A novel second-order linear scheme for the Cahn-Hilliard-Navier-Stokes equations
A novel second-order linear scheme for the Cahn-Hilliard-Navier-Stokes equations
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DOI：
10.1016/j.jcp.2020.109782
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发表时间：
2020-12
期刊：
影响因子：
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通讯作者：
Lizhen Chen;Jia Zhao
中科院分区：
文献类型：
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作者：
Lizhen Chen;Jia Zhao
本文研究了Cahn-Hilliard方程与不可压Navier-Stokes方程的耦合问题，通常称为Cahn-Hilliard-Navier-Stokes（CHNS）系统。CHNS系统已被广泛地用于研究二元流体混合物的动力学。利用改进的蛙跳时间推进方法，我们提出了一种新的数值算法求解CHNS系统在一个有效的和准确的方式。这个新提出的方案有几个优点。首先，所提出的方案在时间和空间上是线性的，使得在每个时间推进步骤中仅需要求解线性代数系统，使得其非常有效。此外，数值解的存在性和唯一性的任何时间步长的大小得到保证。此外，该方案是无条件能量稳定的，在时间上具有二阶精度，在空间上具有谱精度，因此可以使用相对较大的时间和空间网格尺寸来获得可靠的数值解。给出了无条件能量稳定性和解的存在唯一性的严格证明。最后，我们给出了几个数值例子来测试所提出的数值算法，并说明其精度和效率。研究了Cahn-Hilliard方程与Cahn-Hilliard-Navier-Stokes方程在粗化动力学方面的差异。
In this paper, we consider the Cahn-Hilliard equation coupled with the incompressible Navier-Stokes equation, usually known as the Cahn-Hilliard-Navier-Stokes (CHNS) system. The CHNS system has been widely embraced to investigate the dynamics of a binary fluid mixture. By utilizing the modified leap-frog time-marching method, we propose a novel numerical algorithm for solving the CHNS system in an efficient and accurate manner. This newly proposed scheme has several advantages. First of all, the proposed scheme is linear in time and space, such that only a linear algebraic system needs to be solved at each time-marching step, making it extremely efficient. Also, the existence and uniqueness of numerical solutions are guaranteed for any time step size. In addition, the scheme is unconditionally energy stable with second-order accuracy in time and spectral accuracy in space, such that relatively large temporal and spatial mesh sizes can be used to obtain reliable numerical solutions. The rigorous proofs for the unconditional energy stable property and solution existence and uniqueness are given. Furthermore, we present several numerical examples to test the proposed numerical algorithm and illustrate its accuracy and efficiency. The differences of coarsening dynamics between the Cahn-Hilliard equation and the Cahn-Hilliard-Navier-Stokes equations have been investigated as well.