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A well-balanced scheme for an isolated gravitational hydrodynamic system is defined as a scheme which exactly preserves an isothermal hydrostatic solution. In this paper, a well-balanced gas-kinetic symplecticity-preserving BGK (SP-BGK) scheme is developed. In the construction of the scheme, the gravitational potential is modeled as a piecewise constant function inside each cell with a potential jump at the cell interface. In the process of designing such a scheme, the energy conservation, Liouville's theorem, and the symplecticity-preserving property of a Hamiltonian flow play important roles in the description of particles penetration and reflection from a potential barrier. More importantly, the use of the symplecticity-preserving property is crucial in the evaluation of the moments of a postinteraction gas distribution function with a potential jump in terms of the moments of preinteraction distribution function. The SP-BGK method is the first well-balanced shock-capturing gas-kinetic scheme for the Navier-Stokes equation. A few theorems are proved for this scheme, which include the necessity to use an exact Maxwellian for keeping the isothermal hydrostatic state, the total mass and energy (the sum of kinetic, thermal, and gravitational ones) conservation, and the well-balanced property of the SP-BGK scheme to keep an isothermal hydrostatic state during the process of particle transport and collision. Many numerical examples are presented to validate the SP-BGK scheme

A Well-Balanced Symplecticity-Preserving Gas-Kinetic Scheme for Hydrodynamic Equations under Gravitational Field