An efficient explicit full-discrete scheme for strong approximation of stochastic Allen-Cahn equation

TL;DR

This paper introduces a new explicit full-discrete scheme for the stochastic Allen-Cahn equation, achieving higher convergence rates in time and space, with practical implementation and confirmed by numerical experiments.

Contribution

A novel explicit full-discrete scheme combining spectral Galerkin and nonlinearity-tamed exponential integrator methods for stochastic Allen-Cahn equations.

Findings

Convergence rate in time is twice as high as previous methods.

Error bounds are explicitly derived for both space and time discretizations.

Numerical results confirm theoretical convergence rates.

Abstract

In Becker and Jentzen (2019) and Becker et al. (2017), an explicit temporal semi-discretization scheme and a space-time full-discretization scheme were, respectively, introduced and analyzed for the additive noise-driven stochastic Allen-Cahn type equations, with strong convergence rates recovered. The present work aims to propose a different explicit full-discrete scheme to numerically solve the stochastic Allen-Cahn equation with cubic nonlinearity, perturbed by additive space-time white noise. The approximation is easily implementable, performing the spatial discretization by a spectral Galerkin method and the temporal discretization by a kind of nonlinearity-tamed accelerated exponential integrator scheme. Error bounds in a strong sense are analyzed for both the spatial semi-discretization and the spatio-temporal full discretization, with convergence rates in both space and time explicitly identified. It turns out that the obtained convergence rate of the new scheme is, in the temporal direction, twice as high as existing ones in the literature. Numerical results are finally reported to confirm the previous theoretical findings.