A class of unconditionally energy stable relaxation schemes for gradient flows

In this paper, a novel class of unconditionally energy stable schemes are constructed for solving gradient flow models by combining the relaxed scalar auxiliary variable (SAV) approach with the linear multistep technique. The main idea is first to rewrite the original model as an equivalent system, and then the linear multistep method in time, together with the relaxation strategy is employed to compute the solution of the reformulated system. The main benefit of the proposed schemes is that it can achieve second-order temporal accuracy and strictly unconditional energy stability, as well as a quite close to, and in many cases equal to, the original energy than the baseline SAV approaches. Finally, ample numerical experiments for the Allen-Cahn equation and Cahn-Hilliard equation are presented to confirm the efficiency, accuracy and strictly unconditional energy stability of the proposed schemes.