Strong convergence of a fully discrete scheme for stochastic Burgers equation with fractional-type noise

TL;DR

This paper develops and analyzes a fully discrete numerical scheme for the stochastic Burgers equation driven by fractional Brownian motion, proving its strong convergence and boundedness of moments.

Contribution

It introduces a nonlinear-tamed accelerated exponential Euler method for fractional noise and proves its strong convergence with moment bounds.

Findings

The scheme is strongly convergent in probability.

Boundedness of moments for semi-discrete and full-discrete approximations.

Exponential integrability of the stochastic convolution is established.

Abstract

We investigate numerical approximations for the stochastic Burgers equation driven by an additive cylindrical fractional Brownian motion with Hurst parameter $H \in (\frac{1}{2}, 1)$. To discretize the continuous problem in space, a spectral Galerkin method is employed, followed by the presentation of a nonlinear-tamed accelerated exponential Euler method to yield a fully discrete scheme. By showing the exponential integrability of the stochastic convolution of the fractional Brownian motion, we present the boundedness of moments of semi-discrete and full-discrete approximations. Building upon these results and the convergence of the fully discrete scheme in probability proved by a stopping time technique, we derive the strong convergence of the proposed scheme.