Energy stability of exponential time differencing schemes for the nonlocal Cahn‐Hilliard equation

Abstract

The nonlocal Cahn‐Hilliard equation has attracted much attention these years. Despite the advantage of describing more practical phenomena for modeling phase transitions of microstructures in materials, the nonlocal operator in the equation brings a lot of extra computational costs compared with the local Cahn‐Hilliard equation. Thus high order time integration schemes are needed in numerical simulations. In this paper, we propose two classes of exponential time differencing (ETD) schemes for solving the nonlocal Cahn‐Hilliard equation. We first use the Fourier collocation method to discretize the spatial domain, and then the ETD‐based multistep and Runge‐Kutta schemes are adopted for the time integration. In particular, some specific multistep and Runge‐Kutta schemes up to fourth order are constructed. We rigorously establish the energy stabilities of the multistep schemes up to fourth order and the second order Runge‐Kutta scheme, which show that the first order ETD and the second order Runge‐Kutta schemes unconditionally decrease the original energy. We also theoretically prove the mass conservations of the proposed schemes. Several numerical experiments in two and three dimensions are carried out to test the temporal convergence rates of the schemes and to verify their mass conservations and energy stabilities. The long time simulations of coarsening dynamics are also performed to verify the power law for the energy decay.