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

Understand

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.