A novel Lagrange Multiplier approach with relaxation for gradient flows

TL;DR

This paper introduces a new Lagrange Multiplier method called zero-factor (ZF) and its relaxed version (RZF) for gradient flows, achieving unconditionally energy stable schemes with improved efficiency and accuracy.

Contribution

The paper develops the ZF and RZF methods, linking them to SAV-type schemes and enhancing stability and computational simplicity for gradient flow problems.

Findings

ZF schemes are unconditionally energy stable without extra assumptions.

RZF schemes are stable with respect to a modified energy closer to the original.

Numerical examples show improved efficiency and accuracy of the proposed methods.

Abstract

In this paper, we propose a novel Lagrange Multiplier approach, named zero-factor (ZF) approach to solve a series of gradient flow problems. The numerical schemes based on the new algorithm are unconditionally energy stable with the original energy and do not require any extra assumption conditions. We also prove that the ZF schemes with specific zero factors lead to the popular SAV-type method. To reduce the computation cost and improve the accuracy and consistency, we propose a zero-factor approach with relaxation, which we named the relaxed zero-factor (RZF) method, to design unconditional energy stable schemes for gradient flows. The RZF schemes 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 calculation process. 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 way to capture the sharp dissipation of nonlinear free energy. Several numerical examples are provided to demonstrate the improved efficiency and accuracy of the proposed method.