Analysis of finite-difference schemes based on exact and approximate solution of Riemann problem

The Riemann problem of one-dimensional arbitrary discontinuity breakdown for parameters of unsteady gas flow is considered as applied to the design of Godunov-type numerical methods. The problem is solved in exact and approximate statements (Osher-Solomon difference scheme used in shock capturing numerical methods): the intensities (the ratio of static pressures) and flow velocities on the sides of the resulting breakdowns and waves are determined, and then the other parameters are calculated in all regions of the flow. Comparison of calculation results for model flows by exact and approximate solutions is performed. The concept of velocity function is introduced. The dependence of the velocity function on the breakdown intensity is investigated. A special intensity at which isentropic wave creates the same flow rate as the shock wave is discovered. In the vicinity of this singular intensity approximate methods provide the highest accuracy. The domain of applicability for the approximate Osher-Solomon solution is defined by performing test calculations. The results are presented in a form suitable for usage in the numerical methods. The results obtained can be used in the high-resolution numerical methods.