Local Discontinuous Galerkin Method and High Order Semi-Implicit Scheme for the Phase Field Crystal Equation

Abstract. In this paper, we present a local discontinuous Galerkin (LDG) method and two unconditionally energy stable schemes for the phase field crystal (PFC) equation. The semidiscrete energy stability of the LDG method is proved first. The PFC equation is a sixth order nonlinear partial differential equation (PDE), which leads to the severe time step restriction (Δt = O(Δx6)) of explicit time discretization methods to maintain stability. Due to this, we introduce semi-implicit first order and second order time discretization methods which are based on the convex splitting principle of a discrete energy and prove the corresponding unconditional energy stabilities. To improve the temporal accuracy, the spectral deferred correction (SDC) method and a high order semi-implicit Runge–Kutta method combining with the first–order convex splitting method are adopted for the PFC equation with constant and degenerate mobility, respectively. The equations at the implicit time level are nonlinear and we employ an efficient nonlinear multigrid solver to solve the equations. In particular, we show the multigrid solver has optimal complexity numerically. Numerical results are also given to illustrate that the combination of the LDG method for spatial approximation, SDC, and high order semi-implicit time marching methods with the multigrid solver provides an efficient and practical approach when solving the PFC equation.