Higher order approximation for stochastic wave equation

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.

arXiv comment29 pages, 2 figures

References in corpus (8)

• Ergodic Properties of Fractional Brownian-Langevin Motion
• Boundary problems for the fractional and tempered fractional operators
• Galerkin Finite Element Approximations for Stochastic Space-Time Fractional Wave Equations
• Higher order strong approximations of semilinear stochastic wave equation with additive space-time white noise
• Approximating Stochastic Evolution Equations with Additive White and Rough Noises
• Finite element approximations for second order stochastic differential equation driven by fractional Brownian motion
• Numerical approximation for fractional diffusion equation forced by a tempered fractional Gaussian noise
• Strong convergence order for the scheme of fractional diffusion equation driven by fractional Gaussion noise

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• Strong convergence order for the scheme of fractional diffusion equation driven by fractional Gaussion noise
• An inverse random source problem for the time-space fractional diffusion equation driven by fractional Brownian motion
• Difference methods for time discretization of stochastic wave equation
• High-accuracy time discretization of stochastic fractional diffusion equation
• A unified convergence analysis for the fractional diffusion equation driven by fractional Gaussion noise with Hurst index $H\in (0,1)$