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