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

TL;DR

This work tackles the numerical solution of the stochastic Burgers equation driven by additive cylindrical fractional Brownian noise with $H \in (\frac{1}{2},1)$. It develops a fully discrete scheme combining a spectral Galerkin spatial discretization with a nonlinear-tamed accelerated exponential Euler time integrator. The authors prove strong convergence by first establishing convergence in probability via stopping times and then obtaining moment bounds through exponential integrability of the stochastic convolution. Numerical experiments corroborate the theory and illustrate the impact of the Hurst parameter on convergence behavior, highlighting the method's potential for capturing long-time turbulence-like dynamics. The study advances numerical analysis for SPDEs with non-globally monotone drift and rough fractional-type noise, with implications for turbulence modeling and stochastic fluid dynamics.

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.

Figures (1)

• Figure 1: Left: $H=0.95$. Middle: $H=0.7$. Right: $H=0.55$.

Theorems & Definitions (23)

• Lemma 2.1
• Theorem 2.3
• Remark 2.4
• Lemma 3.1
• proof
• Corollary 3.2
• proof
• Lemma 3.3
• Theorem 3.4
• Lemma 3.5