A Novel Lagrange Multiplier Approach with Relaxation for Gradient Flows (CSIAM Transactions on Applied Mathematics, 2024)
In this paper, we propose a novel Lagrange multiplier approach, namedzero-factor (ZF) approach to solve a series of gradient flow problems. The numericalschemes based on the new algorithm are unconditionally energy stable with the original energy and do not require any extra assumption conditions. We also prove thatthe ZF schemes with specific zero factors lead to the popular SAV-type method. Toreduce the computation cost and improve the accuracy and consistency, we proposea zero-factor approach with relaxation, which we named the relaxed zero-factor (RZF)method, to design unconditional energy stable schemes for gradient flows. The RZFschemes can be proved to be unconditionally energy stable with respect to a modified energy that is closer to the original energy, and provide a very simple calculationprocess. The variation of the introduced zero factor is highly consistent with the nonlinear free energy which implies that the introduced ZF method is a very efficient wayto capture the sharp dissipation of nonlinear free energy. Several numerical examplesare provided to demonstrate the improved efficiency and accuracy of the proposedmethod.