Stability and convergence analysis for the implicit-explicit method to the Cahn-Hilliard equation

Abstract

Implicit-explicit methods have been successfully used for the efficient numerical simulation of phase field problems such as the Cahn-Hilliard equation or thin film type equations. Due to the lack of maximum principle and stiffness caused by the effect of small dissipation coefficient, most existing theoretical analysis relies on adding additional stabilization terms, mollifying the nonlinearity or introducing auxiliary variables which implicitly either changes the structure of the problem or trades accuracy for stability in a subtle way. In this work, we introduce a robust theoretical framework to analyze directly the stability and accuracy of the standard implicit-explicit approach without stabilization or any other modification. We take the Cahn-Hilliard equation as a model case and provide a rigorous stability and convergence analysis for the original semi-discrete scheme under certain time step constraints. These settle several questions which have been open since the work of Chen and Shen [Comput. Phys. Comm. 108 (1998), pp. 147–158].