  We have generalized a method for the numerical solution of hyperbolic systems
of equations using a dynamic Voronoi tessellation of the computational domain.
The Voronoi tessellation is used to generate moving computational meshes for
the solution of multi-dimensional systems of conservation laws in finite-volume
form. The mesh generating points are free to move with arbitrary velocity, with
the choice of zero velocity resulting in an Eulerian formulation. Moving the
points at the local fluid velocity makes the formulation effectively
Lagrangian. We have written the TESS code to solve the equations of
compressible hydrodynamics and magnetohydrodynamics for both relativistic and
non-relativistic fluids on a dynamic Voronoi mesh. When run in Lagrangian mode,
TESS is significantly less diffusive than fixed mesh codes and thus preserves
contact discontinuities to high precision while also accurately capturing
strong shock waves. TESS is written for Cartesian, spherical and cylindrical
coordinates and is modular so that auxilliary physics solvers are readily
integrated into the TESS framework and so that the TESS framework can be
readily adapted to solve general systems of equations. We present results from
a series of test problems to demonstrate the performance of TESS and to
highlight some of the advantages of the dynamic tessellation method for solving
challenging problems in astrophysical fluid dynamics.
