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

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.