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I am working on the discrete theory of compressible fluids dynamics, i.e., numerically solving and simulating the compressible fluids , we are interested in the way using discrete exterior calculus, my question is: Is there any work on the discrete theory (especially using discrete exterior calculus) of compressible fluids?

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There's unpublished work by Gay-Balmaz and Pavlov, Variational Discretization of Compressible Fluids, described here, with an instructive summary of the difficulties involved in extending the discrete theory from incompressible to compressible fluids.

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