Physica 12D (1984) 408-425 North-Holland, Amsterdam ADAPTIVE MESH TECHNIQUES FOR FRONTS IN STAR FORMATION Karl-Heinz A. WINKLER* and Michael L. NORMAN zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON Max-Planck-Institut fiir Physik und Astrophysik, Institut fiir Astrophysik, Garching bei Miinchen, Fed. Rep. Germany and Michael J. NEWMAN* X-Division, Los Alamos National Laboratory University of California, Los Alamos, USA We present an implicit, adaptive-grid, finite-difference technique specifically designed to locate and track arbitrary fronts, interfaces and structures in a radiation hydro flow. The adaptive mesh is constructed in such a way that the “average” change of the Row-variables per grid zone is the same throughout the entire grid. A local grid refinement of up to a factor of IO6 compared to an equidistant mesh has been obtained on a machine with 14 decimal digits per word. Crucial to this type of approach is the use of a stiffness operator, which prevents tangling of the mesh. The elliptic equation for the grid motion is solved implicitly coupled with the equations for the flow variables. Per timestep very steep and narrow flow features can travel many times their own width in true space. However, with respect to the adaptive mesh the motion of such flow features is always less than one zone. Thus, the Courant-Friedrichs-Lewy [l] (CFL)-condition’s accuracy limitation is always fulfilled although its stability criterion does not pose any timestep limitations for our implicit numerical method. In fact, the choice of timesteps is based purely on physical and accuracy considerations. Typical applications run on Courant numbers ranging from 0.1 to 10”. The nerformance of this numerical scheme is demonstrated on a planar 1-D shocktube and the formation of a one solar mass protostar in spherical symmetry. 1. Introduction Within the context of ideal compressible fluid dynamics, shock fronts and contact discontinuities appear as true flow discontinuities. At these dis- continuities the differential form of the Euler equations mathematically breaks down. However, the integral form of the equations still holds and can be used to derive the Rankine-Hugoniot jump conditions which connect both sides of a fluid discontinuity. Accordingly, one numerical ap- proach for computing discontinuous fluid flows is to use interface following and/or front tracking algorithms. Such methods were used e.g. by Mc- Bryan [2], Youngs [3], and Smarr, Norman, and Winkler [4], this conference, and are authorita- tively summarized by Hyman [5] in his conference *X-Division Consultant, Los Alamos National Laboratory. *Visitor, Max-Planck-Institut fttr Astrophysik. paper “Moving Mesh Methods for Shocks and Interfaces”. However, front tracking is not always feasible or desirable. Fgr example, in flows in which a large number of fluid discontinuities are generated and/or interact with each other, this numerical approach tends to become progressively com- plicated and cumbersome. In addition to these complications, typical astrophysical flows are also governed by a variety of external forces and mi- crophysical processes which impose complicated physical structures onto the flow with time and length scales alien to pure ideal fluid dynamics. In particular, the formation of stars out of collapsing interstellar gas clouds develops unique flow pat- terns due to self-gravity, non-local radiation effects, dissociation and ionization, and finally nuclear burning. These features require special numerical techniques for their proper treatment, some of which are the main topic of this paper. But first, 0167-2789/84/$03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)