An efficient and robust Lagrange multiplier approach with a penalty term for phase-field models

Abstract

In this paper, we propose and analyze a variant of Lagrange multiplier approach with a penalty term for phase-field models. More precisely, we develop two types of linear second-order decoupled and unconditionally energy stable time-stepping schemes for phase-field models based on the Crank-Nicolson (CN) and the second order backward differentiation formulations (BDF2), where the scalar multiplier variable is evaluated by solving a nonlinear algebraic equation. Comparing with the Lagrange multiplier method (Cheng et al. (2020) [6]) for phase-field models, a scalar penalty term is first introduced to improve the time-step size constraint for the convergence of Newton's iterative algorithm for the nonlinear algebraic equation, which allows us to obtain robust and efficient numerical schemes for phase-field models. Moreover, we only use first order time-stepping scheme to approximate the scalar multiplier variable, which will not affect the second order accuracy of the unknown phase function with some special Lagrange multiplier functions. Meanwhile, it plays an essential role in the derivation of energy stability of the proposed BDF2-based scheme on the nonuniform temporal mesh. In addition, the energy stability of the CN scheme is rigorously established. Furthermore, we deduce that the proposed BDF2 scheme is energy stable with the adjacent time-step ratios no more than 4.8645. Finally, a series of numerical tests are presented to numerically validate the efficiency and stability of our proposed Lagrange multiplier approach.