A Relaxed Lagrange Multiplier Approach for Phase Field Models

Abstract

This paper introduces a novel relaxed Lagrange multiplier (RLM) method for designing efficient and energy-stable numerical schemes for phase-field models. The proposed approach reformulates the original model by introducing a time-dependent Lagrange multiplier (r(t)), whose evolution is governed by an ordinary differential equation involving a relaxation parameter (α>0). This relaxation technique slows down the evolution of the multiplier, thereby improving the consistency between the modified and original systems after temporal discretization, while avoiding the nonlinear algebraic equations arising in the original Lagrange multiplier (LM) method. We construct first- and second-order temporal discretizations based on the RLM approach and rigorously prove their unconditional stability with respect to a modified energy. Furthermore, we establish the boundedness of the numerical Lagrange multiplier and show that it converges to 1 as the relaxation parameter tends to zero. We also demonstrate that the modified energy converges to the original energy as the time-step size (τ) tends to zero. Extensive numerical experiments confirm the theoretical results, demonstrating second-order accuracy, unconditional energy stability, and significantly improved computational efficiency, with approximately half the computational cost of comparable scalar auxiliary variable (SAV) and LM methods. The RLM method resolves the consistency issue associated with the SAV approach without requiring the nonlinear free energy to be bounded from below, thereby providing a robust and highly efficient alternative for simulating phase-field models.

Published in Computer Methods in Applied Mechanics and Engineering