Arbitrarily High Order and Fully Discrete Extrapolated RK-SAV/DG Schemes for Phase-field Gradient Flows

In this paper, we construct and analyze a fully discrete method for phase-field gradient flows, which uses extrapolated Runge-Kutta with scalar auxiliary variable (RK-SAV) method in time and discontinuous Galerkin (DG) method in space. We propose a novel technique to decouple the system, after which only several elliptic scalar problems with constant coefficients need to be solved independently. Discrete energy decay property of the method is proved for gradient flows. The scheme can be of arbitrarily high order both in time and space, which is demonstrated rigorously for the Allen-Cahn equation and the Cahn-Hilliard equation. More precisely, optimal L-2-error bound in space and qth-order convergence rate in time are obtained for q-stage extrapolated RK-SAV/DG method. Several numerical experiments are carried out to verify the theoretical results.