Fully Conservative Finite-difference Schemes of Arbitrary Order for Compressible Flow

Semi-conservative finite-difference schemes for the equations of compressible flow have been known and used for the last couple of years, [1, 2, 3]. These schemes are based on rewritting the Euler- or Navier-Stokes equations in a discrete form which preserves their skew-symmetry. Thus the construction of conservative finite-difference schemes in space is easily possible with a wide variety of explicit differentiation schemes. However, schemes which are conservative both in space and time have not been widely developed. This is related to the unusual form of the skew-symmetric temporal derivative which arises in the momentum equation. Here we will show how to construct fully conservative discretizations of arbitrary order in space and time without such contraints..