Strong convergence of a full discretization for stochastic wave equation with polynomial nonlinearity and addditive noise

Jianbo Cui, Jialin Hong, Lihai Ji, Liying Sun

Published 2019-09-02Version 1

In this paper, we propose a full discretization for $d$-dimensional stochastic wave equation with both polynomial nonlinearity and additive noise, which is based on the spectral Galerkin method in spatial direction and splitting averaged vector field method in temporal direction. Uniform bounds for high order derivatives of the continuous and the full discrete problem are obtained by constructing and analyzing Lyapunov functionals, which are crucial to derive the strong convergence rate of the proposed scheme. Furthermore, we show the exponential integrability properties of both the exact and numerical solutions, which are another key gradients to analyze the approximate error, due to the averaged energy preserving property of both the spatial and full discretization. Based on these regularity estimates and exponential integrability properties, the strong convergence order in both spatial and temporal direction are obtained.Numerical experiments are presented to verify these theoretical results.