We consider a Navier--Stokes--Allen--Cahn (NSAC) system that governs the compressible motion of a viscous, immiscible two-phase fluid at constant temperature. Weak solutions of the NSAC system dissipate an appropriate energy functional. Based on an equivalent re-formulation of the NSAC system we propose a fully-discrete discontinuous Galerkin (dG) discretization that is mass-conservative, energy-stable, and provides higher-order accuracy in space and second-order accuracy in time. The approach relies on the approach in \cite{Giesselmann2015a} and a special splitting discretization of the derivatives of the free energy function within the Crank-Nicolson time-stepping. Numerical experiments confirm the analytical statements and show the applicability of the approach.

翻译：我们研究一个Navier--Stokes--Allen--Cahn (NSAC) 系统，该系统描述了在恒定温度下粘性、不混溶两相流体的可压缩运动。NSAC系统的弱解耗散一个适当的能量泛函。基于NSAC系统的一个等价重构形式，我们提出了一种全离散的间断伽辽金 (dG) 离散格式，该格式满足质量守恒、能量稳定，并在空间上具有高阶精度、在时间上具有二阶精度。该方法借鉴了\cite{Giesselmann2015a}中的思路，并在Crank-Nicolson时间步进框架内，对自由能函数的导数采用了一种特殊的分裂离散策略。数值实验验证了理论分析结论，并展示了该方法的适用性。