In this work we propose a high-order discretization of the Baer-Nunziato two-phase flow model (Baer and Nunziato, Int. J. Multiphase Flow, 12 (1986), pp. 861-889) with closures for interface velocity and pressure adapted to the treatment of discontinuous solutions, and stiffened gas equations of states. We use the discontinuous Galerkin spectral element method (DGSEM), based on collocation of quadrature and interpolation points (Kopriva and Gassner, J. Sci. Comput., 44 (2010), pp. 136-155). The DGSEM uses summation-by-parts (SBP) operators in the numerical quadrature for approximating the integrals over discretization elements (Carpenter et al., SIAM J. Sci. Comput., 36 (2014), pp. B835-B867; Gassner et al., J. Comput. Phys., 327 (2016), pp. 39-66). Here, we build upon the framework provided in (F. Renac, J. Comput. Phys., 382 (2019), pp. 1-36) for nonconservative hyperbolic systems to modify the integration over cell elements using the SBP operators and replace the physical fluxes with entropy conservative fluctuation fluxes from Castro et al. (SIAM J. Numer. Anal., 51 (2013), pp. 1371-1391), while we derive entropy stable numerical fluxes applied at interfaces. This allows to prove a semi-discrete inequality for the cell-averaged physical entropy, while keeping high-order accuracy. The design of the numerical fluxes also formally preserves the kinetic energy at the discrete level. High-order integration in time is performed using strong stability-preserving Runge-Kutta schemes and we propose conditions on the numerical parameters for the positivity of the cell-averaged void fraction and partial densities. The positivity of the cell-averaged solution is extended to nodal values by the use of an a posteriori limiter. The high-order accuracy, nonlinear stability, and robustness of the present scheme are assessed through several numerical experiments in one and two space dimensions.

在这项工作中，我们提出了Baer-Nunziato两相流模型的高阶离散化（Baer和Nunziato，Int.J。Multiphase Flow，12（1986），第861-889页），并针对界面速度和压力进行了封闭。适用于不连续溶液的处理，以及强化的气体状态方程。我们基于不连续的Galerkin光谱元素方法（DGSEM），基于正交点和内插点的搭配（Kopriva和Gassner，J. Sci。Comput。，44（2010），pp。136-155）。DGSEM在数字正交中使用部分求和（SBP）运算符来近似离散化元素上的积分（Carpenter等人，SIAM J. Sci。Comput。，36（2014），第B835-B867页; Gassner等人等，J.Comput.Phys。，327（2016），第39-66页）。在此，我们以（F.Renac，J.Comput.Phys。，382（2019），pp。1-36），针对非保守双曲系统，使用SBP算符修改单元元素的积分，并用Castro等人的熵保守涨落通量代替物理通量。（SIAM J.Numer.Anal。，51（2013），pp.1371-1391），而我们导出了应用于界面的熵稳定数值通量。这允许证明单元平均物理熵的半离散不等式，同时保持高阶精度。数值通量的设计还正式地将动能保留在离散水平上。时间上的高阶积分是使用强大的保持稳定性的Runge-Kutta方案执行的，并且我们提出了关于数值参数的条件，这些条件用于计算单元平均空隙率和部分密度的正值。通过使用后验限幅器，将细胞平均溶液的正性扩展到节点值。通过一维和二维空间中的几个数值实验，评估了该方案的高阶精度，非线性稳定性和鲁棒性。