A Well-Balanced Kinetic Scheme for Gas Dynamic Equations under Gravitational Field

In this paper, a well-balanced kinetic scheme for the gas dynamic equa- tions under gravitational field is developed. In order to construct such a scheme, the physical process of particles transport through a potential barrier at a cell inter- face is considered, where the amount of particle penetration and reflection is evalu- ated according to the incident particle velocity. This work extends the approach of Perthame and Simeoni for the shallow water equations (Calcolo, 38 (2001), pp. 201- 231) to the Euler equations under gravitational field. For an isolated system, this scheme is probably the only well-balanced method which can precisely preserve an isothermal steady state solution under time-independent gravitational potential. A few numerical examples are used to validate the above approach.