On strong convergence of a fully discrete scheme for solving stochastic strongly damped wave equations

Abstract

This article develops an efficient fully discrete scheme for a stochastic strongly damped wave equation (SSDWE) driven by an additive noise and presents its error estimates in the strong sense. We use the truncated spectral expansion of the noise to get an approximate equation and prove its regularity. Then we establish a spatio-temporal discretization of the approximate equation by a finite element method in space and an exponential trapezoidal scheme in time. We prove that the combination can derive higher strong convergence order in time than the use of the piecewise approximation of the noise and the exponential Euler scheme or the implicit Euler scheme in time. Particularly, the temporal strong convergence order of the fully discrete scheme reaches for the one-dimensional space-time white noise, which overcomes the order barrier one. Moreover, we allow the covariance operator of the noise to be noncommutative with the Dirichlet Laplacian, which weakens the common assumptions on the noise in the literature. Finally, some numerical experiments in different spatial dimensions are presented to support our theoretical findings. By means of the piecewise spectral approximation of the noise, a piecewise version of the fully discrete scheme is constructed to fulfill a long-time simulation.