Convergence Analysis of Exponential Time Differencing Schemes for the Cahn-Hilliard Equation

Abstract

In this paper, we rigorously prove the convergence of fully discrete first- and second-order exponential time differencing schemes for solving the Cahn-Hilliard equation. Our analyses mainly follow the standard procedure with the consistency and stability estimates for numerical error functions, while the technique of higher-order consistency analysis is adopted in order to obtain the uniform L^∞ boundedness of the numerical solutions under some moderate constraints on the time step and spatial mesh sizes. This paper provides a theoretical support for numerical analysis of exponential time differencing and other related numerical methods for phase field models, in which an assumption on the uniform L^∞ boundedness is usually needed.