Auxiliary relaxation method to derive thermodynamically consistent phase field models with constraints and structure preserving numerical approximations

In this paper, we introduce a novel approach for formulating phase field models with constraints. The main idea is to introduce auxiliary variables that regularize and gradually dissipate constraint deviations of the phase variables, which we name the auxiliary relaxation method. It integrates seamlessly with the energy variational framework to ensure thermodynamic consistency in the resulting phase field models. Unlike traditional penalty methods, which introduce high stiffness due to large penalty parameters to enforce constraints in phase field models, our approach reduces system stiffness, allowing larger time step sizes when solving phase field models with constraints numerically, thus improving numerical accuracy and efficiency. We demonstrate the effectiveness and robustness of the proposed auxiliary relaxation method by applying it across several scenarios to derive thermodynamically consistent phase field models with constraints. Furthermore, we introduce a general second-order implicit-explicit Crank-Nicolson scheme, combining the relaxed scalar auxiliary variable method with a stabilization technique to solve these models. Through extensive numerical tests, we validate the capability of our modeling and numerical framework to reliably simulate complex dynamics governed by phase field equations with constraints.