Finite difference methods for stochastic Helmholtz equation driven by white noise

In this paper, we propose two numerical methods for the stochastic Helmholtz equation driven by white noise. We obtain the approximate stochastic problem by approximating the white noise with piecewise constant process, provide some regularity of its solution and the truncation error between the approximate stochastic problem and the original problem. The limitation on the wave number $k$ of the finite difference method (FDM) is analyzed and a stochastic finite difference (SFD) scheme is presented. The error analysis shows that the stochastic finite difference method is efficient with a certain convergence rate. Numerical experiments are provided to examine our theoretical results.