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

Friedrich–Alexander Universit¨at Erlangen–Nürnberg, Cauerstraße 11, 91058 Erlangen, Germany

^* Corresponding author: stefan.metzger@fau.de

Received: 17 March 2025
Accepted: 14 June 2025

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. Metzger [IMA J. Numer. Anal. (2024)]) 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 deduce strong rates of convergence using only the standard regularity results that can be established for solutions to the stochastic Allen–Cahn equation. In particular, we show that the proposed linear scheme exhibits the same optimal rates of convergence that were established in Majee and Prohl [Comput. Methods Appl. Math. 18 (2018) 297–311] for a nonlinear structure preserving scheme. Finally, we provide numerical simulations verifying our theoretical findings and discuss the advantages and shortcomings of the proposed scheme.