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The development of a new smoothed particle hydrodynamics (SPH) method, called adaptive smoothed particle hydrodynamics (ASPH), generalized for cosmology and coupled to the particle mesh (PM) method for solving the Poisson equation, for the gasdynamical simulation of galaxy and large-scale structure formation, will be described. The accurate numerical simulation of the highly nonlinear phenomena of shocks and caustics which occur generically in the process of cosmological structure formation requires enormous dynamic range and resolution. Existing numerical methods require substantial modification in order to achieve the required resolution with current computer technology.