Paper  arXiv:0901.4107

E pur si muove: Galiliean-invariant cosmological hydrodynamical simulations on a moving mesh

Hydrodynamic cosmological simulations at present usually employ either the Lagrangian smoothed particle hydrodynamics (SPH) technique, or Eulerian hydrodynamics on a Cartesian mesh with (optional) adaptive mesh refinement (AMR).  Both of these methods have disadvantages that negatively impact their accuracy in certain situations, for example the suppression of fluid instabilities in the case of SPH, and the lack of Galilean-invariance and the presence of overmixing in the case of AMR.  We here propose a novel scheme which largely 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 unsplit Godunov scheme with an exact Riemann solver. The mesh-generating points can in principle be moved arbitrarily. If they are chosen to be stationary, the scheme is equivalent to an ordinary Eulerian method with second order accuracy. If they instead move with the velocity of the local flow, one obtains a Lagrangian formulation of continuum hydrodynamics that does not suffer from the mesh distortion limitations inherent in other mesh-based Lagrangian schemes. In this mode, our new method is fully Galilean-invariant, unlike ordinary Eulerian codes, a property that is of significant importance for cosmological simulations where highly supersonic bulk flows are common.  In addition, the new scheme can adjust its spatial resolution automatically and continuously, and hence inherits the principal advantage of SPH for simulations of cosmological structure growth. The high accuracy of Eulerian methods in the treatment of shocks is also retained, while the treatment of contact discontinuities improves. We discuss how this approach is implemented in our new code AREPO, both in 2D and 3D, and is parallelized for distributed memory computers. We also discuss techniques for adaptive refinement or derefinement of the unstructured mesh.  We introduce an individual time-step approach for finite volume hydrodynamics, and present a high-accuracy treatment of self-gravity for the gas that allows the new method to be seamlessly combined with a high-resolution treatment of collisionless dark matter.  We use a suite of test problems to examine the performance of the new code and argue that the hydrodynamic moving-mesh scheme proposed here provides an attractive and competitive alternative to current SPH and Eulerian techniques.

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