  This paper extends the gas-kinetic scheme for one-dimensional inviscid
shallow water equations (J. Comput. Phys. 178 (2002), pp. 533-562) to
multidimensional gas dynamic equations under gravitational fields. Four
important issues in the construction of a well-balanced scheme for gas dynamic
equations are addressed. First, the inclusion of the gravitational source term
into the flux function is necessary. Second, to achieve second-order accuracy
of a well-balanced scheme, the Chapman-Enskog expansion of the Boltzmann
equation with the inclusion of the external force term is used. Third, to avoid
artificial heating in an isolated system under a gravitational field, the
source term treatment inside each cell has to be evaluated consistently with
the flux evaluation at the cell interface. Fourth, the multidimensional
approach with the inclusion of tangential gradients in two-dimensional and
three-dimensional cases becomes important in order to maintain the accuracy of
the scheme. Many numerical examples are used to validate the above issues,
which include the comparison between the solutions from the current scheme and
the Strang splitting method. The methodology developed in this paper can also
be applied to other systems, such as semi-conductor device simulations under
electric fields.
