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 vanishes, the slow component converges to the solution of a limiting evolution equation, which is captured when the time-step size vanishes by a limiting scheme. The objective of this work is to prove weak error estimates which are uniform with respect to , in terms of : 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.
Keywords: Stochastic partial differential equations, asymptotic preserving schemes, Euler schemes, infinite dimensional Kolmogorov equations
Charles-Edouard Bréhier 1
![](/static/ptf/img/CC-BY%204.0.png)
@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] The heterogeneous multiscale method, Acta Numer., Volume 21 (2012), pp. 1-87 | DOI | MR | Zbl
[2] 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] 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] Noise-induced phenomena in slow-fast dynamical systems, Probability and Its Applications, Springer, 2006, xiv+276 pages (A sample-paths approach) | MR
[5] Strong and weak orders in averaging for SPDEs, Stochastic Processes Appl., Volume 122 (2012) no. 7, pp. 2553-2593 | DOI | MR | Zbl
[6] 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] 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] 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] 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] 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] 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] 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] 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] 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] Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and Its Applications, 152, Cambridge University Press, 2014, xviii+493 pages | DOI | MR
[16] Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comput., Volume 80 (2011) no. 273, pp. 89-117 | DOI | MR | Zbl
[17] Analysis of multiscale methods for stochastic differential equations, Commun. Pure Appl. Math., Volume 58 (2005) no. 11, pp. 1544-1585 | DOI | MR | Zbl
[18] Asymptotic-preserving schemes for multiscale physical problems, Acta Numer., Volume 31 (2022), pp. 415-489 | DOI | Zbl
[19] Multiple time scale dynamics, Applied Mathematical Sciences, 191, Springer, 2015, xiv+814 pages | DOI | MR
[20] Malliavin calculus and its applications, CBMS Regional Conference Series in Mathematics, 110, American Mathematical Society, 2009, viii+85 pages | DOI | MR
[21] Multiscale methods, Texts in Applied Mathematics, 53, Springer, 2008, xviii+307 pages (Averaging and homogenization) | MR
Cited by Sources: