Unconditionally stable exponential integrator schemes for the 2D Cahn-Hilliard equation

Abstract

Phase field models are gradient flows with their energy naturally dissipating in time. In order to preserve this property, many numerical schemes have been well-studied. In this paper we consider a well-known method, namely the exponential integrator method (EI). In the literature a few works studied several EI schemes for various phase field models and proved the energy dissipation by either requiring a strong Lipschitz condition on the nonlinear source term or certain $L^\infty$ bounds on the numerical solutions (maximum principle). However for phase field models such as the (non-local) Cahn-Hilliard equation, the maximum principle no longer exists. As a result, solving such models via EI schemes remains open for a long time. In this paper we aim to give a systematic approach on applying EI-type schemes to such models by solving the Cahn-Hilliard equation with a first order EI scheme and showing the energy dissipation. In fact second order EI schemes can be handled similarly and we leave the discussion in a subsequent paper. To our best knowledge, this is the first work to handle phase field models without assuming any strong Lipschitz condition or $L^\infty$ boundedness. Furthermore, we will analyze the $L^2$ error and present some numerical simulations to demonstrate the dynamics.