A new Lagrange Multiplier approach for gradient flows

TL;DR

This paper introduces a novel Lagrange Multiplier method for gradient flows that ensures unconditional energy stability, dissipation of the original energy, and does not require the free energy to be bounded below, validated through numerical experiments.

Contribution

The paper presents a new Lagrange Multiplier approach that improves energy stability and applicability over existing SAV methods for gradient flows.

Findings

Schemes dissipate the original energy rather than a modified energy.

The approach does not require the free energy to be bounded from below.

Numerical simulations of a coupled Cahn-Hilliard model align with experimental results.

Abstract

We propose a new Lagrange Multiplier approach to design unconditional energy stable schemes for gradient flows. The new approach leads to unconditionally energy stable schemes that are as accurate and efficient as the recently proposed SAV approach \cite{SAV01}, but enjoys two additional advantages: (i) schemes based on the new approach dissipate the original energy, as opposed to a modified energy in the recently proposed SAV approach \cite{SAV01}; and (ii) they do not require the nonlinear part of the free energy to be bounded from below as is required in the SAV approach. The price we pay for these advantages is that a nonlinear algebraic equation has to be solved to determine the Lagrange multiplier. We present ample numerical results to validate the new approach, and, as a particular example of applications, we consider a coupled Cahn-Hilliard model for block copolymers (BCP), and carry out interesting simulations which are consistent with experiment results.