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Volume 43, Issue 4
Numerical Approximation of the Invariant Distribution for a Class of Stochastic Damped Wave Equations

Ziyi Lei & Siqing Gan

DOI: 10.4208/jcm.2404-m2023-0144

J. Comp. Math., 43 (2025), pp. 976-1015.

Published online: 2025-07

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  • Abstract

We study a class of stochastic semilinear damped wave equations driven by additive Wiener noise. Owing to the damping term, under appropriate conditions on the nonlinearity, the solution admits a unique invariant distribution. We apply semi-discrete and fully-discrete methods in order to approximate this invariant distribution, using a spectral Galerkin method and an exponential Euler integrator for spatial and temporal discretization respectively. We prove that the considered numerical schemes also admit unique invariant distributions, and we prove error estimates between the approximate and exact invariant distributions, with identification of the orders of convergence. To the best of our knowledge this is the first result in the literature concerning numerical approximation of invariant distributions for stochastic damped wave equations.

  • Keywords

Stochastic damped wave equation, Invariant distribution, Exponential integrator, Spectral Galerkin method, Weak error estimates, Infinite dimensional Kolmogorov equations.

  • AMS Subject Headings

60H15, 60H35, 65C30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-43-976, author = {Lei , Ziyi and Gan , Siqing}, title = {Numerical Approximation of the Invariant Distribution for a Class of Stochastic Damped Wave Equations}, journal = {Journal of Computational Mathematics}, year = {2025}, volume = {43}, number = {4}, pages = {976--1015}, abstract = {

We study a class of stochastic semilinear damped wave equations driven by additive Wiener noise. Owing to the damping term, under appropriate conditions on the nonlinearity, the solution admits a unique invariant distribution. We apply semi-discrete and fully-discrete methods in order to approximate this invariant distribution, using a spectral Galerkin method and an exponential Euler integrator for spatial and temporal discretization respectively. We prove that the considered numerical schemes also admit unique invariant distributions, and we prove error estimates between the approximate and exact invariant distributions, with identification of the orders of convergence. To the best of our knowledge this is the first result in the literature concerning numerical approximation of invariant distributions for stochastic damped wave equations.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2404-m2023-0144}, url = {http://global-sci.org/intro/article_detail/jcm/24267.html} }
TY - JOUR T1 - Numerical Approximation of the Invariant Distribution for a Class of Stochastic Damped Wave Equations AU - Lei , Ziyi AU - Gan , Siqing JO - Journal of Computational Mathematics VL - 4 SP - 976 EP - 1015 PY - 2025 DA - 2025/07 SN - 43 DO - http://doi.org/10.4208/jcm.2404-m2023-0144 UR - https://global-sci.org/intro/article_detail/jcm/24267.html KW - Stochastic damped wave equation, Invariant distribution, Exponential integrator, Spectral Galerkin method, Weak error estimates, Infinite dimensional Kolmogorov equations. AB -

We study a class of stochastic semilinear damped wave equations driven by additive Wiener noise. Owing to the damping term, under appropriate conditions on the nonlinearity, the solution admits a unique invariant distribution. We apply semi-discrete and fully-discrete methods in order to approximate this invariant distribution, using a spectral Galerkin method and an exponential Euler integrator for spatial and temporal discretization respectively. We prove that the considered numerical schemes also admit unique invariant distributions, and we prove error estimates between the approximate and exact invariant distributions, with identification of the orders of convergence. To the best of our knowledge this is the first result in the literature concerning numerical approximation of invariant distributions for stochastic damped wave equations.

Lei , Ziyi and Gan , Siqing. (2025). Numerical Approximation of the Invariant Distribution for a Class of Stochastic Damped Wave Equations. Journal of Computational Mathematics. 43 (4). 976-1015. doi:10.4208/jcm.2404-m2023-0144
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