Approximation of the invariant measure for stochastic Allen-Cahn equation via an explicit fully discrete scheme
Provides a numerical method to approximate invariant measures for a class of stochastic PDEs, addressing a known challenge in stochastic dynamics.
The paper develops an explicit fully discrete scheme for the stochastic Allen-Cahn equation and proves its weak convergence in infinite time, enabling numerical approximation of the invariant measure.
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.