Strong convergence rates for full-discrete approximations of the stochastic Allen-Cahn equations on 2D torus

TL;DR

This paper develops and analyzes a full discretization scheme for the stochastic Allen-Cahn equation on a 2D torus, achieving specific convergence rates in space and time.

Contribution

It introduces a combined tamed exponential Euler and spectral Galerkin method for the stochastic Allen-Cahn equation, providing explicit convergence rates.

Findings

Spatial convergence rate of α−δ for α in (0,1/3)

Temporal convergence rate of α/6−δ for α in (0,1/3)

Convergence in the space of negative Hölder continuous functions

Abstract

In this paper we construct space-time full discretizations of stochastic Allen-Cahn equations driven by space-time white noise on 2D torus. The approximations are implemented by tamed exponential Euler discretization in time and spectral Galerkin method in space. We finally obtain the convergence rates with the spatial order of $\alpha-\delta$ and the temporal order of ${\alpha}/{6}-\delta$ in $\mathcal C^{-\alpha}$ for $\alpha\in(0,1/3)$ and $\delta>0$ arbitrarily small.