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The aim of this work is the development of an automatic, adaptive mesh refinement strategy for solving hyperbolic conservation laws in two dimensions. There are two main difficulties in doing this. The first problem is due to the presence of discontinuities in the solution and the effect on them of discontinuities in the mesh. The second problem is how to organize the algorithm to minimize memory and CPU overhead. This is an important consideration and will continue to be important as more sophisticated algorithms that use data structures other than arrays are developed for use on vector and parallel computers. b? 1989 Academic Press, Inc.