A High Order Adaptive Time-Stepping Strategy and Local Discontinuous Galerkin Method for the Modified Phase Field Crystal Equation (Communications in Computational Physics, 2018)
In this paper, we will develop a first order and a second order convex splitting,and a first order linear energy stable fully discrete local discontinuous Galerkin(LDG) methods for the modified phase field crystal (MPFC) equation. In which, thefirst order linear scheme is based on the invariant energy quadratization approach.The MPFC equation is a damped wave equation, and to preserve an energy stability,it is necessary to introduce a pseudo energy, which all increase the difficulty of constructingnumerical methods comparing with the phase field crystal (PFC) equation.Due to the severe time step restriction of explicit time marching methods, we introducethe first order and second order semi-implicit schemes, which are proved to be unconditionallyenergy stable. In order to improve the temporal accuracy, the semi-implicitspectral deferred correction (SDC) method combining with the first order convex splittingscheme is employed. Numerical simulations of the MPFC equation always needlong time to reach steady state, and then adaptive time-stepping method is necessaryand of paramount importance. The schemes at the implicit time level are linear or nonlinearand we solve them by multigrid solver. Numerical experiments of the accuracyand long time simulations are presented demonstrating the capability and efficiencyof the proposed methods, and the effectiveness of the adaptive time-stepping strategy.
Keywords: Adaptive time-stepping; local discontinuous Galerkin method; modified phase field crystal equation; convex splitting; pseudo energy; unconditionally energy stable; spectral deferred correction.