Analytic Error Estimates in Semi-Discretization of the Stochastic Cahn-Hilliard Equation

This study examines the semi-discretization of the stochastic Cahn-Hilliard equation, which represents phase separation phenomena in multi-component mixtures affected by random fluctuations. An analytic error estimate in the L² norm is derived for the solution of the continuous stochastic equation compared to its semi-discretized approximation. The finite difference method is utilized for spatial discretization, ensuring the stability and convergence properties of the numerical scheme. We support our theoretical findings with numerical experiments that confirm the established error estimates and underscore the implications for simulating phase separation in noisy environments. The primary finding indicates that the error diminishes as spatial resolution increases, contingent upon specific smoothness and regularity conditions applied to the initial data and noise. The study presents numerical experiments to validate theoretical findings and examines the implications of results for simulating phase separation in noisy environments. This study enhances the understanding of the dynamics of stochastic phase separation and establishes a solid framework for the advancement of numerical methods for stochastic partial differential equations (SPDEs).