Error Analysis of a SAV-Based Numerical Scheme for a Stochastic Viscous Cahn-Hilliard Equation with Hyperbolic Relaxation and Multiplicative Noise

Abstract We develop and analyze a scalar auxiliary variable (SAV)-based numerical scheme for the stochastic viscous Cahn-Hilliard equation with multiplicative noise. This equation models phase separation under random fluctuations and presents significant analytical and computational challenges due to the strong coupling between the nonlinear chemical potential and state-dependent noise. By introducing a carefully constructed auxiliary variable, the scheme reformulates the system into an equivalent form that enables a linear discretization while preserving a discrete version of the stochastic energy dissipation law. This ensures that the underlying gradient flow structure is faithfully captured despite the presence of multiplicative noise. To address the analytical difficulties arising from stochastic forcing and nonlinear potentials, we derive new regularity estimates for the auxiliary variable and rigorously prove strong convergence of the semi-discrete numerical solution. Numerical experiments confirm the accuracy, stability, and robustness of the proposed method. To the best of our knowledge, this work presents the first rigorous convergence analysis of an SAV-type scheme for the stochastic Cahn-Hilliard equation with multiplicative noise, thereby filling a key gap in the structure-preserving numerical analysis of nonlinear stochastic gradient flows.