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We describe a powerful methodology for numerical solution of 3-Dself-gravitational hydrodynamics problems with extremely high resolution. Ourmethod utilizes the technique of local adaptive mesh refinement (AMR),employing multiple grids at multiple levels of resolution. These grids areautomatically and dynamically added and removed as necessary to maintainadequate resolution. This technology allows for the solution of problems in amanner that is both more efficient and more versatile than other fixed andvariable resolution methods. The application of AMR to simulate the collapseand fragmentation of a molecular cloud, a key step in star formation, isdiscussed. Such simulations involve many orders of magnitude of variation inlength scale as fragments form. In this paper we briefly describe themethodology and present an illustrative application for nonisothermal cloudcollapse. We describe the numerical Jeans condition, a criterion for stabilityof self-gravitational hydrodynamics problems. We show the first well-resolvednonisothermal evolutionary sequence beginning with a perturbed dense molecularcloud core that leads to the formation of a binary system consisting ofprotostellar cores surrounded by distinct protostellar disks. The scale of thedisks, of order 100 AU, is consistent with observations of gaseous diskssurrounding single T-Tauri stars and debris disks surrounding systems such as$\beta$ Pictoris.Comment: 10 pages, 6 figures (color postscript). To appear in the proceedings  of Numerical Astrophysics 1998, Tokyo, March 10-13, 199

Gravitational Collapse and Fragmentation in Molecular Clouds with Adaptive Mesh Refinement Hydrodynamics