High order linearly implicit methods for semilinear evolution PDEs
Dujardin, Guillaume1 ; Lacroix-Violet, Ingrid2
1Univ. Lille, Inria, CNRS, UMR 8524 - Laboratoire Paul Painlevé F-59000 Lille, France
2Université de Lorraine, CNRS, IECL, F-54000 Nancy, France
The SMAI Journal of computational mathematics, Tome 10 (2024), pp. 325-354

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.

Publié le :
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

Dujardin, Guillaume  1   ; Lacroix-Violet, Ingrid  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 
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     title = {High order~linearly implicit methods for semilinear evolution {PDEs}},
     journal = {The SMAI Journal of computational mathematics},
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     year = {2024},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
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Dujardin, Guillaume; Lacroix-Violet, Ingrid. High order linearly implicit methods for semilinear evolution PDEs. The SMAI Journal of computational mathematics, Tome 10 (2024), pp. 325-354. doi: 10.5802/smai-jcm.111

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

[2] Akrivis, Georgios; Dougalis, Vassilios A. 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 | MR | Zbl | DOI | Numdam

[3] Akrivis, Georgios; Makridakis, Charalambos; Nochetto, Ricardo H. 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] Allaire, Grégoire; Craig, Alan Numerical Analysis and Optimization: An Introduction to Mathematical Modelling and Numerical Simulation, Numerical Mathematics and Scientific Computation, Oxford University Press, 2007 | MR | DOI

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

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

[7] Besse, Christophe; Bidégaray, Brigitte; Descombes, Stéphane 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 | MR | DOI | Zbl

[8] Besse, Christophe; Carles, Rémi; Méhats, Florian 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/... | Zbl | DOI | MR

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

[10] Besse, Christophe; Dujardin, Guillaume; Lacroix-Violet, Ingrid 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 | Zbl | DOI | MR

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

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

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

[14] Cheng, Qing; Shen, Jie 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 | Zbl | DOI | MR

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

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

[18] Descombes, Stéphane 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 | MR

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

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

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

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

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

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

[25] Hairer, Ernst 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 | Zbl | DOI | MR

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

[27] Hochbruck, Marlis; Ostermann, Alexander 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) | Zbl | DOI | MR

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

[29] Klein, Christian 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 | MR

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

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

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

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

[34] Shen, Jie; Xu, Jie 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 | MR

[35] Weideman, J. André C.; Herbst, Ben M. 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 | MR

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