论文标题
熵稳定的高阶不连续的Galerkin光谱元素方法，用于Baer-Nunziato两相流模型
An entropy stable high-order discontinuous Galerkin spectral element method for the Baer-Nunziato two-phase flow model
论文作者
论文摘要
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 and Nunziato，Int。J。Multiphase Flow，12（1986），第861-889页，闭合，用于界面速度和压力，以适应不连续的溶液和状态固化的气体固化溶液和僵硬的气体平衡。我们基于正交和插值点的套筒（Kopriva and Gassner，J。Sci。Comput。，44（2010），第136-155页），我们使用不连续的Galerkin光谱元素法（DGSEM）（DGSEM）。 DGSEM在数值数字中使用逐个局部的总和（SBP）运算符来近似离散元素的积分（Carpenter等，Siam J.Sci。Comput。，36（2014），第36（2014），第B835-B867; b835-B867; Gassner et al。在这里，我们建立在（F. Renac，J。Comput。Phys。，382（2019），第1-36页）中提供的非保守双曲线系统的框架，以使用SBP操作员修改单元元素的集成，并用Castro的熵保守波动替换物理磁通量。 （Siam J.Numer。Anal。，51（2013），第1371-1391页），而我们得出在界面上应用的熵稳定通量。这允许为细胞平均物理熵建立半污垢不等式，同时精确。数值通量的设计还正式保留了半污垢水平的动能。使用强稳定性的runge-kutta方案进行高阶整合时间，我们就数值参数提出条件，以使细胞平均空隙分数和部分密度的阳性。通过使用A后验限限，将细胞平均溶液的阳性扩展到节点值。通过在一个和两个空间维度中的几个数值实验评估了本方案的高级准确性，非线性稳定性和鲁棒性。
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 fluxes applied at interfaces. This allows to establish a semi-discrete inequality for the cell-averaged physical entropy, while being high-order accurate. The design of the numerical fluxes also formally preserves the kinetic energy at the semi-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.
