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We describe a modified form of Smoothed Particle Hydrodynamics (SPH) in whichthe specific thermal energy equation is based on a compatibly differencedformalism, guaranteeing exact conservation of the total energy. We compare theerrors and convergence rates of the standard and compatible SPH formalisms onanalytic test problems involving shocks. We find that the new compatibleformalism reliably achieves the expected first-order convergence in such tests,and in all cases improves the accuracy of the numerical solution over thestandard formalism.

A compatibly differenced total energy conserving form of SPH