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

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

Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray--Scott equations

Hausenblas E, Randrianasolo TA, Thalhammer M. Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray--Scott equations. Journal of Computational and Applied Mathematics. 2019;364: 112335.Mathematical models based on systems of reaction-diffusion equations provide fundamental tools for the description and investigation of various processes in biology, biochemistry, and chemistry; in specific situations, an appealing characteristic of the arising nonlinear partial differential equations is the formation of patterns, reminiscent of those found in nature. The deterministic Gray–Scott equations constitute an elementary two-component system that describes autocatalytic reaction processes; depending on the choice of the specific parameters, complex patterns of spirals, waves, stripes, or spots appear. In the derivation of a macroscopic model such as the deterministic Gray–Scott equations from basic physical principles, certain aspects of microscopic dynamics, e.g. fluctuations of molecules, are disregarded; an expedient mathematical approach that accounts for significant microscopic effects relies on the incorporation of stochastic processes and the consideration of stochastic partial differential equations. The present work is concerned with a theoretical and numerical study of the stochastic Gray–Scott equations driven by independent spatially time-homogeneous Wiener processes. Under suitable regularity assumptions on the prescribed initial states, existence and uniqueness of the solution processes is proven. Numerical simulations based on the application of a time-adaptive first-order operator splitting method and the fast Fourier transform illustrate the formation of patterns in the deterministic case and their variation under the influence of stochastic nois

Superconvergence of Strang splitting for NLS in TdT^d

In this paper we investigate the convergence properties of semi-discretized approximations by Strang splitting method applied to fast-oscillating nonlinear Schr¨odinger equations. In a first step and for further use, we briefly adapt a known convergence result for Strang method in the context of NLS on $T^d$ for a large class of nonlinearities. In a second step, we examine how errors depend on the length of the period $\varepsilon$, the solutions being considered on intervals of fixed length (independent of the period). Our main contribution is to show that Strang splitting with constant step-sizes is unexpectedly more accurate by a factor $\varepsilon$ as compared to established results when the step-size is chosen as an integer fraction of the period, owing to an averaging effect

Improved error estimates for splitting methods applied to highly-oscillatory nonlinear Schrödinger equations

International audienceIn this work, the error behavior of operator splitting methods is analyzed for highly-oscillatory differential equations. The scope of applications includes time-dependent nonlinear Schrödinger equations, where the evolution operator associated with the principal linear part is highly-oscillatory and periodic in time. In a first step, a known convergence result for the second-order Strang splitting method applied to the cubic Schrödinger equation is adapted to a wider class of nonlinearities. In a second step, the dependence of the global error on the decisive parameter 0 < ε < < 1, defining the length of the period, is examined. The main result states that, compared to established error estimates, the Strang splitting method is more accurate by a factor ε, provided that the time stepsize is chosen as an integer fraction of the period. This improved error behavior over a time interval of fixed length, which is independent of the period, is due to an averaging effect. The extension of the convergence result to higher-order splitting methods and numerical illustrations complement the investigations

Generalization of splitting methods based on modified potentials to nonlinear evolution equations of parabolic and Schr\"odinger type

The present work is concerned with the extension of modified potential operator splitting methods to specific classes of nonlinear evolution equations. The considered partial differential equations of Schr{\"o}dinger and parabolic type comprise the Laplacian, a potential acting as multiplication operator, and a cubic nonlinearity. Moreover, an invariance principle is deduced that has a significant impact on the efficient realisation of the resulting modified operator splitting methods for the Schr{\"o}dinger case.} Numerical illustrations for the time-dependent Gross--Pitaevskii equation in the physically most relevant case of three space dimensions and for its parabolic counterpart related to ground state and excited state computations confirm the benefits of the proposed fourth-order modified operator splitting method in comparison with standard splitting methods. The presented results are novel and of particular interest from both, a theoretical perspective to inspire future investigations of modified operator splitting methods for other classes of nonlinear evolution equations and a practical perspective to advance the reliable and efficient simulation of Gross--Pitaevskii systems in real and imaginary time.Comment: 30 pages, 6 figure