Quad-SAV scheme for gradient systems

Abstract

In this work we develop a new Scalar Auxiliary Variable scheme for general gradient flows. The main novelty of our scheme is that it always produces positive approximations for the nonlinear part of the free energy without any stabilizations. We prove unconditional stability of our scheme for general gradient systems and establish error estimates for the fully discrete scheme in the case of Allen–Cahn and Cahn–Hilliard equations. Several representative numerical examples are presented to demonstrate the accuracy and the efficiency of the proposed scheme.