一种二阶全离散线性能量稳定的二元可压缩粘性流体模型数值方案
A Second Order Fully-discrete Linear Energy Stable Scheme for a Binary Compressible Viscous Fluid Model
AI总结：
本文提出了一种二阶全离散线性数值方案，用于二元可压缩粘性流体混合物的热力学一致的水动力相场模型，该模型基于广义Onsager原理推导。该方案在 staggered 网格上保持质量守恒和能量耗散，通过能量二次化方法和Crank-Nicolson方法进行离散化，并通过数值实验验证了其收敛性和实用性。
摘要：
我们提出了一种线性、二阶全离散数值方案，用于在staggered网格上解决由广义Onsager原理推导出的二元可压缩粘性流体混合物的热力学一致水动力相场模型。该水动力模型不仅具有变分结构，还保证了质量和线性动量守恒以及能量耗散。我们首先使用能量二次化方法将模型重新表述为等价形式，然后利用Crank-Nicolson方法在时间上离散化，得到半离散的偏微分方程系统。所推导的数值方案在半离散水平上保持质量守恒和能量耗散定律。然后，我们使用二阶有限差分方法在空间上离散化半离散PDE系统，得到一个全离散方案，该方案尊重离散的能量耗散定律。我们证明了由全离散方案产生的线性系统的唯一可解性。通过网格细化和两个关于两聚合物流体因spinodal分解导致相分离以及气液混合物界面演化的数值实验，展示了新方案的收敛性和在应用中的实用性。
英文摘要：
We present a linear, second order fully discrete numerical scheme on a staggered grid for a thermodynamically consistent hydrodynamic phase field model of binary compressible fluid flow mixtures derived from the generalized Onsager Principle. The hydrodynamic model not only possesses the variational structure, but also warrants the mass, linear momentum conservation as well as energy dissipation. We first reformulate the model in an equivalent form using the energy quadratization method and then discretize the reformulated model to obtain a semi-discrete partial differential equation system using the Crank-Nicolson method in time. The numerical scheme so derived preserves the mass conservation and energy dissipation law at the semi-discrete level. Then, we discretize the semi-discrete PDE system on a staggered grid in space to arrive at a fully discrete scheme using the 2nd order finite difference method, which respects a discrete energy dissipation law. We prove the unique solvability of the linear system resulting from the fully discrete scheme. Mesh refinements and two numerical examples on phase separation due to the spinodal decomposition in two polymeric fluids and interface evolution in the gas-liquid mixture are presented to show the convergence property and the usefulness of the new scheme in applications.
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