A scalar auxiliary variable-based semi-implicit scheme for stochastic Cahn--Hilliard equation

TL;DR

This paper introduces a stable semi-implicit numerical scheme for the stochastic Cahn--Hilliard equation using a scalar auxiliary variable approach, achieving optimal convergence and energy preservation.

Contribution

The paper develops a novel SSAV-based semi-implicit scheme for stochastic Cahn--Hilliard equations, incorporating Itô corrections and proving optimal convergence and energy law preservation.

Findings

Achieves strong convergence order of 0.5 for the scheme.

Demonstrates energy asymptotic preservation.

Validates theoretical results through numerical experiments.

Abstract

In this paper, we present a novel semi-implicit numerical scheme for the stochastic Cahn--Hilliard equation driven by multiplicative noise. By reformulating the original equation into an equivalent stochastic scalar auxiliary variable (SSAV) system, our method enables an efficient and stable treatment of polynomial nonlinearities in a semi-implicit fashion. In order to accurately capture the impact of stochastic perturbations, we carefully incorporate It\^o correction terms into the SSAV approximation. Leveraging the smoothing properties of the underlying semigroup and the $H^{-1}$-dissipative structure of the nonlinear term, we establish the optimal strong convergence order of one-half for the proposed scheme in the trace-class noise case. Moreover, we show that the modified SAV energy asymptotically preserves the energy evolution law. Finally, numerical experiments are provided to validate the theoretical results and to explore the influence of noise near the sharp-interface limit.