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

Alonso‐Mallo, Isaías; Cano, Begoña; Reguera, Nuria

doi:10.1093/imanum/drx047

Cited by 13 publications

(30 citation statements)

References 22 publications

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“…Assuming that u 0 ∈ D(A) and ∂u 0 = v 0 , the solution of (8) is given by (see e.g. [1]) u(t) = e tA 0 (u 0 − K(0)v 0 ) + K(0)(v 0 + v 1 t) − t 0 e sA 0 K(0)v 1 ds. (9) Notice that (9) is well defined for any u 0 ∈ X and v 0 , v 1 ∈ Y ; therefore, it may be considered as a generalized solution of (8) even when ∂u 0 = v 0 or u 0 / ∈ D(A).…”

Section: Preliminariesmentioning

confidence: 99%

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

Alonso‐Mallo

Cano

Reguera

2019

Journal of Computational and Applied Mathematics

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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...

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Section: Preliminariesmentioning

confidence: 99%

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

Alonso‐Mallo

Cano

Reguera

2019

Journal of Computational and Applied Mathematics

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“…It should be noted, however, that the convergence is much more predictable for the third order correction. That is, we do not observe the erratic 1] for the unmodified Strang splitting as well as the second order TDBC and CEC corrected Strang splitting applied to equation (21) with f (u) = e u−1 are shown. The space discretization is conducted by using a second order upwind finite difference stencil with 500 grid points.…”

Section: Dispersive Problemmentioning

confidence: 63%

A comparison of boundary correction methods for Strang splitting

Einkemmer¹,

Ostermann²

2018

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper we consider splitting methods in the presence of non-homogeneous boundary conditions. In particular, we consider the corrections that have been described and analyzed in Einkemmer, Ostermann 2015 and Alonso-Mallo, Cano, Reguera 2016. The latter method is extended to the non-linear case, and a rigorous convergence analysis is provided. We perform numerical simulations for diffusion-reaction, advection-reaction, and dispersion-reaction equations in order to evaluate the relative performance of these two corrections. Furthermore, we introduce an extension of both methods to obtain order three locally and evaluate under what circumstances this is beneficial.

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“…Here (17) is understood in a generalized sense, in the same way that e it∆0 u 0 is understood when ∂u 0 ̸ = 0 (see [6,17]). Notice also that the last equality just comes from (1) considering that…”

Section: Time Semidiscretizationmentioning

confidence: 99%

“…Moreover, they have been usually used when considering homogeneous boundary conditions and the analysis has been performed under that assumption. Just some recent research [5,6,13] has been done to include non-homogeneous boundary conditions. Moreover, the techniques which are suggested there manage to avoid order reduction for both homogeneous and nonhomogeneous boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

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Avoiding order reduction when integrating nonlinear Schrödinger equation with Strang method

Cano

Reguera

2017

Journal of Computational and Applied Mathematics

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In this paper a technique is suggested to avoid order reduction when using Strang method to integrate nonlinear Schrödinger equation subject to time-dependent Dirichlet boundary conditions. The computational cost of this technique is negligible compared to that of the method itself, at least when the timestepsize is fixed. Moreover, a thorough error analysis is given as well as a modification of the technique which allows to conserve the symmetry of the method while retaining its second order.

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

Cited by 13 publications

References 22 publications

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

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

A comparison of boundary correction methods for Strang splitting

Avoiding order reduction when integrating nonlinear Schrödinger equation with Strang method

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