Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations

TL;DR

This paper establishes the first known strong convergence rates for full-discrete numerical schemes approximating stochastic Allen-Cahn equations driven by space-time white noise, including sharp error bounds.

Contribution

It provides the first proof of strong convergence rates and error bounds for full-discrete approximations of SPDEs with superlinear nonlinearities like the stochastic Allen-Cahn equation.

Findings

Proved strong convergence rates for full-discrete schemes.

Established lower bounds showing rates are essentially sharp.

Extended analysis to space-time white noise driven SPDEs.

Abstract

The scientific literature contains a number of numerical approximation results for stochastic partial differential equations (SPDEs) with superlinearly growing nonlinearities but, to the best of our knowledge, none of them prove strong or weak convergence rates for full-discrete numerical approximations of space-time white noise driven SPDEs with superlinearly growing nonlinearities. In particular, in the scientific literature there exists neither a result which proves strong convergence rates nor a result which proves weak convergence rates for full-discrete numerical approximations of stochastic Allen-Cahn equations. In this article we bridge this gap and establish strong convergence rates for full-discrete numerical approximations of space-time white noise driven SPDEs with superlinearly growing nonlinearities such as stochastic Allen-Cahn equations. Moreover, we also establish lower bounds for strong temporal and spatial approximation errors which demonstrate that our strong convergence rates are essentially sharp and can, in general, not be improved.