Numerical methods for hyperbolic SPDEs: a Wiener chaos approach

Abstract

In this paper we propose a novel numerical scheme based on the Wiener chaos expansion for solving hyperbolic stochastic partial differential equations (PDEs). Through the Wiener chaos expansion the stochastic PDE is reduced to an infinite hierarchy of deterministic PDEs which is then truncated to a finite system of PDEs, that can be addressed by standard techniques. A priori and a posteriori convergence results for the method are provided. The proposed method is applied to solve the stochastic wave equation with additive noise and the stochastic Klein–Gordon wave equation with multiplicative noise and the results are compared to those derived by the Monte Carlo method. The main advantages of the proposed scheme is that it provides almost identical results and is significantly faster than the Monte Carlo simulation method, providing a convenient way to compute numerically the statistical moments of the solution.