Semi-Lagrangian Approximation of Conservation Laws of Gas Flow in a Channel with Backward Step

In the chapter, the numerical modeling of a supersonic flow of a viscous heat-conducting gas in a flat channel with the backward step is considered. A numerical algorithm is proposed for the initial boundary-value problem for the Navier-Stokes equations. These equations are modified and supplemented by new boundary conditions to provide the conservation law for the full energy: kinetic and inner. Then the combination of the Lagrangian approximation for the transfer operators and the conforming finite element method for other terms provides an efficient algorithm. Particular attention has been given to the approximation providing the conservation laws for mass and full energy at discrete level. Test calculations have been performed for a wide range of Mach and Reynolds numbers.