Hydrodynamic simulations on a moving Voronoi mesh
  At the heart of any method for computational fluid dynamics lies the question
of how the simulated fluid should be discretized. Traditionally, a fixed
Eulerian mesh is often employed for this purpose, which in modern schemes may
also be adaptively refined during a calculation. Particle-based methods on the
other hand discretize the mass instead of the volume, yielding an approximately
Lagrangian approach. It is also possible to achieve Lagrangian behavior in
mesh-based methods if the mesh is allowed to move with the flow. However, such
approaches have often been fraught with substantial problems related to the
development of irregularity in the mesh topology. Here we describe a novel
scheme that eliminates these weaknesses. It is based on a moving unstructured
mesh defined by the Voronoi tessellation of a set of discrete points. The mesh
is used to solve the hyperbolic conservation laws of ideal hydrodynamics with a
finite volume approach, based on a second-order Godunov scheme with an exact
Riemann solver. A particularly powerful feature of the approach is that the
mesh-generating points can in principle be moved arbitrarily. If they are given
the velocity of the local flow, a highly accurate Lagrangian formulation of
continuum hydrodynamics is obtained that is free of mesh distortion problems,
while it is at the same time fully Galilean-invariant, unlike ordinary Eulerian
codes. We describe the formulation and implementation of our new Voronoi-based
hydrodynamics, and we discuss a number of illustrative test problems that
highlight its performance in practical applications.
