A General Numerical Fluid Dynamics Algorithm for Astrophysical Applications

Fini te difference simulation of f l u i d flows under astrophysical conditions is often complicated by factors such as complex gas physics, the occurrence of dynamics at widely d i f fer ing length scales, and the necessity of using imp l i c i t d i f ference equations. This report describes a simple, general, and e f f i c i en t algorithm for solving one-dimensional, spherical ly symmetric problems using a variat ion of the ICED-ALE method. A conuuter code named VEGA has been wr i t ten based on this algorithm, and the early ji^ges of the collapse of a onesolar mass protostel lar cloud are presented as a sample solution. I . INTRODUCTION The application of the equations of f luid dynamics to astrophysical problems opens the possibility of studying complex dynamical phases of stellar evolution in considerable detail. However, the numerical algorithms in the astronomical literature tend to have restrictive stabil ity l imits; rezoning, when done at a l l , is done in a primitive, ad hoc fashion. In an attempt to overcome these di f f iculties we have implemented a variation on the YAQUI code of Hirt, Amsden, and Cook. die report here the basic outline of our method and the progress we have made in applying i t to astrophysics. In our code, VEGA, the conservation equations are solved in two steps: In Phase I the solution is obtained in the coordinate system moving with the f lu id; this is called the Lagrangian phase. In Phase I I , the rezone or convective phase, the convection terms are added. By rezoning we mean the movement of mesh vertices to maintain a reasonable grid spacing. This is accomplished by defining a grid velocity as some arbitrary fraction of the fluid velocity. This fraction can be a function of space and time. In the limit that the grid velocity is zero the calculation is Eulerian. I f the grid velocity is set equal to the fluid velocity, the calculation is Lagrangian. This technique removes the ad hoc nature and numerical diff icult ies of rezoning by procedures such as the insertijin and deletion of grid points. J; Attempts to obtain numerical {solutions to problems in stellar evolution must cop|: with limits on the time step imposed by the wave s'peeds associated with the problem. In an explicit calculation one simply updates the flow variables for each cell in terms of quantities available at the beginning of the calculational cycle. This approach imposes stabil ity requirements that no waves (sound waves, elastic waves, thermal waves, etc.) can travel more than one cell per cycle. This is the well-known Courant condit ion, which is usually expressed as c 6t/6x < 1.. where c is the wave speed. To remove this restriction, we employ an implicit scheme that inc'ides in the solution of the equations as much advanced time information as possible. This requires either an iterative scheme to reach a consistent set of advanced time values or a direct solution technique applied to a linearized set of difference equations.