Strong Approximation of Stochastic Allen-Cahn Equation with White Noise

TL;DR

This paper proves an optimal strong convergence rate for a fully discrete numerical scheme solving the stochastic Allen-Cahn equation driven by white noise, using transformation, spectral Galerkin, and backward Euler methods.

Contribution

It introduces a novel approach transforming the stochastic equation into a regular random equation and establishes the convergence rate for the discretization scheme.

Findings

Optimal strong convergence rate achieved.

Numerical experiments confirm theoretical results.

Method applicable to stochastic PDEs with white noise.

Abstract

We establish an optimal strong convergence rate of a fully discrete numerical scheme for second order parabolic stochastic partial differential equations with monotone drifts, including the stochastic Allen-Cahn equation, driven by an additive space-time white noise. Our first step is to transform the original stochastic equation into an equivalent random equation whose solution possesses more regularity than the original one. Then we use the backward Euler in time and spectral Galerkin in space to fully discretize this random equation. By the monotone assumption, in combination with the factorization method and stochastic calculus in martingale-type 2 Banach spaces, we derive a uniform maximum norm estimation and a H\"older-type regularity for both stochastic and random equations. Finally, the strong convergence rate of the proposed fully discrete scheme under the $l_t^\infty L^2_\omega L^2_x \cap l_t^q L^q_\omega L^q_x$-norm is obtained. Several numerical experiments are carried out to verify the theoretical result.