Strong error estimates for a fully discrete SAV scheme for the stochastic Allen--Cahn equation with multiplicative noise

TL;DR

This paper develops and analyzes a fully discrete scalar auxiliary variable scheme for the stochastic Allen--Cahn equation with multiplicative noise, achieving optimal strong convergence rates and demonstrating stability and efficiency.

Contribution

It introduces an augmented scalar auxiliary variable scheme that extends applicability to stochastic PDEs with poor temporal regularity and proves optimal convergence rates.

Findings

The scheme achieves the same optimal convergence rates as nonlinear structure-preserving methods.

Numerical simulations confirm the theoretical convergence rates.

The method is unconditionally stable and efficient for stochastic Allen--Cahn equations.

Abstract

We investigate the numerical approximation of the stochastic Allen--Cahn equation with multiplicative noise on a periodic domain. The considered scheme uses a recently proposed augmented variant of scalar auxiliary variable method for the discretization with respect to time. While scalar auxiliary variable methods in general allow for the construction of unconditionally stable, efficient linear schemes, the considered augmented version (cf. [S. Metzger, 2024, IMA J. Numer. Anal.]) additionally compensates for the typically poor temporal regularity of solutions to stochastic partial differential equations and hence extends the range of applicability of the scheme. In this work, we establish strong rates of convergence and show that the proposed linear scheme exhibits the same optimal rates of convergence that were established in [A. K. Majee & A. Prohl, 2018, Comput. Methods Appl. Math.] for a nonlinear structure preserving scheme. Finally, we provide numerical simulations verifying our theoretical findings.