Energy-Decaying Extrapolated RK--SAV Methods for the Allen--Cahn and Cahn--Hilliard Equations

We construct and analyze a class of extrapolated and linearized Runge--Kutta (RK) methods, which can be of arbitrarily high order, for the time discretization of the Allen--Cahn and Cahn--Hilliard phase field equations, based on the scalar auxiliary variable (SAV) formulation. We prove that the proposed q-stage RK--SAV methods have qth-order convergence in time and satisfy a discrete version of the energy decay property. Numerical examples are provided to illustrate the discrete energy decay property and accuracy of the proposed methods.