Strong convergence rate of an exponentially integrable scheme for stochastic nonlinear wave equation

Abstract

In this paper, we present an exponentially integrable numerical method for stochastic wave equation with cubic nonlinearity and additive space-time noise. We first apply the spectral Galerkin method to discretize the original equation and show that this spatial discretization possesses an energy evolution law and certain exponential integrability property. Then the exponential integrability property of the exact solution is deduced by proving the strong convergence of the semi-discretization. To propose a fully discrete numerical method which could inherit both the energy evolution law and the exponential integrability, we use the splitting technique and averaged vector field method in the temporal direction. Combining these structure-preserving properties with regularity estimates of the exact and the numerical solutions, we obtain the strong convergence rate of the proposed scheme. Finally, numerical experiments verify the theoretical results.