In recent years a number of shock-capturing numerical algorithms based on Total Variation Diminishing (TVD) schemes have been developed for the solution of the Euler equations in gas dynamics. These upwind methods are conservative, second-order accurate in time and space, and make use of the general coordinate systems to treat the complex flow geometries. Nonlinear flux limiters are employed to yield oscillation-free sharp shock profiles. In this paper, to study the numerical properties and performance of the TVD schemes in a comparative manner, we have chosen three current popular methods: the Osher scheme, the Yang's and the van-Leer's. In the Osher scheme, numerical dissipation operator is obtained from the solution of an Approximate Riemann problem. In the Yang's and van-Leer's, on the other hand, it is obtained from the flux-splitting. These methods are applied first to a steady problem, the supersonic internal flow through a channel having a 2% circular arc bump on a wall, with the channel inlet Mach number $M_infty$ =1.4. As an unsteady application, a normal shock moving through a similar channel having a 15% circular arc bump (shock speed $M_s$=2.0) is considered. The results are discussed and the numerical characteristics of these methods are carefully differentiated in the text.
