Operator splitting based structure-preserving numerical schemes for the mass-conserving convective Allen-Cahn equation
Operator splitting based structure-preserving numerical schemes for the mass-conserving convective Allen-Cahn equation
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DOI：
10.1016/j.jcp.2022.111695
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发表时间：
2022-10
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影响因子：
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通讯作者：
Rihui Lan;Jingwei Li;Yongyong Cai;L. Ju
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文献类型：
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作者：
Rihui Lan;Jingwei Li;Yongyong Cai;L. Ju
质量守恒的对流Allen-Cahn（MCAC）方程是流场耦合多相流体相场模拟的重要组成部分。它继承了经典Allen-Cahn方程的最大值界原理（MBP），即在适当的初始和边界条件下，时间相关的解在所有时间内保持一个一致的绝对值逐点界.在本文中，我们发展了两个基于算子分裂方法的MCAC方程的保结构数值格式。特别地，在每个时间步长的MCAC方程的推进被分成两个（时间上的一阶分裂）或三个（时间上的二阶分裂）阶段，并且每个阶段由质量守恒AC方程或输运方程组成。质量守恒的AC部分，然后通过使用经典的有限体积近似在空间和稳定的指数时间差分在时间上的离散化和由此产生的系统可以有效地解决通过快速傅立叶变换为基础的算法。输运部分采用显式强稳定性Runge-Kutta迭代法和满足极大值原理的有限体积法求解。最佳的误差估计来自建议的全离散计划，以及保存的离散MBP和守恒的质量。各种数值例子也被提出来验证理论结果和证明所提出的计划的性能。
The mass-conserving convective Allen-Cahn (MCAC) equation is an important component of phase field modeling for the multiphase fluid coupled with a flow field. It inherits the maximum bound principle (MBP) from the classic Allen-Cahn equation, in the sense that the time-dependent solution under appropriate initial and boundary conditions preserves a uniform point-wise bound in the absolute value for all time. In this paper, we develop two structure-preserving numerical schemes for the MCAC equation based on the operator splitting approach. In particular, the advancing of the MCAC equation at each time step is split into two (first-order splitting in time) or three (second-order splitting in time) stages, and each of the stages consists of either a mass-conserving AC equation or a transport equation. The mass-conserving AC part is then discretized by using the classic finite volume approximation in space and the stabilized exponential time differencings in time and the resulting system can be efficiently solved via fast Fourier transform based algorithms. The transport part is solved by explicit strong stability preserving Runge-Kutta substeppings combined with a maximum-principle-satisfying finite volume method. Optimal error estimates are derived for the proposed fully-discrete schemes, as well as preservation of the discrete MBP and conservation of the mass. Various numerical examples are also presented to verify the theoretical results and demonstrate the performance of the proposed schemes.