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

摘要

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