Aim of the study: This study examines numerical methods for solving the problems in gas dynamics, which are based on an exact or approximate solution to the problem of breakdown of an arbitrary discontinuity (the Riemann problem). Results: Comparative analysis of finite difference schemes for the Euler equations integration is conducted on the basis of an exact or approximate solution to the problem of an arbitrary discontinuity breakdown. An approach to the numerical solution of the Euler equations governing inviscid compressible gas flow is developed on the basis of the finite volume method and finite difference schemes for flow calculation of various degrees of accuracy. Calculation results show that monotonic derivative correction provides numerical solution uniformity in the breakdown neighborhood. On one hand, it prevents the formation of new extremum points, thereby providing monotonicity, but on the other hand, it causes smoothing of existing minimums and maximums and accuracy loss. Conclusions: The developed numerical calculation method makes it possible to perform high-accuracy calculations of flows with strong non-stationary shock and detonation waves and no nonphysical solution oscillations on the shock wave front.