Entropy conserving/stable schemes for a vector-kinetic model of hyperbolic systems

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 mostly of entropy conserving/stable schemes for macroscopic model. 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 Tadmor [40]. 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.