Relaxed Lagrange Multiplier (RLM) Schemes for Phase Field Models Preserving the Relaxed Original Energy Dissipation Law

Jul 16, 2026

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Phase-field models simulate multiphase phenomena through a diffuse interface governed by a free-energy functional and an energy-dissipation law. This paper develops a family of relaxed Lagrange multiplier (RLM) numerical schemes designed to preserve a relaxed form of the original energy dissipation rate. Unlike SAV and IEQ methods that dissipate modified energies using auxiliary variables, RLM schemes closely track the original dissipation. The paper proves energy stability, constructs first- and second-order variants, and shows unique solvability and efficient linear solves. Numerical experiments confirm convergence and accurate interface dynamics.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages37

Reading Time1 h 33 min

Introduction

Phase-field modeling and energy dissipation

Energy-stable numerical schemes

Convex splitting and stabilization

Exponential time differencing (ETD)

IEQ and SAV approaches

Relaxed SAV/IEQ and consistency issues

Lagrange multiplier (LM) method and motivation

arXiv :2607 .00355v1 [math .NA] 1 Jul 2026 Relaxed Lagrange Multiplier (RLM) Schemes for Phase Field Models Preserving the Relaxed Original Energy Dissipation Law Xiaobo Jing∗ Jia Zhao† Abstract Phase-field models are typically derived from variational principles for a free-energy functional and are widely used to simulate complex multiphase phenomena in science and engineering. A central goal in designing numerical schemes for these models is to preserve the underlying energy-dissipation law. In this paper, we propose a class of relaxed Lagrange multiplier (RLM) schemes for phase field models. In contrast to popular scalar auxiliary variable (SAV) and invariant energy quadratization (IEQ) methods, which dissipate a modified energy involving auxiliary variables, the RLM schemes dissipate a relaxed version of the original energy and closely track the original energy dissipation rate. Compared with the classical Lagrange multiplier (LM) approach, the RLM schemes ensure that the resulting discrete system is uniquely solvable over a broad range of time steps. The key idea is to augment the LM formulation with a relaxation term, yielding a scalar quadratic equation for the multiplier with an explicit closed-form solution. The resulting schemes are linear and efficient because each time step requires solving only two linear systems with constant coefficients, at a cost comparable to that of SAV schemes. We construct both first-order and second-order variants and prove their energy stability. Numerical experiments verify the expected convergence rates and demonstrate that the RLM schemes accurately capture interface dynamics. Key words: Allen–Cahn Equation, Cahn–Hilliard Equation, Relaxed Lagrange Multiplier, Energy Stable, Phase Field 1 Introduction Phase-field methods are widely used to model multiphase problems across many areas of science and engineering. In materials science, they are used to simulate microstructural evolution processes such as solidification, grain growth, and phase transformations [1,3,29,43] . In fluid dynamics, they are used to model multiphase flows, including droplet dynamics and bubble formation [2,32,34] . Some other examples include fracture mechanics, biological systems, and pattern formation [5] . Phase-field models are usually derived from variational principles for a free-energy functional that governs the system dynamics via an energy-dissipation law [18,35] . The core idea of phase-field methods is to introduce a diffuse interface variable whose thickness is controlled by an artificial parameter ε . Instead of explicitly tracking sharp interfaces, we solve PDEs for the phase variable over the entire domain, and the interface is recovered from level sets of the phase variable. Although this formulation is conceptually simple, it presents significant numerical challenges. The small ∗ School of Mathematics, Southeast University, Nanjing 210096, Jiangsu Province, P.R. China. Email: [xiaobo@seu.edu.cn](xiaobo@seu.edu.cn) †Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA. Email: [jia.zhao@ua.edu](jia.zhao@ua.edu)[ ](jia.zhao@ua.edu)(corresponding author) parameter ε yields stiff systems that require sufficiently fine spatial and temporal resolution, and the nonlinear potential terms should be carefully treated to ensure stability [40] . Moreover, numerical methods should preserve the underlying energy dissipation principle. Schemes that preserve this property are called energy-stable, and those that maintain stability regardless of the time step are termed unconditionally energy-stable. Several classes of energy-stable numerical schemes have been developed for phase-field models. The convex splitting method [16,17,37] separates the free energy into convex and concave components, treating them implicitly and explicitly, respectively. Stabilization techniques add artificial damping terms to enable explicit treatment of nonlinear terms, yielding linear sch

FAQs

What is the main contribution of the proposed RLM schemes?

The paper proposes relaxed Lagrange multiplier (RLM) schemes that dissipate a relaxed version of the original free energy and closely match the original energy dissipation rate, unlike SAV/IEQ that dissipate modified energies with auxiliary variables.

How do RLM schemes improve over the classical LM approach?

Compared with the classical Lagrange multiplier (LM) method, the RLM schemes ensure unique solvability of the resulting discrete system for a broad range of time steps.

What do the numerical experiments show?

Numerical experiments verify the expected convergence rates and demonstrate that the RLM schemes accurately capture interface dynamics.

Jul 16, 2026

Upload by Logic