The moment of entropy equation for vector-BGK model results in the entropy equation for macroscopic model. However, this is usually not the case in numerical methods because the current literature consists only of entropy conserving/stable schemes for macroscopic model (to the best of our knowledge). In this paper, we attempt to fill this gap by developing an entropy conserving scheme for vector-kinetic model, and we show that the moment of this results in an entropy conserving scheme for macroscopic model. With the numerical viscosity of entropy conserving scheme as reference, the entropy stable scheme for vector-kinetic model is developed in the spirit of [33]. We show that the moment of this scheme results in an entropy stable scheme for macroscopic model. The schemes are validated on several benchmark test problems for scalar and shallow water equations, and conservation/stability of both kinetic and macroscopic entropies are presented.

翻译：向量BGK模型的熵方程矩过程可导出宏观模型的熵方程。然而在数值方法中，现有文献仅涉及宏观模型的熵守恒/稳定格式（据我们所知），二者通常并不等价。本文尝试填补这一空白，通过构建向量-动力学模型的熵守恒格式，并证明该格式的矩过程可导出宏观模型的熵守恒格式。以熵守恒格式的数值粘性为参考，我们遵循[33]的思路发展了向量-动力学模型的熵稳定格式，同时证明该格式的矩过程可导出宏观模型的熵稳定格式。通过标量方程和浅水方程的多个基准测试问题验证了数值格式，并展示了动力学熵与宏观熵的守恒性/稳定性。