Improving Numerical Error Bounds Near Sharp Interface Limit for Stochastic Reaction-Diffusion Equations

TL;DR

This paper develops numerical methods for stochastic reaction-diffusion equations that achieve error bounds with polynomial dependence on the interface thickness parameter, addressing a key challenge in simulating complex geometric surface evolutions.

Contribution

It introduces a novel approach using the asymptotic log-Harnack inequality to obtain polynomial error bounds for fully discrete schemes in stochastic reaction-diffusion equations.

Findings

Established polynomial weak error bounds with respect to interface thickness

Applied spectral Galerkin and tamed Euler schemes with explicit error dependence

Overcame exponential growth issues in error analysis

Abstract

In the study of geometric surface evolutions, stochastic reaction-diffusion equation provides a powerful tool for capturing and simulating complex dynamics. A critical challenge in this area is developing numerical approximations that exhibit error bounds with polynomial dependence on $\vv^{-1}$, where the small parameter $\vv>0$ represents the diffuse interface thickness. The existence of such bounds for fully discrete approximations of stochastic reaction-diffusion equations remains unclear in the literature. In this work, we address this challenge by leveraging the asymptotic log-Harnack inequality to overcome the exponential growth of $\vv^{-1}$. Furthermore, we establish the numerical weak error bounds under the truncated Wasserstein distance for the spectral Galerkin method and a fully discrete tamed Euler scheme, with explicit polynomial dependence on $\vv^{-1}$.