In general, high order splitting methods suffer from an order reduction phenomena when applied to the time integration of partial differential equations with non-periodic boundary conditions. In the last decade, there were introduced several modifications to prevent the second order Strang splitting method from such a phenomena. In this article, inspired by these recent corrector techniques, we introduce a splitting method of order three for a class of semilinear parabolic problems that avoids order reduction in the context of non-periodic boundary conditions. We give a proof for the third order convergence of the method in a simplified linear setting and confirm the result by numerical experiments. Moreover, we show numerically that the high order convergence persists for an order four variant of a splitting method, and also for a nonlinear source term.
Keywords: High order splitting methods, diffusion-reaction equation, non-homogeneous boundary conditions, order reduction phenomena
Ramona Häberli 1
Ramona Häberli. Overcoming the order barrier two in splitting methods when applied to semilinear parabolic problems with non-periodic boundary conditions. The SMAI Journal of computational mathematics, Volume 12 (2026), pp. 269-288. doi: 10.5802/smai-jcm.149
@article{SMAI-JCM_2026__12__269_0,
author = {Ramona H\"aberli},
title = {Overcoming the order barrier two in splitting methods when applied to semilinear parabolic problems with non-periodic boundary conditions},
journal = {The SMAI Journal of computational mathematics},
pages = {269--288},
year = {2026},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {12},
doi = {10.5802/smai-jcm.149},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.149/}
}
TY - JOUR AU - Ramona Häberli TI - Overcoming the order barrier two in splitting methods when applied to semilinear parabolic problems with non-periodic boundary conditions JO - The SMAI Journal of computational mathematics PY - 2026 SP - 269 EP - 288 VL - 12 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.149/ DO - 10.5802/smai-jcm.149 LA - en ID - SMAI-JCM_2026__12__269_0 ER -
%0 Journal Article %A Ramona Häberli %T Overcoming the order barrier two in splitting methods when applied to semilinear parabolic problems with non-periodic boundary conditions %J The SMAI Journal of computational mathematics %D 2026 %P 269-288 %V 12 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.149/ %R 10.5802/smai-jcm.149 %G en %F SMAI-JCM_2026__12__269_0
[1] Numerical analysis for the two-dimensional Fisher-Kolmogorov-Petrovski-Piskunov equation with mixed boundary condition, J. Appl. Math. Comput., Volume 68 (2022) no. 6, pp. 3589-3614 | DOI | MR | Zbl
[2] Avoiding order reduction when integrating linear initial boundary value problems with Lawson methods, IMA J. Numer. Anal., Volume 37 (2017), pp. 2091-2119 | DOI | Zbl | MR
[3] Avoiding order reduction when integrating reaction-diffusion boundary value problems with exponential splitting methods, J. Comput. Appl. Math., Volume 357 (2019), pp. 228-250 | DOI | Zbl | MR
[4] Comparison of efficiency among different techniques to avoid order reduction with Strang splitting, Numer. Methods Partial Differ. Equations, Volume 37 (2021) no. 1, pp. 854-873 | DOI | Zbl | MR
[5] Integrating semilinear wave problems with time-dependent boundary values using arbitrarily high-order splitting methods, Mathematics, Volume 9 (2021) no. 10, pp. 854-873 | DOI
[6] Solving the Schrödinger eigenvalue problem by the imaginary time propagation technique using splitting methods with complex coefficients, J. Chem. Phys., Volume 139 (2013) no. 124117, pp. 1-11 | DOI
[7] Strang splitting method for semilinear parabolic problems with inhomogeneous boundary conditions: a correction based on the flow of the nonlinearity, SIAM J. Sci. Comput., Volume 42 (2020) no. 3, p. A1913-A1934 | DOI | MR | Zbl
[8] On the necessity of negative coefficients for operator splitting schemes of order higher than two, Appl. Numer. Math., Volume 54 (2005) no. 1, pp. 23-37 | DOI | MR | Zbl
[9] Optimized high order splitting methods for some classes of parabolic equations, Math. Comput., Volume 82 (2013) no. 283, pp. 1559-1576 | MR | Zbl | DOI
[10] Exponential quadrature rules without order reduction for integrating linear initial boundary value problems, SIAM J. Numer. Anal., Volume 56 (2018) no. 3, pp. 1187-1209 | DOI | Zbl | MR
[11] Splitting methods with complex times for parabolic equations, BIT Numer. Math., Volume 49 (2009), pp. 487-508 | DOI | MR | Zbl
[12] Convergence of a splitting method of high order for reaction-diffusion systems, Math. Comput., Volume 70 (2001) no. 236, pp. 1481-1501 | DOI | MR | Zbl
[13] Efficient boundary corrected Strang splitting, Appl. Math. Comput., Volume 332 (2018), pp. 76-89 | DOI | MR | Zbl
[14] Overcoming order reduction in diffusion-reaction splitting. Part 1: Dirichlet boundary conditions, SIAM J. Sci. Comput., Volume 37 (2015) no. 3, p. A1577-A1592 | DOI | MR | Zbl
[15] Overcoming order reduction in diffusion-reaction splitting. Part 2: Oblique boundary conditions, SIAM J. Sci. Comput., Volume 38 (2016) no. 6, p. A3741-A3757 | DOI | MR | Zbl
[16] Partial differential equations, Graduate Studies in Mathematics, 19, American Mathematical Society, 1997, xxii+586 pages | MR
[17] Analysis of exponential splitting methods for inhomogeneous parabolic equations, IMA J. Numer. Anal., Volume 35 (2015), pp. 161-178 | DOI | Zbl | MR
[18] Operator splitting and approximate factorization for taxis-diffusion-reaction models, Appl. Numer. Math., Volume 42 (2002) no. 1-3, pp. 159-176 | DOI | MR | Zbl
[19] N-th order operator splitting schemes and nonreversible systems, SIAM J. Numer. Anal., Volume 33 (1996) no. 1, pp. 349-367 https://www.jstor.org/stable/2158438 | DOI | MR | Zbl
[20] Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, Springer Series in Computational Mathematics, 31, Springer, 2010, xviii+644 pages Reprint of the second edition (2006) | MR | Zbl
[21] A second-order positivity preserving scheme for semilinear parabolic problems, Appl. Numer. Math., Volume 62 (2012) no. 10, pp. 1428-1435 | DOI | MR | Zbl
[22] High order splitting methods for analytic semigroups exist, BIT Numer. Math., Volume 49 (2009), pp. 527-542 | DOI | MR | Zbl
[23] Numerical solution of time-dependent advection-diffusion-reaction equations, Springer Series in Computational Mathematics, 33, Springer, 2003, x+471 pages | DOI | MR | Zbl
[24] Analytic semigroups and reaction-diffusion problems, 2005 (Internet Seminar)
[25] Analytic semigroups and optimal regularity in parabolic problems, Modern Birkhäuser Classics, Springer, 1995, xviii+424 pages | MR | Zbl
[26] Splitting methods, Acta Numer., Volume 11 (2002), pp. 341-434 | DOI | MR | Zbl
[27] Direct methods in the theory of elliptic equations, Springer Monographs in Mathematics, Springer, 1967, xvi+372 pages | MR
[28] A Krylov subspace algorithm for evaluating the -functions appearing in exponential integrators, ACM Trans. Math. Softw., Volume 38 (2012) no. 3, 22, 19 pages | DOI | MR | Zbl
[29] Characterizing and minimizing the operator split error for Fisher’s equation, Appl. Math. Lett., Volume 19 (2006), pp. 604-612 | MR | Zbl | DOI
[30] Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations, Phys. Lett. A, Volume 146 (1990) no. 6, pp. 319-323 | DOI | MR
[31] Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics, 25, Springer, 2006, xii+370 pages | MR | Zbl
[32] Construction of higher order symplectic integrators, Phys. Lett. A, Volume 150 (1990) no. 5-7, pp. 262-268 | DOI | MR | Zbl
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