Strong convergence of a fully discrete scheme for stochastic Burgers equation with fractional-type noise
Provides a rigorous convergence result for numerical approximation of a stochastic PDE with fractional noise, which is a challenging problem in stochastic analysis.
The paper proves strong convergence of a fully discrete scheme for the stochastic Burgers equation driven by fractional Brownian motion with Hurst parameter > 1/2, using a spectral Galerkin method and nonlinear-tamed accelerated exponential Euler method.
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.