JOURNAL OF COMPUTATIONAL MATHEMATICS

Browse issues
Uniform weak error estimates for an asymptotic preserving scheme applied to a class of slow-fast parabolic semilinear SPDEs
Charles-Edouard Bréhier1
1 Universite de Pau et des Pays de l’Adour, E2S UPPA, CNRS, LMAP, Pau, France
The SMAI Journal of computational mathematics, Volume 10 (2024), pp. 175-228.

We study an asymptotic preserving scheme for the temporal discretization of a system of parabolic semilinear SPDEs with two time scales. Owing to the averaging principle, when the time scale separation ϵ vanishes, the slow component converges to the solution of a limiting evolution equation, which is captured when the time-step size Δt vanishes by a limiting scheme. The objective of this work is to prove weak error estimates which are uniform with respect to ϵ, in terms of Δt: the scheme satisfies a uniform accuracy property. This is a non trivial generalization of the recent article [10] in an infinite dimensional framework. The fast component is discretized using the modified Euler scheme for SPDEs introduced in the recent work [8]. Proving the weak error estimates requires delicate analysis of the regularity properties of solutions of infinite dimensional Kolmogorov equations. Numerical experiments illustrate the asymptotic preserving property and the uniform weak error estimates.

Published online:
DOI: 10.5802/smai-jcm.110
Classification: 60H35, 65C30, 60H15
Keywords: Stochastic partial differential equations, asymptotic preserving schemes, Euler schemes, infinite dimensional Kolmogorov equations

Charles-Edouard Bréhier 1

1 Universite de Pau et des Pays de l’Adour, E2S UPPA, CNRS, LMAP, Pau, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{SMAI-JCM_2024__10__175_0,
     author = {Charles-Edouard Br\'ehier},
     title = {Uniform weak error estimates for an asymptotic preserving scheme applied to a class of slow-fast parabolic semilinear {SPDEs}},
     journal = {The SMAI Journal of computational mathematics},
     pages = {175--228},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {10},
     year = {2024},
     doi = {10.5802/smai-jcm.110},
     language = {en},
     url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.110/}
}
TY  - JOUR
AU  - Charles-Edouard Bréhier
TI  - Uniform weak error estimates for an asymptotic preserving scheme applied to a class of slow-fast parabolic semilinear SPDEs
JO  - The SMAI Journal of computational mathematics
PY  - 2024
SP  - 175
EP  - 228
VL  - 10
PB  - Société de Mathématiques Appliquées et Industrielles
UR  - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.110/
DO  - 10.5802/smai-jcm.110
LA  - en
ID  - SMAI-JCM_2024__10__175_0
ER  - 
%0 Journal Article
%A Charles-Edouard Bréhier
%T Uniform weak error estimates for an asymptotic preserving scheme applied to a class of slow-fast parabolic semilinear SPDEs
%J The SMAI Journal of computational mathematics
%D 2024
%P 175-228
%V 10
%I Société de Mathématiques Appliquées et Industrielles
%U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.110/
%R 10.5802/smai-jcm.110
%G en
%F SMAI-JCM_2024__10__175_0
Charles-Edouard Bréhier. Uniform weak error estimates for an asymptotic preserving scheme applied to a class of slow-fast parabolic semilinear SPDEs. The SMAI Journal of computational mathematics, Volume 10 (2024), pp. 175-228. doi : 10.5802/smai-jcm.110. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.110/

[1] Assyr Abdulle; Weinan E; Björn Engquist; Eric Vanden-Eijnden The heterogeneous multiscale method, Acta Numer., Volume 21 (2012), pp. 1-87 | DOI | MR | Zbl

[2] Assyr Abdulle; Grigorios A. Pavliotis Numerical methods for stochastic partial differential equations with multiple scales, J. Comput. Phys., Volume 231 (2012) no. 6, pp. 2482-2497 | DOI | MR | Zbl

[3] Nathalie Ayi; Erwan Faou Analysis of an asymptotic preserving scheme for stochastic linear kinetic equations in the diffusion limit, SIAM/ASA J. Uncertain. Quantif., Volume 7 (2019) no. 2, pp. 760-785 | DOI | MR | Zbl

[4] Nils Berglund; Barbara Gentz Noise-induced phenomena in slow-fast dynamical systems, Probability and Its Applications, Springer, 2006, xiv+276 pages (A sample-paths approach) | MR

[5] Charles-Edouard Bréhier Strong and weak orders in averaging for SPDEs, Stochastic Processes Appl., Volume 122 (2012) no. 7, pp. 2553-2593 | DOI | MR | Zbl

[6] Charles-Edouard Bréhier Analysis of an HMM time-discretization scheme for a system of stochastic PDEs, SIAM J. Numer. Anal., Volume 51 (2013) no. 2, pp. 1185-1210 | DOI | MR | Zbl

[7] Charles-Edouard Bréhier Orders of convergence in the averaging principle for SPDEs: the case of a stochastically forced slow component, Stochastic Processes Appl., Volume 130 (2020) no. 6, pp. 3325-3368 | DOI | MR | Zbl

[8] Charles-Edouard Bréhier Analysis of a Modified Regularity-Preserving Euler Scheme for Parabolic Semilinear SPDEs: Total Variation Error Bounds for the Numerical Approximation of the Invariant Distribution, Found. Comput. Math. (2024)

[9] Charles-Edouard Bréhier; Arnaud Debussche Kolmogorov equations and weak order analysis for SPDEs with nonlinear diffusion coefficient, J. Math. Pures Appl., Volume 119 (2018), pp. 193-254 | DOI | MR | Zbl

[10] Charles-Edouard Bréhier; Shmuel Rakotonirina-Ricquebourg On Asymptotic Preserving Schemes for a Class of Stochastic Differential Equations in Averaging and Diffusion Approximation Regimes, Multiscale Model. Simul., Volume 20 (2022) no. 1, pp. 118-163 | DOI | MR | Zbl

[11] Sandra Cerrai Second order PDE’s in finite and infinite dimension, Lecture Notes in Mathematics, 1762, Springer, 2001, x+330 pages (A probabilistic approach) | DOI | MR

[12] Sandra Cerrai A Khasminskii type averaging principle for stochastic reaction-diffusion equations, Ann. Appl. Probab., Volume 19 (2009) no. 3, pp. 899-948 | DOI | MR | Zbl

[13] Sandra Cerrai Averaging principle for systems of reaction-diffusion equations with polynomial nonlinearities perturbed by multiplicative noise, SIAM J. Math. Anal., Volume 43 (2011) no. 6, pp. 2482-2518 | DOI | MR | Zbl

[14] Sandra Cerrai; Mark Freidlin Averaging principle for a class of stochastic reaction-diffusion equations, Probab. Theory Relat. Fields, Volume 144 (2009) no. 1-2, pp. 137-177 | DOI | MR | Zbl

[15] Giuseppe Da Prato; Jerzy Zabczyk Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and Its Applications, 152, Cambridge University Press, 2014, xviii+493 pages | DOI | MR

[16] Arnaud Debussche Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comput., Volume 80 (2011) no. 273, pp. 89-117 | DOI | MR | Zbl

[17] Weinan E; Di Liu; Eric Vanden-Eijnden Analysis of multiscale methods for stochastic differential equations, Commun. Pure Appl. Math., Volume 58 (2005) no. 11, pp. 1544-1585 | DOI | MR | Zbl

[18] Shi Jin Asymptotic-preserving schemes for multiscale physical problems, Acta Numer., Volume 31 (2022), pp. 415-489 | DOI | Zbl

[19] Christian Kuehn Multiple time scale dynamics, Applied Mathematical Sciences, 191, Springer, 2015, xiv+814 pages | DOI | MR

[20] David Nualart Malliavin calculus and its applications, CBMS Regional Conference Series in Mathematics, 110, American Mathematical Society, 2009, viii+85 pages | DOI | MR

[21] Grigorios A. Pavliotis; Andrew M. Stuart Multiscale methods, Texts in Applied Mathematics, 53, Springer, 2008, xviii+307 pages (Averaging and homogenization) | MR

Cited by Sources: