The paper shows researchers currently apply standard approximation methods when using the finite volume method in problems of discontinuous gas-dynamic flows. In addition, the topic is relevant due to development of a methodology for theoretical and practical studies of gas dynamics and heat transfer during gas flow in a pipe based on a mathematical model using new methods for approximating discontinuous functions. The paper describes S.K. Godunov’s circuit, based on the approximation of flows at the cells boundaries of the difference grid using the exact solution of the Riemann problem of the gas-dynamic discontinuity decay. The authors point out a serious error in the calculations during approximation using Fourier series, as the Gibbs effect occurs. The paper presents Professor S.V. Alyukov’s approximation scheme applied to a discontinuous gas-dynamic flow for the first time. The authors’ analysis of the exact solutions and calculated data showed a difference of 4‒5% between them. The results confirm the possibility of using a new methodological approach to solving the problems of gas dynamics of discontinuous flows in their mathematical modeling and use of difference schemes.