Approximation of the invariant measure for stochastic Allen-Cahn equation via an explicit fully discrete scheme

TL;DR

This paper introduces an explicit fully discrete numerical scheme combining spectral Galerkin and tamed exponential Euler methods to approximate the invariant measure of the stochastic Allen-Cahn equation.

Contribution

It develops a novel numerical approach with weak error analysis over infinite time, enabling accurate approximation of the invariant measure.

Findings

The scheme is proven to be effective for the stochastic Allen-Cahn equation.

Weak error bounds are established using Malliavin calculus.

Numerical results demonstrate the scheme's accuracy in approximating the invariant measure.

Abstract

In this paper we propose an explicit fully discrete scheme to numerically solve the stochastic Allen-Cahn equation. The spatial discretization is done by a spectral Galerkin method, followed by the temporal discretization by a tamed accelerated exponential Euler scheme. Based on the time-independent boundedness of moments of numerical solutions, we present the weak error analysis in an infinite time interval by using Malliavin calculus. This provides a way to numerically approximate the invariant measure for the stochastic Allen-Cahn equation.