Approximation of stochastic advection-diffusion equation using compact finite difference technique

In this paper, we propose a new method for solving the stochastic advection-diffusion equation of Ito type. In this work, we use a compact finite difference approximation for discretizing spatial derivatives of the mentioned equation and semi-implicit Milstein scheme for the resulting linear stochastic system of differential equation. The main purpose of this paper is the stability investigation of the applied method. finally, some numerical examples are provided to show the accuracy and efficiency of the proposed technique.