicted in Table \\ref{tabletime} are available through several Herschel programmes. The distribution of bolometric temperatures in Table~\\ref{tabletime} along with the\ndata employed here are inconsistent. Over 80\\% of observed clumps have temperatures below 30\\,K (bottom lines in Table~\\ref{tabletime}) whereas the models predict a much more even number distribution with temperature. It is clear that only a small fraction of the Herschel cores on the very high mass tracks will go on to form such massive stars. \n\n A resolution to this problem is straightforward:\nthe most massive forming star observed in the clumps is a factor of about 3 smaller than that used in the literature to calculate tracks. \nIn Table~\\ref{tabletime2}, we present re-calculated \nnumber distributions on the assumption that the clump masses are indeed large but are actually being heated by protostars which will form much lower mass stars. The new tracks are illustrated in Fig.~\\ref{newtracks}. Extremely good correspondences are apparent. Note that the initial clump masses are now 4.7 times larger, corresponding to stars of three times the mass according to Equation~\\ref{clumpmass}.\n\nThis interpretation of the statistics, in which only 10--15\\% of the initial clump mass ends up in stars, is independent of the accretion model. As shown on the panels of\nFig.~\\ref{tbolversustime2}, it is very difficult to distinguish between the models for the high mass clumps.\n\nThe new interpretation is consistent with the data relating star clusters to the most massive stars as presented by \\citet{2010MNRAS.401..275W}.\nTheir Fig.~3 shows that there is a minimum mass of the most-massive star for a star cluster of a given size which is approximately three times lower than the average stellar mass. \nThis minimum mass would, of course, be the most likely if drawn randomly from a distribution corresponding to the Initial mass Function. \nWe thus recommend that far-infrared data be interpreted by tracks as shown in Fig.~\\ref{newtracks}.\n\n Fig.~\\ref{newtracks} also demonstrates that the evolutionary phase of the observed sample of Herschel protostars is more advanced than previously interpreted\nbecause the final mass of the stars had been overestimated. The assertion here is that there are far fewer embedded protostars with mass exceeding 30~M$_\\odot$. \n\n\n\\begin{figure\n\\includegraphics[width=8.7cm]{d-6-12-c-new.eps}\n\\caption{Revised L$_{bol}$--M$_{clump}$ tracks for the Constant Accretion model with\nfast cold accretion . \nThe six solid lines correspond to systematically increasing final stellar masses of 8, 10, 15, 30, 50 and 100\\,M$_\\odot$. \nThe Constant Accretion Model is taken here with final stellar mass, accretion timescale and accretion rates provided in Table~\\ref{maintable} but the initial clump has\nbeen increased by a factor of 4.7 corresponding to an initial clump mass 9.4 times larger than the final cluster mass.\nThe data are from the Herschel $l = 30^\\circ$ field as analysed by \\citet{2013A&A...549A.130V} with protostellar clumps (triangles) and pre-stellar clumps (crosses; \ntaken as bound starless clumps if the temperature is used to derive the relevant internal measure of pressure).\\label{newtracks} }\n\\end{figure}\n\n\n\n\\subsection{Radiative Feedback: hot accretion}\n\nAs the protostar matures, the rapid increase in luminosity leads to a high surface temperature and a high number of extreme ultraviolet photons, $N_{Ly}$, capable of generating \na surrounding source of free-free radio emission as quantified in Sub-section~\\ref{radiation}. With this interpretation, we can compare two distinct indicators of evolution:\n$M_{clump}\/L_{bol}$ and $N_{Ly}$\/$L_{bol}$. Remarkably, both these quantities are, at least in principle, distance independent. It is apparent from\nFig.~\\ref{lyman5} that there is a significant difference between the two extreme models with the Accelerated Accretion tracks for the most massive stars occupying a wider region whereas the Power Law Model predicts much less variation.\n\nWe can now suppose that the inner accretion is not spherical but streams onto a limited area of the star, forming accretion hot spots on the surface. \nTaking a fraction $f_{acc}$ of the accretion to free-fall on to a fraction $f_{hot}$ of the surface area, yields a hot spot temperature $T_{hot}$ given by\n\\begin{equation}\n T_{hot}^4(R,t) = \\frac{L_{int} + f_{acc} L_{acc}\/f_{hot} } {4{\\pi}{\\sigma}R_*^2}.\n\\end{equation} \nContributions to the Lyman flux from the hot spot and the rest of the surface are then added. As shown in Fig.\\ref{hotspots},\nthe behaviour is different with the the accretion luminosity generating significant early ultraviolet emission. Moreover, the hotspots are very important Lyman emittors for the stars of mass 10 -- 20~M$_\\odot$ and can dominate the radio emission during the early phases of star formation. \n\n\n\\begin{figure\n\\includegraphics[width=8.7cm]{d-5-17-h.eps}\n\\includegraphics[width=8.7cm]{d-1-17-h.eps}\n\\caption{The ratio of Lyman photon flux to bolometric luminosity, $N_{Ly}$\/$L_{bol}$, against the evolutionary measure $M_{clump}\/L_{bol}$. \nIn this diagram, any large distance ambiguity is excluded. The Accelerating Accretion Model (upper panel) and Power Law Model (lower panel) tracks are displayed\nfor the Hot Accretion model. We also now implicitly assume the 4.7 times higher clump mass deduced in Sub-section \\ref{herschel}. Here, and in the following figures, the model tracks correspond to stars of final mass 100 (full) , 50 (dotted), 30 (dashed) , 15 (dot-dashed) and 10 M$_\\odot$ (3-dot-dashed). \\label{lyman5}}\n\\end{figure}\n\\begin{figure\n\\includegraphics[width=8.7cm]{d-5-17-h-hot.eps}\n\\includegraphics[width=8.7cm]{d-1-17-h-hot.eps}\n\\caption{Hot Spot Accretion. The ratio of Lyman photon flux to bolometric luminosity, $N_{Ly}$\/$L_{bol}$, against the evolutionary measure $M_{clump}\/L_{bol}$. \nA hot spot covering 5\\% of the surface upon which 75\\% of the accretion luminosity is emitted is assumed here. \nIn this diagram, any large distance ambiguity is excluded. The Accelerating Accretion Model (upper panel) and Power Law Model (lower panel) tracks are displayed.. \\label{hotspots}}\n\\end{figure}\n\nTo compare to available data, we plot in Fig.~\\ref{lognolyhot} the Lyman photon flux and bolometric luminosity.for a typical hot accretion model, with and without\nhot spots. The observed data are taken from from \\citet{2013A&A...550A..21S} and are clearly at variance with this model: there remains a significant number of data point lying above the tracks. This problem was discussed by \\citet{2013ApJS..208...11L}, \\citet{2013MNRAS.tmp.2012U} and \\citet{2013A&A...550A..21S} on comparing data to the expected Lyman flux from ZAMS stars. \\citet{2013A&A...550A..21S} speculated that one resolution could be if there was an extra component from the accretion. The lower panel, however, demonstrates that this is not sufficient in the case where the star itself has formed through spherical accretion: the bloated protostar is too large to permit a significant release of extreme ultraviolet photons through free-fall on to the surface. This conclusion applies to all accretion models discussed here.\n\n\\begin{figure\n\\includegraphics[width=8.7cm]{d-5-20-h-nohs.eps}\n\\includegraphics[width=8.7cm]{d-5-20-h-hs.eps}\n\\caption{The Lyman photon flux, $N_{Ly}$, against $L_{bol}$ for the Accelerated Accretion Model with hot accretion. The upper panel displays the tracks with\nspherical accretion while the lower panel shows the result of channelled accretion on to hot spots covering 5\\% of the surface upon which 75\\% of the accretion luminosity is emitted. \nThe data points are taken from \\citet{2013A&A...550A..21S}, derived from ATCA 18\\,GHz observations. The thick red line corresponds to the relationship for ZAMS stars, taken from\n\\citep{1973AJ.....78..929P}. The model tracks correspond to stars of final mass \n100 (full) , 50 (dotted), 30 (dashed) , 15 (dot-dashed) and 10 M$_\\odot$ (3-dot-dashed). \\label{lognolyhot}}\n\\end{figure}\n\n \n\\subsection{Radiative Feedback: cold accretion}\n\nCold accretion generates a young star with a considerably smaller radius through the early phases. Hence cold accretion does indeed generate more Lyman photons earlier as shown in the top panel of Fig~\\ref{lyman-cold} although not greatly different from the hot accretion examples (note the different axial scales). \n\n\\begin{figure\n\\includegraphics[width=8.7cm]{d-6-17-c-nohs.eps}\n\\includegraphics[width=8.7cm]{d-6-17-c-hs.eps}\n\\caption{Cold accretion. The ratio of Lyman photon flux to bolometric luminosity, $N_{Ly}$\/$L_{bol}$, against the evolutionary measure $M_{clump}\/L_{bol}$. \nIn this diagram, any large distance ambiguity is excluded. The Constant Accretion Model with accretion settling on to the star's surface (upper panel) and with 75\\%\nbeing funnelled on to 5\\% of the surface at the free-fall speed (lower panel) are displayed. \\label{lyman-cold}}\n\\end{figure}\n\nWe now again suppose that the inner accretion is guided by the magnetic field to form accretion hot spots on the surface. As shown in the lower panel of Fig~\\ref{lyman-cold},\nthe behaviour is very different: the accretion luminosity dominates the early UV emission. Moreover, the hotspots are very important Lyman emittors for the stars of mass \n10 -- 20~M$_\\odot$ and will dominate the radio emission during the early phases of star formation. \n\nFig.~\\ref{lognolycold} compares the Constant Accretion Model to the ATCA data. While cold accretion does not generate sufficient extreme ultraviolet (top panel) if a considerable fraction of the accreting material is funnelled on to accretion hotspots, then the data can be very well interpreted (lower panel). This implies that the star has formed through a disk rather than spherical infall. However, at some stage,\nthe magnetic field becomes sufficiently strong so that material is diverted and funnelled from the inner disk radius to effectively free-fall on to the surface. The star, of course, has\npreviously formed through the cold accretion and so maintains the relatively small radius. The enhanced Lyman flux is a result of the high accretion on to a small growing protostar. The subsequent temporary extreme fall in the Lyman flux occurs as the bolometric luminosity falls and the star expands. \n\nWe find that the conclusion that funnelled accretion onto a compact protostar is occurring is independent of the chosen accretion model, as illustrated in Fig.~\\ref{lognolycold2} for two extreme accretion types. Good fits to the ATCA data require hotspot surface areas of less than 3\\% -- 5\\% for\nmass fractions of 50\\% and 75\\% respectively. These ranges are consistent with the fractions deduced from observations of young stars \\citep{1998ApJ...509..802C}. However,\nit should be noted that while we concentrate on explaining the enigmatic high Lyman flux, most observed sources are either consistent with ZAMS or are underluminous. Some of these sources \nare not consistent with that expected on taking into account the additional low-Lyman flux of the associated stellar cluster \\citep{2013ApJS..208...11L,2013MNRAS.tmp.2012U}.\nIn the present context, these sources can be the result of either (1) distributed accretion over the surface and\/or (2) single stars in the early-bloating or late Kelvin-Helmholtz contraction phases.\n\n The anomolously high Lyman fluxes only require the formation of intermediate mass stars. High accretion rates are required to explain some data points. However, as can be seen from Fig.~\\ref{lognolycold2}, the protostar need only grow up to the beginning of the bloating phase in order to account for the Lyman flux through hot spot accretion. \n \n\\begin{figure\n\\includegraphics[width=8.7cm]{d-6-20-c-nohs.eps}\n\\includegraphics[width=8.7cm]{d-6-20-c-hs.eps}\n\\caption{The Lyman photon flux, $N_{Ly}$, against $L_{bol}$ for the Constant Accretion Model with a compact protostar formed via Cold Accretion. The upper panel displays the tracks with\npure disk accretion while the lower panel shows the result of channelled accretion on to hot spots covering 5\\% of the surface upon which 75\\% of the accretion luminosity is emitted. \nThe data are taken from \\citet{2013A&A...550A..21S}, derived from ATCA 18\\,GHz observations. The thick red line corresponds to the relationship for ZAMS stars, taken from\n\\citet{1973AJ.....78..929P}. The model tracks correspond to stars of final mass \n100 (full) , 50 (dotted), 30 (dashed) , 15 (dot-dashed) and 10 M$_\\odot$ (3-dot-dashed). \\label{lognolycold}}\n\\end{figure}\n\n\\begin{figure\n\\includegraphics[width=8.7cm]{d-5-20-c-hs.eps}\n\\includegraphics[width=8.7cm]{d-1-20-c-hs.eps}\n\\caption{The Lyman photon flux, $N_{Ly}$, against $L_{bol}$ for the hot spot model combined with with the Accelerated Accretion Model (upper panel) \nand the Power Law Accretion Model (lower panel) with a compact protostar formed via Cold Accretion. The results of channelled accretion on to hot spots covering 5\\% of the surface upon which 75\\% of the accretion luminosity is emitted are displayed.\nThe data are taken from \\citet{2013A&A...550A..21S}, derived from ATCA 18\\,GHz observations. The thick red line corresponds to the relationship for ZAMS stars, taken from\n\\citep{1973AJ.....78..929P}. The model tracks correspond to stars of final mass \n100 (full) , 50 (dotted), 30 (dashed) , 15 (dot-dashed) and 10 M$_\\odot$ (3-dot-dashed). \\label{lognolycold2}}\n\\end{figure}\n\n\n\n\n\\section{Conclusions}\n\nA model for massive stars has been constructed by piecing together models for the protostellar structure, the inflow from a large clump and the \nradiation feedback. The framework requires the accretion rate from the clump to be specified. In this first work we consider a specific subset of possible flows. \nWe consider both hot and cold accretion scenarios, identified as the limiting cases for spherical free-fall and disk accretion, respectively. We assume the fiducial cases\npresented by \\citet{2009ApJ...691..823H} in the\n`Hot Accretion' scenario and \\citep{2010ApJ...721..478H} for the `Cold Accretion' structure \n but it should be noted that there is considerable uncertainty, depending on the assumed initial interior state and the physics of the radiation feedback\n(see also \\citet{2013ApJ...772...61K}). \n\nStrongly variable accretion rates have been investigated by \\citet{2012MNRAS.424..457S} as well as \\citet{2013ApJ...772...61K} by utilising\nhydrodynamic simulations. In both these works, the puffy extended nature of the protostars is evident.\nHere we only consider smooth evolutions and\n do not consider accretion outbursts or pulsations, or the jet and outflow properties. We also do not consider binary formation, geometry and inclination effects,\nor the evolution of the size of the H\\,{\\small II} region. Hence, this first work sets up the fundamental algorithms and compares results for two recent diagnostic tools. \n\nModels for the formation of massive stars through accretion can be tested by comparing predictions to a range of observational parameters.\nThese include the bolometric and extreme ultraviolet luminosities, the envelope and disk mass, and the outflow momentum and energy.\nWe here determine possible evolutionary tracks on assuming the variation with time of the accretion rate from a molecular clump on to the star and calculate how the star, envelope and outflow simultaneously evolve. This is achieved by making analytical prescriptions for the components based on current knowledge. We update and extend previous models and confirm previous conclusions that the clump mass must far exceeds the accreted mass, most of it being converted into a surrounding cluster of low-mass objects or dispersed. \n \nWe find that Accelerated Accretion is not favoured on the basis of the L$_{bol}$--M$_{clump}$ diagnostic diagram which does not directly provide a test to differentiate the models.\nOnly a slow accretion model can be distinguished in which the star and clump evolve on the same time scale, which is not pursued since it seems unlikely given\nthe contrasting sound-crossing time scales between cores and clumps. This is mainly because the protostar tends to accrete most of its mass within a short time span in all the other models. \n\nInstead, we show that the time spent within each range of bolometric temperature can be closely related to the underlying accretion model. As shown in Table~\\ref{tabletime},\nAccelerated Accretion generates relatively more sources at low bolometric temperatures. Sets of far-infrared Herschel data covering the temperature range from 20\\,K -- 70\\,K\nshould provide some insight. However, modelling and observations of the bolometric temperature both remain problematic especially at the low temperatures which can be dominated by unbound non-stellar and pre stellar objects in addition to AGB stars \\citep{2013A&A...549A.130V}. However, we find a solution which fits the data in which the initial clump mass is four to five times larger than that necessary to generate the associated star cluster corresponding to the mass of the most massive star. We thus generate revised evolutionary tracks which are consistent with statistics for the bolometric temperature. In these revised models, the star remains deeply embedded throughout its formation and the bolometric temperature distribution is no longer a sensitive diagnostic to differentiate between accretion models.\n\nAccretion models may be better tested through a complete sample of hotter \nprotostars with temperatures in the range 50\\,K to 100\\,K. In addition, as with their low-mass counterparts, large periodic accretion variations could dominate the statistics. \n\nThe Accelerated Accretion Model has been advanced in the literature because we expect the gravitational sphere of influence of a central object to grow as its mass grows.\nHere, however, with the large accretion rates often assumed, we take an inner envelope to already exist with sufficient mass to meet the later needs of the star and outflow.\nIn this case, the accretion rate depends on how fast mass can flow in from the envelope, through the disk, rather than how massive the central protostar has become.\n\nFinally, we have investigated Lyman fluxes as deduced observationally from radio fluxes. Observationally, it has been shown that objects of luminosity $\\sim$ 10,000~L$_\\odot$ can possess very high Lyman fluxes, inconsistent with their expected stellar temperatures \\citep{2013A&A...550A..21S}. We have shown here that the problem is resolvable if the protostar is relatively compact, formed through cold accretion via a disk. However, the present accretion must\ninvolve a funnelling free-fall mechanism onto a fraction of the stellar surface estimated to be less than 10\\%. The mechanism for this remains unknown but if the evidence that massive protostars are configured in the same way as low-mass stars continues to grow, then accretion via magnetic flux tubes and jets driven by magneto-centrifugal processes are conceivable.\n\nThe above explanation of the excess Lyman photon flux requires evolution under the Cold Accretion scenario. Hot Accretion falls far short even with the inclusion of hot-spots, as demonstrated in Fig.~\\ref{lognolyhot}. Cold accretion was defined as accretion on to the photosphere with no back-heating, so that the accreting material has the same entropy as that of the photosphere \\citep{2010ApJ...721..478H}. This is a limiting case which may be difficult to realise. It could be generally expected that the rapid mass accretion should be somewhat hot because a fraction of the entropy should be advected into the stellar interior \\citep{2011arXiv1106.3343H}. Taking larger protostellar radii will reduce the Lyman flux; taking radii 50\\% higher than that predicted for cold accretion, however, does not significantly alter the qualitative of fit to the ATCA data while doubling the radius has a considerable effect. Accurate results will require the implementation of a full\nstellar evolution code. \n\nThe major objective here has been to construct a consistent model which links the components. In the following works, we will investigate the consequences of mass\noutflows, accretion variations, maser production, thermal radio jets and H~{\\small II} regions with the purpose of determining how their evolutions are coordinated. \n\n\\section{Acknowledgements}\n\nI wish to thank Riccardo Cesaroni, Davide Elia, Sergio Molinari and Alvaro Sanchez-Monge for their encouragement and comment.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}