A family of structure-preserving exponential time differencing Runge–Kutta schemes for the viscous Cahn–Hilliard equation

Abstract

We consider the numerical approximations of the viscous Cahn–Hilliard equation with either the Ginzburg–Landau polynomial potential or Flory–Huggins logarithmic potential. One challenge in solving such a fourth-order-in-space system is to develop accurate temporal discretization to preserve the energy stability, mass conservation, and maximum principle. We resolve this issue by developing a family of first- and second-order time marching schemes based on exponential time differencing Runge–Kutta (ETDRK) methods. We prove that the proposed schemes unconditionally preserve the three characteristics of the viscous Cahn–Hilliard equation. Because of the preservation of maximum principle, an error estimate in the infinity-norm is derived for the fully discrete system. Various numerical experiments are performed to verify the theoretical results and demonstrate the excellent performance of the proposed schemes.