Strong approximation for fractional wave equation forced by fractional Brownian motion with Hurst parameter $H\in(0,\frac{1}{2})$

TL;DR

This paper develops a numerical scheme for fractional stochastic wave equations driven by fractional Brownian motion with Hurst parameter H in (0, 1/2), achieving a strong convergence rate of 1/2 + H and demonstrating its effectiveness through numerical experiments.

Contribution

It introduces a novel time discretization method for fractional stochastic wave equations driven by fractional Brownian motion, with proven convergence rates and improved analysis techniques.

Findings

Convergence rate of 1/2 + H for the proposed scheme.

Effective approximation of stochastic convolution using integration by parts.

Numerical experiments confirm theoretical convergence and efficiency.

Abstract

We consider the time discretization of fractional stochastic wave equation with Gaussian noise, which is negatively correlated. Major obstacles to design and analyze time discretization of stochastic wave equation come from the approximation of stochastic convolution with respect to fractional Brownian motion. Firstly, we discuss the smoothing properties of stochastic convolution by using integration by parts and covariance function of fractional Brownian motion. Then the regularity estimates of the mild solution of fractional stochastic wave equation are obtained. Next, we design the time discretization of stochastic convolution by integration by parts. Combining stochastic trigonometric method and approximation of stochastic convolution, the time discretization of stochastic wave equation is achieved. We derive the error estimates of the time discretization. Under certain assumptions, the strong convergence rate of the numerical scheme proposed in this paper can reach $\frac{1}{2}+H$. Finally, the convergence rate and computational efficiency of the numerical scheme are illustrated by numerical experiments.