Modal Discontinuous Galerkin Simulations for Grad’s 13 Moment Equations: Application to Riemann Problem in Continuum-Rarefied Flow Regime

Abstract

For real-world engineering applications, there is a great deal of interest in developing effective models and simulations of continuum-rarefied gas flows. In this study, the numerical simulations of Grad’s 13 (G13) moment equations with application to the Riemann problem in a wide range of continuum-rarefied flow regimes are presented. This work emphasizes numerical robustness and wave phenomena in the G13 system to provide a building block for regularized 13 moment systems, and high-order Grad’s models. For this purpose, a high-order modal discontinuous Galerkin solver is developed for solving one-dimensional G13 moment equations. For spatial discretization, hierarchical modal basis functions premised on orthogonal-scaled Legendre polynomials are used. The proposed approach reduces the G13 systems into a framework of ordinary differential equations in time, which are addressed by an explicit third-order SSP Runge-Kutta algorithm. Three Riemann test cases, including the shock tube, two shock waves and two rarefaction waves, are examined in continuum-rarefied flow regimes. In the G13 system, the arising characteristic waves and dissipation phenomena are investigated in depth. The numerical results demonstrate that every Riemann problem does not have a solution for the G13 system because of loss of hyperbolicity of the system.