A high-order discontinuous Galerkin finite element method for a quadrature-based moment-closure model
The Euler equations are a system of nonlinear partial differential equations that prescribe the evolution of mass density, velocity, and pressure of a gas in thermodynamic equilibrium. In order to extend the validity of the Euler equations beyond thermodynamic equilibrium, equations for higher moments must be added to the system. The core difficulty with expanding the Euler system is that every new moment evolution equation that is added requires knowledge of the next moment. This problem is known as the moment-closure problem. In this work we study a particular strategy for closing the moment hierarchy: quadrature-based moment-closures. In particular, we review existing approaches that close the moment hierarchy by assuming that the underlying distribution is the sum of two delta functions, two Gaussian distributions, or two B-splines. Next we develop a closure based on three delta functions (tri-delta), where one of the delta functions is located at a prescribed location. This leads to a Gauss-Radau-type quadrature rule. We derive exact formulas that relate the positions and weights of the three delta functions to the primitive variables: mass density, velocity, pressure, heat flux, and kurtosis. We also derive exact conditions that simultaneously guarantee that the underlying system of partial differential equations remain hyperbolic and that the inversion problem from primitive variables to Gauss-Radau quadrature weights and points is solvable. Furthermore, we prove that the region in solution space for which these conditions are satisfied is convex. Finally, we develop a high-order discontinuous Galerkin finite element method to solve this system with a moment-realizability limiter that guarantees that the numerical solution remains in this convex hyperbolic/moment-realizable region.