Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear evolution equations of parabolic type

[EN] The main objective of this work is to provide a stability and error analysis of high-order commutator-free quasi-Magnus (CFQM) exponential integrators. These time integration methods for nonautonomous linear evolution equations are formed by products of exponentials involving linear combinations of the defining operator evaluated at certain times. In comparison with other classes of time integration methods, such as Magnus integrators, an inherent advantage of CFQM exponential integrators is that structural properties of the operator are well preserved by the arising linear combinations. Employing the analytical framework of sectorial operators in Banach spaces, evolution equations of parabolic type and dissipative quantum systems are included in the scope of applications. In this context, however, numerical experiments show that CFQM exponential integrators of nonstiff order five or higher proposed in the literature suffer from poor stability properties. The given analysis delivers insight that CFQM exponential integrators are well defined and stable only if the coefficients occurring in the linear combinations satisfy a positivity condition and that an alternative approach for the design of stable high-order schemes relies on the consideration of complex coefficients. Together with suitable local error expansions, this implies that a high-order CFQM exponential integrator retains its nonstiff order of convergence under appropriate regularity and compatibility requirements on the exact solution. Numerical examples confirm the theoretical result and illustrate the favourable behaviour of novel schemes involving complex coefficients in stability and accuracy.Ministerio de Economia y Competitividad (Spain) through projects MTM2013-46553-C3 and MTM2016-77660-P (AEI/FEDER, UE) to S.B. and F.C.Blanes Zamora, S.; Casas, F.; Mechthild Thalhammer (2018). Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear evolution equations of parabolic type. IMA Journal of Numerical Analysis. 38(2):743-778. https://doi.org/10.1093/imanum/drx012S74377838

Geometric integration of non-autonomous Hamiltonian problems

Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of dimensions. We show that one possible extension of symplectic methods in the autonomous setting to the non-autonomous setting is obtained by using canonical transformations. Many existing methods fit into this framework. We also perform experiments which indicate that for exponential integrators, the canonical and symmetric properties are important for good long time behaviour. In particular, the theoretical and numerical results support the well documented fact from the literature that exponential integrators for non-autonomous linear problems have superior accuracy compared to general ODE schemes.Comment: 20 pages, 3 figure

Higher-order exponential integrators for quasi-linear parabolic problems. Part II: Convergence

In this work, the convergence analysis of explicit exponential time integrators based on general linear methods for quasi-linear parabolic initial boundary value problems is pursued. Compared to other types of exponential integrators encountering rather severe order reductions, in general, the considered class of exponential general linear methods provides the possibility of constructing schemes that retain higher-order accuracy in time when applied to quasi-linear parabolic problems. In view of practical applications, the case of variable time step sizes is incorporated. The convergence analysis is based upon two fundamental ingredients. The needed stability bounds, obtained under mild restrictions on the ratios of subsequent time step sizes, have been deduced in the recent work [C. González and M. Thalhammer, SIAM J. Numer. Anal., 53 (2015), pp. 701--719]. The core of the present work is devoted to the derivation of suitable local and global error representations. In conjunction with the stability bounds, a convergence result is established.Ministerio de Economía, Industria y Competitividad, proyecto MTM2013-46553-C3-1-P y Austrian Science Fund (FWF), projects P21620-N13 and P28645-N35

High-order commutator-free quasi-Magnus exponential integrators for non-autonomous linear evolution equation

The class of commutator-free quasi-Magnus (CFQM) exponential integrators provides a favourable alternative to standard Magnus integrators, in particular for large-scale applications arising in the time integration of non-autonomous linear evolution equations. The schemes are given by compositions of several exponentials that comprise certain linear combinations of the values of the defining operator at specified nodes. Due to the fact that previously proposed CFQM exponential integrators of order five or higher involve negative coefficients in the linear combinations, severe instabilities are observed for spatially semi-discretised parabolic equations or for master equations describing dissipative quantum systems. In order to remedy this issue, two different approaches for the design of efficient time integrators of orders four, five, and six are pursued: (i) the study of CFQM exponential integrators involving complex coefficients that satisfy a positivity condition, and (ii) the study of unconventional methods in the sense that an additional exponential involving a commutator of higher order with respect to the time stepsize occurs. Numerical experiments confirm that the identified novel time integrators are superior to other integrators of the same family previously proposed in the literature

Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear Schrodinger equations

[EN] This work is devoted to the derivation of a convergence result for high-order commutator-free quasi-Magnus (CFQM) exponential integrators applied to nonautonomous linear Schrodinger equations; a detailed stability and local error analysis is provided for the relevant special case where the Hamilton operator comprises the Laplacian and a regular space-time-dependent potential. In the context of nonautonomous linear ordinary differential equations, CFQM exponential integrators are composed of exponentials involving linear combinations of certain values of the associated time-dependent matrix; this approach extends to nonautonomous linear evolution equations given by unbounded operators. An inherent advantage of CFQM exponential integrators over other time integration methods such as Runge-Kutta methods or Magnus integrators is that structural properties of the underlying operator family are well preserved; this characteristic is confirmed by a theoretical analysis ensuring unconditional stability in the underlying Hilbert space and the full order of convergence under low regularity requirements on the initial state. Due to the fact that convenient tools for products of matrix exponentials such as the Baker-Campbell-Hausdorff formula involve infinite series and thus cannot be applied in connection with unbounded operators, a certain complexity in the investigation of higher-order CFQM exponential integrators for Schrodinger equations is related to an appropriate treatment of compositions of evolution operators; an effective concept for the derivation of a local error expansion relies on suitable linearisations of the evolution equations for the exact and numerical solutions, representations by the variation-ofconstants formula and Taylor series expansions of parts of the integrands, where the arising iterated commutators determine the regularity requirements on the problem data.Ministerio de Economia y Competitividad (Spain) (project MTM2016-77660-P (AEI/FEDER, UE) to S.B., F.C. and C.G.).Blanes Zamora, S.; Casas, F.; González, C.; Thalhammer, M. (2021). Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear Schrodinger equations. IMA Journal of Numerical Analysis. 41(1):594-617. https://doi.org/10.1093/imanum/drz058S59461741

Positivity-preserving methods for ordinary differential equations

[EN] Many important applications are modelled by differential equations with positive solutions. However, it remains an outstanding open problem to develop numerical methods that are both (i) of a high order of accuracy and (ii) capable of preserving positivity. It is known that the two main families of numerical methods, Runge-Kutta methods and multistep methods, face an order barrier. If they preserve positivity, then they are constrained to low accuracy: they cannot be better than first order. We propose novel methods that overcome this barrier: second order methods that preserve positivity unconditionally and a third order method that preserves positivity under very mild conditions. Our methods apply to a large class of differential equations that have a special graph Laplacian structure, which we elucidate. The equations need be neither linear nor autonomous and the graph Laplacian need not be symmetric. This algebraic structure arises naturally in many important applications where positivity is required. We showcase our new methods on applications where standard high order methods fail to preserve positivity, including infectious diseases, Markov processes, master equations and chemical reactions.The authors thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme "Geometry, compatibility and structure preservation in computational differential equations" when work on this paper was undertaken. This work was supported by EPSRC grant EP/R014604/1. S.B. has been supported by project PID2019-104927GB-C21 (AEI/FEDER, UE).Blanes Zamora, S.; Iserles, A.; Macnamara, S. (2022). Positivity-preserving methods for ordinary differential equations. ESAIM Mathematical Modelling and Numerical Analysis. 56(6):1843-1870. https://doi.org/10.1051/m2an/20220421843187056