A modified semi--implict Euler-Maruyama Scheme for finite element discretization of SPDEs with additive noise
It provides a more accurate numerical method for solving stochastic partial differential equations, which is important for researchers in computational mathematics and physics.
The paper introduces a modified semi-implicit Euler-Maruyama scheme for finite element discretization of SPDEs with additive noise, proving better convergence properties than the standard method through analysis and numerical results for linear and nonlinear equations.
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.