The high-order approximation of SPDEs with multiplicative noise via amplitude equations

TL;DR

This paper develops high-order approximation methods for SPDEs with multiplicative noise using amplitude equations, improving convergence rates by incorporating second-order amplitude equations and validating results with stochastic Allen-Cahn equations.

Contribution

It introduces a second-order amplitude equation approach for better approximation of SPDE solutions, surpassing first-order methods in convergence accuracy.

Findings

Improved convergence rate for SPDE approximations.

Successful application to stochastic Allen-Cahn equation.

Numerical validation supporting theoretical results.

Abstract

The emphasis of this paper is to investigate the high-order approximation of a class of SPDEs with cubic nonlinearity driven by multiplicative noise with the help of the amplitude equations. The highlight of our work is that we improve the convergence rate between the real solutions and the approximate ones. Precisely, previous results often focused on deriving the approximate solutions via the first-order amplitude equations. However, the approximate solutions are constructed by the first-order amplitude equations and the second-order ones in this paper. And, we rigorously prove that such approximate solutions enjoy improved convergence property. In order to illustrate this demonstration more intuitively, we apply our main theorem to stochastic Allen-Cahn equation, and provide numerical analysis.