Conservative finite volume schemes for fluid flow discretize the spatial
domain into control volumes and store cell averages of the flow
variables. Cartesian meshes for domains with complicated boundaries are
easy to generate but give rise to cut cells with arbitrarily small
volumes.
Explicit integration schemes over such meshes have a time step
restriction proportional to the smallest cell volume. We present an
mplementation of the kinetic scheme for gas dynamics by Perthame [SIAM
J. Num. Anal., 27:1305--1421]
on arbitrary Cartesian meshes. The formulation
allows a time step based on the underlying regular cell size, and
retains L1-stability, positivity and second order convergence.
From the point of view of implementation, this scheme relies on a
fundamental operation very different from that of flux based schemes
which leads to further interesting questions of its own.
Numerical convergence studies on irregular grids in one and two space
dimensions are presented.
|