A convergent stochastic scalar auxiliary variable method

TL;DR

This paper introduces an extended scalar auxiliary variable method for stochastic PDEs, enabling unconditionally stable, higher-order, linear schemes with proven convergence and practical numerical demonstrations.

Contribution

It extends the scalar auxiliary variable approach to stochastic PDEs by incorporating higher order terms, resulting in a stable, convergent, and practical numerical scheme.

Findings

The scheme is unconditionally energy stable.

Convergence to strong solutions is rigorously proven.

Numerical simulations confirm the scheme's effectiveness.

Abstract

We discuss an extension of the scalar auxiliary variable approach, which was originally introduced by Shen et al. ([Shen, Xu, Yang, J. Comput. Phys., 2018]) for the discretization of deterministic gradient flows. By introducing an additional scalar auxiliary variable, this approach allows to derive a linear scheme, while still maintaining unconditional stability. Our extension augments the approximation of the evolution of this scalar auxiliary variable with higher order terms, which enables its application to stochastic partial differential equations. Using the stochastic Allen--Cahn equation as a prototype for nonlinear stochastic partial differential equations with multiplicative noise, we propose an unconditionally energy stable, linear, fully discrete finite element scheme based on our augmented scalar auxiliary variable method. Recovering a discrete version of the energy estimate and establishing Nikolskii estimates with respect to time, we are able to prove convergence of discrete solutions towards pathwise unique martingale solutions by applying Jakubowski's generalization of Skorokhod's theorem. A generalization of the Gy\"ongy--Krylov characterization of convergence in probability to quasi-Polish spaces finally provides convergence of fully discrete solutions towards strong solutions of the stochastic Allen--Cahn equation. Finally, we present numerical simulations underlining the practicality of the scheme and the importance of the introduced augmentation terms.