Higher Order Approximation for Stochastic Space Fractional Wave Equation Forced by an Additive Space-Time Gaussian Noise

Abstract

The infinitesimal generator (fractional Laplacian) of a process obtained by subordinating a killed Brownian motion catches the power-law attenuation of wave propagation. This paper studies the numerical schemes for the stochastic wave equation with fractional Laplacian as the space operator, the noise term of which is an infinite dimensional Brownian motion or fractional Brownian motion (fBm). Firstly, we establish the regularity of the mild solution of the stochastic fractional wave equation. Then a spectral Galerkin method is used for the approximation in space, and the space convergence rate is improved by postprocessing the infinite dimensional Gaussian noise. In the temporal direction, when the time derivative of the mild solution is bounded in the sense of mean-squared $L^p$-norm, we propose a modified stochastic trigonometric method, getting a higher strong convergence rate than the existing results, i.e., the time convergence rate is bigger than $1$. Particularly, for time discretization, the provided method can achieve an order of $2$ at the expenses of requiring some extra regularity to the mild solution. The theoretical error estimates are confirmed by numerical experiments.