Approximation of stochastic advection diffusion equations with finite difference scheme

In this paper, a high-order and conditionally stable stochastic difference scheme is proposed for the numerical solution of Ito^ stochastic advection diffusion equation with one dimensional white noise process. We applied a finite difference approximation of fourth-order for discretizing space spatial derivative of this equation. The main properties of deterministic difference schemes, i.e. consistency, stability and convergence, are developed for the stochastic case. It is shown through analysis that the proposed scheme has these properties. Numerical results are given to demonstrate the computational efficiency of the stochastic scheme.