Time integration of dissipative stochastic PDEs

TL;DR

This paper investigates numerical methods for solving stochastic reaction-diffusion PDEs, emphasizing the preservation of mean-square dissipativity through stochastic time integration schemes, supported by theoretical analysis and numerical experiments.

Contribution

It demonstrates the conservative properties of stochastic θ-methods and stochastic θ-IMEX methods in preserving dissipativity, considering spatial and temporal discretization effects.

Findings

Stochastic θ-methods preserve mean-square dissipativity.

Numerical experiments confirm theoretical predictions.

Spatial and temporal step sizes influence dissipativity preservation.

Abstract

The paper is focused on the numerical solution of stochastic reaction-diffusion problems. A special attention is addressed to the conservation of mean-square dissipativity in the time integration of the spatially discretized problem, obtained by means of finite differences. The analysis highlights the conservative ability of stochastic $\theta$-methods and stochastic $\theta$-IMEX methods, emphasizing the roles of spatial and temporal stepsizes. A selection of numerical experiments is provided, confirming the theoretical expectations.