Numerical ergodicity of a full‐discrete exponential Euler scheme for stochastic Burgers–Huxley equation

Abstract

In this paper, we mainly complete the research on approximate study of invariant measures for one‐dimensional stochastic Burgers–Huxley equations driven by additive Gaussian noise. A fully discretization scheme is proposed based on the spectral Galerkin method and the exponential Euler scheme in both spatial and temporal directions, respectively. Moreover, we establish the unique ergodicity for both the spatial semi‐discretization and the full discretization of numerical solutions, along with providing rigorous error estimations for invariant measures under approximate conditions. Some numerical experiments are presented to verify the theoretical findings.