Entropy-conservative spatial discretization of the multidimensional quasi-gasdynamic system of equations

The multidimensional quasi-gasdynamic system written in the form of mass, momentum, and total energy balance equations for a perfect polytropic gas with allowance for a body force and a heat source is considered. A new conservative symmetric spatial discretization of these equations on a nonuniform rectangular grid is constructed (with the basic unknown functions—density, velocity, and temperature—defined on a common grid and with fluxes and viscous stresses defined on staggered grids). Primary attention is given to the analysis of entropy behavior: the discretization is specially constructed so that the total entropy does not decrease. This is achieved via a substantial revision of the standard discretization and applying numerous original features. A simplification of the constructed discretization serves as a conservative discretization with nondecreasing total entropy for the simpler quasi-hydrodynamic system of equations. In the absence of regularizing terms, the results also hold for the Navier–Stokes equations of a viscous compressible heat-conducting gas.