We present a detailed description and verification of a discontinuous Galerkin finite element method (DG) for the multi-component chemically reacting compressible Navier-Stokes equations that retains the desirable properties of DG, namely discrete conservation and high-order accuracy in smooth regions of the flow. Pressure equilibrium between adjacent elements is maintained through the consistent evaluation of the thermodynamics model and the resulting weak form, as well as the proper choice of nodal basis. As such, the discretization does not generate unphysical pressure oscillations in smooth regions of the flow or at material interfaces where the temperature is continuous. Additionally, we present an hp-adaptive DG method for solving systems of ordinary differential equations, DGODE, which is used to resolve the temporal evolution of the species concentrations due to stiff chemical reactions. The coupled solver is applied to several challenging test problems including multi-component shocked flows as well as chemically reacting detonations, deflagrations, and shear flows with detailed kinetics. We demonstrate that the discretization does not produce unphysical pressure oscillations and, when applicable, we verify that it maintains discrete conservation. The solver is also shown to reproduce the expected temperature and species profiles throughout a detonation as well as the expected two-dimensional cellular detonation structure. We also demonstrate that the solver can produce accurate, high-order, approximations of temperature and species profiles without artificial stabilization for the case of a one-dimensional pre-mixed flame. Finally, high-order solutions of two- and three-dimensional multi-component chemically reacting shear flows, computed without any additional stabilization, are presented.

我们提出了一种不连续Galerkin有限元方法（DG）的详细描述和验证，该方法用于多组分化学反应的可压缩Navier-Stokes方程，该方程保留了DG的理想特性，即离散的守恒性和高光滑度的光滑区域流。相邻元件之间的压力平衡是通过热力学模型的一致评价将得到的弱的形式，以及节点基础的适当选择维持。这样，离散化不会在流动的平滑区域或温度连续的材料界面处产生不自然的压力振荡。此外，我们提出了一个马力自适应DG方法求解常微分方程组DGODE，该方法用于解决由于剧烈化学反应而引起的物种浓度随时间变化的问题。耦合求解器适用于一些具有挑战性的测试问题，包括多组分冲击流以及具有详细动力学的化学反应爆轰，爆燃和剪切流。我们证明了离散化不会产生非自然的压力振荡，并且在适用时，我们验证了离散化保持不变。还显示了求解器可在整个爆轰过程中以及在预期的二维蜂窝爆炸过程中再现预期的温度和物种分布。我们还证明了求解器可以产生准确的高阶，对于一维预混火焰，无需人工稳定即可近似获得温度和物质分布的近似值。最后，给出了二维和三维多组分化学反应剪切流的高阶解，该解无需进行任何额外的稳定即可计算出。