BETHE-Hydro: An Arbitrary Lagrangian-Eulerian Multi-dimensional   Hydrodynamics Code for Astrophysical Simulations
  In this paper, we describe a new hydrodynamics code for 1D and 2D astrophysical simulations, BETHE-hydro, that uses time-dependent, arbitrary, unstructured grids. The core of the hydrodynamics algorithm is an arbitrary Lagrangian-Eulerian (ALE) approach, in which the gradient and divergence operators are made compatible using the support-operator method. We present 1D and 2D gravity solvers that are finite differenced using the support-operator technique, and the resulting system of linear equations are solved using the tridiagonal method for 1D simulations and an iterative multigrid-preconditioned conjugate-gradient method for 2D simulations. Rotational terms are included for 2D calculations using cylindrical coordinates. We document an incompatibility between a subcell pressure algorithm to suppress hourglass motions and the subcell remapping algorithm and present a modified subcell pressure scheme that avoids this problem. Strengths of this code include a straightforward structure, enabling simple inclusion of additional physics packages, the ability to use a general equation of state, and most importantly, the ability to solve self-gravitating hydrodynamic flows on time-dependent, arbitrary grids. In what follows, we describe in detail the numerical techniques employed and, with a large suite of tests, demonstrate that BETHE-hydro finds accurate solutions with 2$^{nd}$-order convergence. 
Jeremiah W. Murphy (1) and Adam Burrows (2,1) ((1) Steward   Observatory, University of Arizona, (2) Department of Astrophys...