Summary: The ability of previously proposed modifications of Godunov’s scheme to develop physically justified numerical solutions of the equations of inviscid gas dynamics is tested. Modifications constructed using the approach proposed by V. P. Kolgan [“The way to apply principle of derivative minimal value for generating finite-element schemes for calculating discontinuous solutions of gas dynamics” (Russian), Uch. Zap. TsAGI 3, 68–77 (1972)] to improve the accuracy of solutions in spatial variables are considered. The problems of a coaxial supersonic flow around a semi-infinite rectangle and a circular cylinder are solved. The calculations are performed on a uniform rectangular grid. It is shown that the numerical solution can correspond to both a steady and pulsating flow with different shapes of shock waves in front of the end wall, depending on the viscosity of the numerical scheme and the initial parameter distribution.