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An algorithm for the computation of steady and unsteady inviscid, compressible, two-dimensional flows is presented. Grid generation utilizes the quadtree spatial subdivision algorithm, allowing automatic meshing of boundaries of arbitrary geometry, and allowing h-adaptivity to geometric and solution features. Internal boundaries and interfaces may move, deform, coalesce, and disintegrate arbitrarily. The flow-solver is of the High-Resolution Gudonov type. For moving boundaries, a novel feature of the algorithm is that the grid is stationary and boundaries are allowed to move across grid-lines. This is enabled by merging cells in the vicinity of boundaries to form composite cells that are topologically invariant during individual motion steps. This merging also eliminates the small-cell time-step constraint introduced by grid non-conformality. The motion of boundaries may be prescribed, or computed from aerodynamic, body, or contact forces in a coupled fashion. Two computations illustrating the capabilities of the algorithm are presented. [ABSTRACT FROM AUTHOR]