The scalar auxiliary variable (SAV) approach for gradient flows

Abstract

We propose a new approach, which we term as scalar auxiliary variable (SAV) approach, to construct efficient and accurate time discretization schemes for a large class of gradient flows. The SAV approach is built upon the recently introduced IEQ approach. It enjoys all advantages of the IEQ approach but overcomes most of its shortcomings. In particular, the SAV approach leads to numerical schemes that are unconditionally energy stable and extremely efficient in the sense that only decoupled equations with constant coefficients need to be solved at each time step. The scheme is not restricted to specific forms of the nonlinear part of the free energy, so it applies to a large class of gradient flows. Numerical results are presented to show that the accuracy and effectiveness of the SAV approach over the existing methods.