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.

翻译：本文针对带源项的十矩高斯闭合方程，提出了一种二阶精确的直接欧拉广义黎曼问题（GRP）格式。给出了与稀疏波、接触间断和剪切波相关的广义黎曼不变量，并得到了一维精确黎曼求解器。随后，直接利用广义黎曼不变量和Rankine-Hugoniot跳跃条件，在欧拉框架下解析局部GRP的左右非线性波（稀疏波和激波），进而推导出一维直接欧拉GRP格式。由于涉及更多的物理变量和基本波，这些过程更为复杂、技术性强且非平凡。本文展示了一些一维和二维数值实验，以验证所提GRP格式的精度和高分辨率，其中为简便起见，二维直接欧拉GRP格式采用Strang分裂方法给出。需要强调的是，本文首次构造了多个二维黎曼问题的算例。