A fast explicit time-splitting spectral scheme for the viscous Cahn-Hilliard equation with nonlocal diffusion operator

Abstract

The viscous Cahn-Hilliard equation (VCH) is a parameter-dependent model encapsulated Cahn-Hilliard (CH) model and constrained Allen-Cahn (CAC) model. In this paper, we propose a fast explicit time-splitting spectral scheme for the VCH with nonlocal diffusion operator. The numerical scheme is constructed based on the second-order symmetric time-splitting method, which results in a splitting of the original problem into nonlocal linear part and nonlinear part, respectively. The nonlocal linear subproblem can be solved exactly by applying the Fourier spectral discretization in space and thus has no stability restriction on the time-step size. The nonlinear subproblem is solved via a second-order strong stability preserving Runge–Kutta method. The new scheme is time symmetric, fully explicit and conserves the discrete mass exactly. Theoretical analysis shows that the stability of the scheme depends on both the viscosity coefficient and the range of nonlocal interactions. In addition, a rigorously convergence analysis of the scheme is established, and the convergence rate is proved to be second-order in time and spectral-order in space, respectively. Four numerical experiments are performed to demonstrate the effectiveness of the proposed scheme.