A Symplecticity-preserving Gas-kinetic Scheme for Gas Dynamic Equations under External Forcing Field

In this paper, based on the BGK equation, Liouville's theorem, and symplecticity preserving property of a Hamiltonian flow, a well-balanced kinetic scheme for the compressible Navier-Stokes equations under external forcing field is developed. In order to construct such a scheme, the physical process of particles transport through a potential barrier at a cell interface will be modeled with penetration and reflection according to incident particle velocity, and the symplectic property of phase space volume. For the simulation of a gravitational gas system, the current scheme has both a well-balanced property for keeping hydrostatic solution and the shock capturing capability for unsteady flow. In this paper, we will present a symplecticity preserving BGK scheme with the inclusion of particle collision in the gas evolution stage.