Regularization effect of noise on fully discrete approximation for stochastic reaction-diffusion equation near sharp interface limit

TL;DR

This paper demonstrates that noise can regularize stochastic reaction-diffusion equations, enabling polynomial error bounds for numerical approximations near sharp interface limits, and introduces a novel fully discrete scheme with proven error bounds.

Contribution

It shows how noise regularizes stochastic phase field equations, establishing polynomial error bounds and proposing a new numerical scheme with rigorous analysis.

Findings

Error bounds depend polynomially on 1/ε near sharp interface limit

Proposed polynomial taming fully discrete scheme achieves these bounds

Method extends to other numerical approximations for semilinear SPDEs

Abstract

To capture and simulate geometric surface evolutions, one effective approach is based on the phase field methods. Among them, it is important to design and analyze numerical approximations whose error bound depends on the inverse of the diffuse interface thickness (denoted by $\frac 1\epsilon$) polynomially. However, it has been a long-standing problem whether such numerical error bound exists for stochastic phase field equations. In this paper, we utilize the regularization effect of noise to show that near sharp interface limit, there always exists the weak error bound of numerical approximations, which depends on $\frac 1\epsilon$ at most polynomially. To illustrate our strategy, we propose a polynomial taming fully discrete scheme and present novel numerical error bounds under various metrics. Our method of proof could be also extended to a number of other fully numerical approximations for semilinear stochastic partial differential equations (SPDEs).