论文标题
不连续的可压缩多组分流模型的不连续的盖尔金光谱元素方法
A discontinuous Galerkin spectral element method for a nonconservative compressible multicomponent flow model
论文作者
论文摘要
在这项工作中，我们提出了Shyue对基于伽玛的可压缩多组分模型的准确，健壮且稳定的离散化[J.计算。 Phys。，142（1998），208-242]，其中每个组件遵循状态僵硬的气体方程（EOS）。我们在这里扩展了Renac提出的框架[J.计算。物理。 382（2019），1-26]和Coquel等。 [J。计算。物理。 431（2021）110135]用于双曲系统的离散化，具有通量和非守护产物，用于在多个空间维度中具有弯曲元件的非结构化网格。该框架依赖于不连续的Galerkin光谱元素方法（DGSEM），使用正交和插值点的搭配。我们通过离散化元素来修改积分，在这种元素中，我们通过两点数值波动替换物理通量和非保守产品。这项工作的贡献是三倍。首先，我们分析了半混凝土DGSEM离散化，并证明该方案是高阶精度，自由流的保存，并且在排除材料接口时熵稳定。其次，我们设计了一个使用HLLC求解器的三点方案，该方案不需要用于近似非保守产品的根发现算法。事实证明，该方案是稳定的，对于凸熵，熵稳定，在材料界面上保存均匀的状态，满足了特定熵的最低原理，并且在EOS参数上满足了最大原理。第三，将HLLC求解器应用于DGSEM方案中的界面，而我们考虑积分中的两种波动，而不是离散化元素：保存材料界面和熵保守。使用SSP runge-kutta方案进行时间积分。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 an accurate, robust, and stable discretization of the gamma-based compressible multicomponent model by Shyue [J. Comput. Phys., 142 (1998), 208-242] where each component follows a stiffened gas equation of state (EOS). We here extend the framework proposed in Renac [J. Comput. Phys. 382 (2019), 1-26] and Coquel et al. [J. Comput. Phys. 431 (2021) 110135] for the discretization of hyperbolic systems, with both fluxes and nonconservative products, to unstructured meshes with curved elements in multiple space dimensions. The framework relies on the discontinuous Galerkin spectral element method (DGSEM) using collocation of quadrature and interpolation points. We modify the integrals over discretization elements where we replace the physical fluxes and nonconservative products by two-point numerical fluctuations. The contributions of this work are threefold. First, we analyze the semi-discrete DGSEM discretization and prove that the scheme is high-order accurate, free-stream preserving, and entropy stable when excluding material interfaces. Second, we design a three-point scheme with a HLLC solver that does not require a root-finding algorithm for approximating the nonconservative products. The scheme is proved to be robust and entropy stable for convex entropies, preserves uniform states across material interfaces, satisfies a discrete minimum principle on the specific entropy and maximum principles on the EOS parameters. Third, the HLLC solver is applied at interfaces in the DGSEM scheme, while we consider two kinds of fluctuations in the integrals over discretization elements: material interface preserving and entropy conservative. Time integration is performed using SSP Runge-Kutta schemes. The high-order accuracy, nonlinear stability, and robustness of the present scheme are assessed through several numerical experiments in one and two space dimensions.
