An efficient and robust Lagrange multiplier approach with a penalty term for phase-field models
An efficient and robust Lagrange multiplier approach with a penalty term for phase-field models
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DOI：
10.1016/j.jcp.2023.112236
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发表时间：
2023-05
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影响因子：
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通讯作者：
Dianming Hou;Yuexin Ning;Chao Zhang
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作者：
Dianming Hou;Yuexin Ning;Chao Zhang
在本文中，我们提出并分析了带有相场模型惩罚项的拉格朗日乘子方法的变体。更准确地说，我们基于 Crank-Nicolson (CN) 和二阶向后微分公式 (BDF2) 为相场模型开发了两种类型的线性二阶解耦和无条件能量稳定时间步进方案，其中标量乘子变量通过求解非线性代数方程来评估。与相场模型的拉格朗日乘子法（Cheng et al. (2020) [6]）相比，首先引入标量惩罚项来改善非线性代数方程牛顿迭代算法收敛的时间步长约束，这使得我们能够获得鲁棒且高效的相场模型数值格式。而且，我们仅使用一阶时间步长方案来逼近标量乘子变量，这不会影响具有某些特殊拉格朗日乘子函数的未知相位函数的二阶精度。同时，它在非均匀时间网格上所提出的基于 BDF2 的方案的能量稳定性推导中起着至关重要的作用。此外，CN方案的能量稳定性是严格建立的。此外，我们推断所提出的 BDF2 方案是能量稳定的，相邻时间步长比不超过 4.8645。最后，提出了一系列数值测试，以数值验证我们提出的拉格朗日乘子方法的效率和稳定性。
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.