A modified semi--implict Euler-Maruyama Scheme for finite element discretization of SPDEs with additive noise

TL;DR

This paper introduces a modified semi-implicit Euler-Maruyama scheme for SPDEs with additive noise, demonstrating improved convergence in finite element discretizations through theoretical analysis and numerical experiments.

Contribution

A novel modified scheme for SPDEs that enhances convergence properties over standard methods, applicable to finite element discretizations.

Findings

Proven convergence in the $L^{2}$ norm for specific SPDEs.

Numerical results show improved convergence compared to standard methods.

Effective for both linear and nonlinear stochastic PDEs.

Abstract

We consider the numerical approximation of a general second order semi--linear parabolic stochastic partial differential equation (SPDE) driven by additive space-time noise. We introduce a new modified scheme using a linear functional of the noise with a semi--implicit Euler--Maruyama method in time and in space we analyse a finite element method (although extension to finite differences or finite volumes would be possible). We prove convergence in the root mean square $L^{2}$ norm for a diffusion reaction equation and diffusion advection reaction equation. We present numerical results for a linear reaction diffusion equation in two dimensions as well as a nonlinear example of two-dimensional stochastic advection diffusion reaction equation. We see from both the analysis and numerics that the proposed scheme has better convergence properties than the standard semi--implicit Euler--Maruyama method.