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

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 Itô 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.

💡 Research Summary

This paper introduces a novel semi‑implicit time‑discretization scheme for the stochastic Cahn–Hilliard equation with multiplicative noise, based on a stochastic scalar auxiliary variable (SSAV) formulation. The authors first rewrite the original fourth‑order stochastic PDE into an equivalent system that couples the order parameter φ with a scalar auxiliary variable r(t)=√Eₚ(φ(t)), where Eₚ denotes the p‑th power of the Ginzburg–Landau energy. This reformulation isolates the nonlinear drift term as r(t)·F′(φ) and separates the linear fourth‑order operator –A² (the Laplacian squared with homogeneous Dirichlet boundary conditions).

On each time interval