Avoiding order reduction when integrating reaction–diffusion boundary value problems with exponential splitting methods

Abstract

In this paper, we suggest a technique to avoid order reduction in time when integrating reaction-diffusion boundary value problems under non-homogeneous boundary conditions with exponential splitting methods. More precisely, we consider Lie-Trotter and Strang splitting methods and Dirichlet, Neumann and Robin boundary conditions. Beginning from an abstract framework in Banach spaces, a thorough error analysis after full discretization is performed and some numerical results are shown which corroborate the theoretical results. * over vectors [13], they constitute an effective tool to integrate such problems in a stable way.In this paper, we will center on the first-order Lie-Trotter and second-order Strang methods. The order reduction which turns up with these methods when integrating linear problems with homogeneous boundary conditions was recently studied in [12]. In [1] we have suggested a technique to deal with non-homogeneous boundary conditions in linear problems. That technique has some similarities to that suggested in [3] for other exponential-type methods, which are Lawson ones. With that procedure, we managed to avoid order reduction completely in linear problems.The aim of the present paper is to generalize that technique to nonlinear reactiondiffusion problems and to prove that order reduction can also be completely avoided.There are other results in the literature concerning this problem or a more specific one. For example, in [7], a generalized Strang method is suggested for the specific nonlinear Schrödinger equation. However, in that paper, an abstract formulation of the problem is not given (as it is here), Neumann or Robin type boundary conditions are not considered, parabolic problems for which a summation-by-parts argument can be applied are not included and finally, Lie-Trotter method is not analyzed. On the other hand, in [10,11], a completely different technique is suggested to avoid order reduction with the same methods and nonlinear problems than here, but the analysis for the local and global error is just performed in time. The error coming from the space discretization and the numerical approximation in time of the nonlinear and smooth part are not included and therefore, a practical implementation of the technique for the practitioners is not justified. However, in the present paper, the exact formulas to be implemented are described in (40)-(41) for Lie-Trotter and in (58), (60) and (62) for Strang. Moreover, the analysis is performed under quite general assumptions on the space discretization and time integration of the nonlinear part. For that, we use the maximum norm, which facilitates its applicability to quite general problems.The paper is structured as follows. Section 2 gives some preliminaries on the abstract setting of the problem, on the assumptions of regularity which are required for the solution to be approximated and on Lie-Trotter and Strang methods. Section 3 describes the technique to avoid order reduction after time integration with Lie-Trotter method and explains h...

Avoiding order reduction when integrating linear initial boundary value problems with exponential splitting methods

“…Although the previous analysis is valid for other types of boundary conditions, we consider here, for the sake of simplicity, an abstract spatial discretization which is suitable for Dirichlet boundary conditions. (Look at [5] for a complete analysis of a similar technique with Neumann or Robin boundary conditions for nonlinear problems where the nonlinear part is a smooth operator. With linear problems, although both operators are unbounded, the analysis there would be extended in a simpler way because the boundary conditions can always be exactly calculated in terms of data instead of just approximately, as it happens in [5].…”

“…Although it is not an aim of this paper, there are already results on applying a similar technique to nonlinear problems [5,7], and [11] tries to compare with the technique in [9,10] for them.…”

Exponential Quadrature Rules Without Order Reduction for Integrating Linear Initial Boundary Value Problems

“…As for (ii), a summation-by-parts argument like that given in [5] for splitting exponential methods also applies here because of hypothesis (13) and the fact that f ∈ C s+2 . Finally, (iii) follows in the same way as in the proof of Theorem 1.…”

Comparison of efficiency among different techniques to avoid order reduction with Strang splitting

“…On the other hand, in [5], another technique is suggested in which appropriate boundary conditions are suggested for each part of the splitting. The analysis there considers both the space and time discretization.…”

“…The conclusions of this work are sketched in Section 5. Finally, in an appendix, a thorough error analysis (including numerical differentiation) is given for one of the implementations of the base technique which was suggested in [5], but which modifications were not included in the analysis.…”