High order linearly implicit methods for semilinear evolution PDEs

Guillaume Dujardin; Ingrid Lacroix-Violet

doi:10.5802/smai-jcm.111

High order linearly implicit methods for semilinear evolution PDEs

Guillaume Dujardin ¹; Ingrid Lacroix-Violet ²

¹ Univ. Lille, Inria, CNRS, UMR 8524 - Laboratoire Paul Painlevé F-59000 Lille, France
² Université de Lorraine, CNRS, IECL, F-54000 Nancy, France

The SMAI Journal of computational mathematics, Volume 10 (2024), pp. 325-354.

Abstract

This paper considers the numerical integration of semilinear evolution PDEs using the high order linearly implicit methods developped in [21] in the ODE setting. These methods use a collocation Runge–Kutta method as a basis, and additional variables that are updated explicitly and make the implicit part of the collocation Runge–Kutta method only linearly implicit. In this paper, we introduce several notions of stability for the underlying Runge–Kutta methods as well as for the explicit step on the additional variables necessary to fit the context of evolution PDE. We prove a main theorem about the high order of convergence of these linearly implicit methods in this PDE setting, using the stability hypotheses introduced before. We use nonlinear Schrödinger equations and heat equations as main examples but our results extend beyond these two classes of evolution PDEs. We illustrate our main result numerically in dimensions 1 and 2, and we compare the efficiency of the linearly implicit methods with other methods from the literature. We also illustrate numerically the necessity of the stability conditions of our main result.

Published online: 2024-11-19

DOI: 10.5802/smai-jcm.111

Classification: 65M12, 65M22, 65M70, 35K05, 35K58, 35Q55
Keywords: numerical analysis, high order methods, linearly implicit methods, collocation methods, Runge–Kutta methods, NLS equation, heat equation

Author's affiliations:

Guillaume Dujardin ¹; Ingrid Lacroix-Violet ²

¹ Univ. Lille, Inria, CNRS, UMR 8524 - Laboratoire Paul Painlevé F-59000 Lille, France
² Université de Lorraine, CNRS, IECL, F-54000 Nancy, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{SMAI-JCM_2024__10__325_0,
     author = {Guillaume Dujardin and Ingrid Lacroix-Violet},
     title = {High order~linearly implicit methods for semilinear evolution {PDEs}},
     journal = {The SMAI Journal of computational mathematics},
     pages = {325--354},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {10},
     year = {2024},
     doi = {10.5802/smai-jcm.111},
     language = {en},
     url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.111/}
}

TY  - JOUR
AU  - Guillaume Dujardin
AU  - Ingrid Lacroix-Violet
TI  - High order linearly implicit methods for semilinear evolution PDEs
JO  - The SMAI Journal of computational mathematics
PY  - 2024
SP  - 325
EP  - 354
VL  - 10
PB  - Société de Mathématiques Appliquées et Industrielles
UR  - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.111/
DO  - 10.5802/smai-jcm.111
LA  - en
ID  - SMAI-JCM_2024__10__325_0
ER  -

%0 Journal Article
%A Guillaume Dujardin
%A Ingrid Lacroix-Violet
%T High order linearly implicit methods for semilinear evolution PDEs
%J The SMAI Journal of computational mathematics
%D 2024
%P 325-354
%V 10
%I Société de Mathématiques Appliquées et Industrielles
%U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.111/
%R 10.5802/smai-jcm.111
%G en
%F SMAI-JCM_2024__10__325_0

Guillaume Dujardin; Ingrid Lacroix-Violet. High order linearly implicit methods for semilinear evolution PDEs. The SMAI Journal of computational mathematics, Volume 10 (2024), pp. 325-354. doi : 10.5802/smai-jcm.111. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.111/

References
Cited by

[1] Georgios Akrivis; Michel Crouzeix Linearly Implicit Methods for Nonlinear Parabolic Equations, Math. Comput., Volume 73 (2004) no. 246, pp. 613-635 http://www.jstor.org/stable/4099792 | DOI | Zbl

[2] Georgios Akrivis; Vassilios A. Dougalis On a class of conservative, highly accurate Galerkin methods for the Schrödinger equation, ESAIM, Math. Model. Numer. Anal., Volume 25 (1991) no. 6, pp. 643-670 | DOI | Numdam | MR | Zbl

[3] Georgios Akrivis; Charalambos Makridakis; Ricardo H. Nochetto A posteriori error estimates for the Crank–Nicolson method for parabolic equations, Math. Comput., Volume 75 (2006) no. 254, pp. 511-531 | DOI | MR | Zbl

[4] Grégoire Allaire; Alan Craig Numerical Analysis and Optimization: An Introduction to Mathematical Modelling and Numerical Simulation, Numerical Mathematics and Scientific Computation, Oxford University Press, 2007 | DOI

[5] Nadine Badr; Frédéric Bernicot; Emmanuel Russ Algebra properties for Sobolev spaces – Applications to semilinear PDE’s on manifolds, J. Anal. Math., Volume 118 (2012) no. 2, pp. 509-544 | DOI | Zbl

[6] Christophe Besse A relaxation scheme for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., Volume 42 (2004) no. 3, pp. 934-952 | DOI | MR | Zbl

[7] Christophe Besse; Brigitte Bidégaray; Stéphane Descombes Order Estimates in Time of Splitting Methods for the Nonlinear Schrödinger Equation, SIAM J. Numer. Anal., Volume 40 (2002) no. 1, pp. 26-40 | arXiv | DOI | Zbl

[8] Christophe Besse; Rémi Carles; Florian Méhats An asymptotic preserving scheme based on a new formulation for NLS in the semiclassical limit, Multiscale Model. Simul., Volume 11 (2013) no. 4, pp. 1228-1260 http://hal.archives-ouvertes.fr/docs/00/75/20/11/pdf/ap.pdf | DOI | MR | Zbl

[9] Christophe Besse; Stéphane Descombes; Guillaume Dujardin; Ingrid Lacroix-Violet Energy-preserving methods for nonlinear Schrödinger equations, IMA J. Numer. Anal., Volume 41 (2020) no. 1, pp. 618-653 | arXiv | DOI | Zbl

[10] Christophe Besse; Guillaume Dujardin; Ingrid Lacroix-Violet High order exponential integrators for nonlinear Schrödinger equations with application to rotating Bose–Einstein condensates, SIAM J. Numer. Anal., Volume 55 (2017) no. 3, pp. 1387-1411 | DOI | MR | Zbl

[11] Kevin Burrage; Willem H. Hundsdorfer; Jan G. Verwer A study of $B$ -convergence of Runge–Kutta methods, Computing, Volume 36 (1986) no. 1-2, pp. 17-34 | DOI | MR | Zbl

[12] Mari Paz Calvo; Javier de Frutos; Julia Novo Linearly implicit Runge–Kutta methods for advection-reaction-diffusion equations, Appl. Numer. Math., Volume 37 (2001) no. 4, pp. 535-549 | DOI | Zbl

[13] François Castella; Philippe Chartier; Stéphane Descombes; Gilles Vilmart Splitting methods with complex times for parabolic equations, BIT, Volume 49 (2009), pp. 487-508 | DOI | Zbl

[14] Qing Cheng; Jie Shen Multiple Scalar Auxiliary Variable (MSAV) Approach and its Application to the Phase-Field Vesicle Membrane Model, SIAM J. Sci. Comput., Volume 40 (2018) no. 6, p. A3982-A4006 | DOI | Zbl

[15] Michel Crouzeix Étude de la stabilité des méthodes de Runge–Kutta appliquées aux équations paraboliques, Publications des séminaires de mathématiques et informatique de Rennes, Volume S4 (1974) no. 3, pp. 1-6 http://eudml.org/doc/274743 | Numdam

[16] Michel Crouzeix; Pierre-Arnaud Raviart Méthodes de Runge–Kutta (1980) (Unpublished lecture notes, Université de Rennes)

[17] Michel Delfour; Michel Fortin; Guy Payre Finite-difference solutions of a nonlinear Schrödinger equation, J. Comput. Phys., Volume 44 (1981) no. 2, pp. 277-288 | DOI | MR | Zbl

[18] Stéphane Descombes Convergence of a Splitting Method of High Order for Reaction-Diffusion Systems, Math. Comput., Volume 70 (2001) no. 236, pp. 1481-1501 http://www.jstor.org/stable/2698737 | DOI | Zbl

[19] Stéphane Descombes; Marc Massot Operator splitting for nonlinear reaction-diffusion systems with an entropic structure: singular perturbation and order reduction, Numer. Math., Volume 97 (2004), pp. 667-698 | DOI | Zbl

[20] Guillaume Dujardin Exponential Runge–Kutta methods for the Schrödinger equation, Appl. Numer. Math., Volume 59 (2009) no. 8, pp. 1839-1857 | DOI | Zbl

[21] Guillaume Dujardin; Ingrid Lacroix-Violet High order linearly implicit methods for evolution equations, ESAIM, Math. Model. Numer. Anal., Volume 56 (2022) no. 3, pp. 743-766 | DOI | Zbl

[22] Guillaume Dujardin; Ingrid Lacroix-Violet $\hat{A}$ - and $\hat{I}$ - stability of Runge–Kutta collocation methods, Appl. Numer. Math., Volume 202 (2024), pp. 158-172 | DOI | Zbl

[23] Angel Durán; Jesús M. Sanz-Serna The numerical integration of relative equilibrium solutions. The nonlinear Schrödinger equation, IMA J. Numer. Anal., Volume 20 (2000) no. 2, pp. 235-261 | DOI | Zbl

[24] Pierre Grisvard Elliptic Problems in Nonsmooth Domains, Society for Industrial and Applied Mathematics, 2011 | DOI

[25] Ernst Hairer Constructive characterization of $A$ -stable approximations to $\exp (z)$ and its connection with algebraically stable Runge–Kutta methods, Numer. Math., Volume 39 (1982) no. 2, pp. 247-258 | DOI | MR | Zbl

[26] Ernst Hairer; Christian Lubich; Gerhard Wanner Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31, Springer, 2002 | DOI

[27] Marlis Hochbruck; Alexander Ostermann Exponential Runge–Kutta methods for parabolic problems, Appl. Numer. Math., Volume 53 (2005) no. 2, pp. 323-339 Tenth Seminar on Numerical Solution of Differential and Differntial-Algebraic Euqations (NUMDIFF-10) | DOI | Zbl

[28] Marlis Hochbruck; Alexander Ostermann Exponential integrators, Acta Numer., Volume 19 (2010), pp. 209-286 | DOI | Zbl

[29] Christian Klein Fourth order time-stepping for low dispersion Korteweg–de Vries and nonlinear Schrödinger equations, Electron. Trans. Numer. Anal., Volume 29 (2007), pp. 116-135 http://eudml.org/doc/117659 | Zbl

[30] Hervé Ledret Numerical Approximation of PDEs (2011-2012) (https://www.ljll.math.upmc.fr/ledret/M1ApproxPDE.html)

[31] Christian Lubich On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations, Math. Comput., Volume 77 (2008) no. 264, pp. 2141-2153 | DOI | MR | Zbl

[32] Christian Lubich; Alexander Ostermann Linearly implicit time discretization of non-linear parabolic equations, IMA J. Numer. Anal., Volume 15 (1995) no. 4, pp. 555-583 | DOI

[33] Yousef Saad Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, 2003 | DOI

[34] Jie Shen; Jie Xu Convergence and Error Analysis for the Scalar Auxiliary Variable (SAV) Schemes to Gradient Flows, SIAM J. Numer. Anal., Volume 56 (2018) no. 5, pp. 2895-2912 | DOI | Zbl

[35] J. André C. Weideman; Ben M. Herbst Split-Step Methods for the Solution of the Nonlinear Schrodinger Equation, SIAM J. Numer. Anal., Volume 23 (1986) no. 3, pp. 485-507 | DOI | Zbl

Cited by Sources: