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?

There's unpublished work by GayBalmaz and Pavlov, Variational Discretization of Compressible Fluids, described here, with an instructive summary of the difficulties involved in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. 

