Analysis of spectral Galerkin method with higher order time discretization for the nonlinear stochastic Fisher's type equation driven by multiplicative noise

Abstract

This paper primarily focuses on developing a high-order-in-time spectral Galerkin approximation method for nonlinear stochastic Fisher's type equations driven by multiplicative noise. For this reason, we first design an improved discretization scheme in time based on the Milstein method, and then propose a spectral Galerkin approximation method in space. We analyze the $H^1$ stability and $L^2$ stability estimations for numerical solutions of the proposed fully discrete spectral Galerkin approximation formulation under reasonable assumptions about the multiplicative noise function $g\left(u\right)$ and the nonlinear multiplicative function $f\left(u\right(x,t\left)\right)$. Additionally, we achieve nearly optimal error convergence orders in both space and time. Especially, the time convergence order almost reaches 1 under the $L^2$ norm. Finally, several numerical experiments are carried out for the stochastic Fisher's type models to validate all the theoretical results, and it can also be seen that the numerical results are consistent with the physical properties of the Fisher's equation.