A second-order direct Eulerian GRP scheme for ten-moment Gaussian closure equations with source terms

This paper proposes a second-order accurate direct Eulerian generalized Riemann problem (GRP) scheme for the ten-moment Gaussian closure equations with source terms. The generalized Riemann invariants associated with the rarefaction waves, the contact discontinuity and the shear waves are given, and the 1D exact Riemann solver is obtained. After that, the generalized Riemann invariants and the Rankine-Hugoniot jump conditions are directly used to resolve the left and right nonlinear waves (rarefaction wave and shock wave) of the local GRP in Eulerian formulation, and then the 1D direct Eulerian GRP scheme is derived. They are much more complicated, technical and nontrivial due to more physical variables and elementary waves. Some 1D and 2D numerical experiments are presented to check the accuracy and high resolution of the proposed GRP schemes, where the 2D direct Eulerian GRP scheme is given by using the Strang splitting method for simplicity. It should be emphasized that several examples of 2D Riemann problems are constructed for the first time.