Approximation of Stochastic Advection–Diffusion Equations with Predictor-Corrector Methods

Abstract

This paper focuses on the development and analysis of predictor-corrector methods for solving stochastic advection–diffusion equations. These equations play a significant role in modeling various physical phenomena where uncertainties are present. We first derive the predictor-corrector schemes and analyze their stability, consistency, and convergence in the mean-square sense. The results indicate that under appropriate conditions, the proposed methods maintain stability and exhibit desirable convergence properties. Additionally, we present a detailed comparison of the stability of these methods with some other existing numerical approaches. Numerical experiments validate the theoretical findings and demonstrate the accuracy and robustness of the methods. Although this study is primarily concerned with linear stochastic partial differential equations, we also discuss the potential extension of these methods to nonlinear cases, providing a foundation for future research in this direction.