Efficient discretization of fractional SPDEs via Galerkin and exponential Euler methods

Abstract

This paper presents an efficient numerical method for solving stochastic partial differential equations (SPDEs) involving infinite-dimensional fractional Brownian motion. Fractional Brownian motion, characterized by a Hurst parameter $H\in (0,1)$, is a Gaussian process widely used to model various real-world phenomena. It is a fundamental model for representing persistent behaviors in physical systems, making it highly effective for simulating correlated noise in quantum mechanics, material science, and astrophysics. Our method combines the Galerkin approach for spatial discretization with the exponential Euler scheme for time discretization to approximate solutions for fractional SPDEs. Specifically, we consider Q-fractional Brownian motion with two distinct types of operator Q. This study aims to establish theoretical results regarding the convergence and error estimates of the proposed method, followed by validation through numerical experiments. The structure of the paper is designed to provide a comprehensive explanation of the problem, analyze spatial and temporal discretization errors, and include a numerical example in subsequent sections.