Full Discretization of Semilinear Stochastic Wave Equations Driven by Multiplicative Noise

Abstract

A fully discrete approximation of the semilinear stochastic wave equation driven by multiplicative noise is presented. A standard linear finite element approximation is used in space, and a stochastic trigonometric method is used for the temporal approximation. This explicit time integrator allows for mean-square error bounds independent of the space discretization and thus does not suffer from a step size restriction as in the often used Störmer--Verlet leapfrog scheme. Furthermore, it satisfies an almost trace formula (i.e., a linear drift of the expected value of the energy of the problem). Numerical experiments are presented and confirm the theoretical results.