**Nikiforov**

Vladimir O.

D.Sc., Prof.

Vladimir O.

D.Sc., Prof.

doi: 10.17586/2226-1494-2015-15-1-139-148

doi: 10.17586/2226-1494-2015-15-1-139-148

# ANALYSIS OF FINITE-DIFFERENCE SCHEMES BASED ON EXACT AND APPROXIMATE SOLUTION OF RIEMANN PROBLEM

**Read the full article**';

**Article in**Russian

**For citation:**Bulat P.V., Volkov K.N., Silnikov M.S., Chernyshev M.V. Analysis of finite-difference schemes based on exact and approximate solution of Riemann problem. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2015, vol. 15, no. 1, pp. 139–148

**Abstract**

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.

**Keywords:**arbitrary discontinuity breakdown, shock wave, Riemann wave, contact discontinuity

**References**

1. Handbook of Shock Waves. Eds. G. Ben-Dor, D. Igra, T. Elperin. NY, Academic Press, 2001, 824 p.

2. Uskov V.N. Begushchie Odnomernye Volny [Running One-Dimensional Waves]. St. Petersburg, BGTU "Voenmekh" Publ., 2000, 220 p.

3. Arkhipova L.P., Uskov V.N. Universal'noe reshenie zadachi ob otrazhenii odnomernykh begushchikh voln ot tverdoi stenki i ego analiz dlya voln uplotneniya [Universal solution of problem of 1D moving waves reflection from a solid wall and analysis for waves seal]. Vestnik of the St. Petersburg University: Mathematics, 2013, no. 2, pp. 77–81.

4. Arkhipova L.P. Skorostnaya funktsiya intensivnosti odnomernoi nestatsionarnoi volny i ee analiz dlya voln uplotneniya [The speed function of intensity of one-dimensional moving wave and it's analysis for shock and compression waves]. Vestnik Samarskogo Gosudarstvennogo Aerokosmicheskogo Universiteta im. Akademika S.P. Koroleva, 2012, no. 3–1(34), pp. 57–62.

5. Godunov S.K. A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics. Sbornik: Mathematics, 1959, vol. 47, no. 8–9, pp. 357–393.

6. Godunov S.K., Zabrodin A.V., Ivanov M.Ya., Kraiko A.N. Chislennoe Reshenie Mnogomernykh Zadach Gazovoi Dinamiki [Numerical Solution of Multi-Dimensional Problems in Gas Dynamics]. Moscow, Nauka Publ., 1976, 400 p.

7. Gel'fand B.E., Sil'nikov M.V., Mikhailin A.I., Orlov A.V. Attenuation of blast overpressures from liquid in an elastic shell. Combustion, Explosion and Shock Waves, 2001, vol. 37, no. 5, pp. 607–612.

8. Silnikov M.V., Mikhaylin A.I. Protection of flying vehicles against blast loads. Acta Astronautica, 2014, vol. 97, no. 1, pp. 30–37. doi: 10.1016/j.actaastro.2013.12.012

9. Osher S., Solomon F. Upwind difference schemes for hyperbolic conservation laws. Mathematics of Computation, 1982, vol. 38, no. 158, pp. 339–374. doi: 10.1090/S0025-5718-1982-0645656-0

10. Volkov K.N. Primenenie i realizatsiya raznostnykh skhem vysokoi razreshayushchei sposobnosti dlya resheniya zadach gazovoi dinamiki na nestrukturirovannykh setkakh [Application and implementation of high-resolution difference schemes for solution of gas dynamics problems on unstructured meshes]. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2014, no. 6, pp. 153–162.

11. Kulikovskii A.G., Pogorelov N.V., Semenov A.Yu. Mathematical Aspects of Numerical Solution of Hyperbolic Systems. Chapman & Hall/CRC Press, 2001, 560 p.

12. Kolgan V.P. Primenenie printsipa minimal'nykh znachenii proizvodnoi k postroeniyu konechno-raznostnykh skhem dlya rascheta razryvnykh reshenii gazovoi dinamiki [Applying the principle of the minimum values of the derivative to the construction of finite-difference schemes for computation of discontinuous solution of gas dynamics]. TsAGI Science Journal, 1972, vol. 3, no. 6, pp. 68–77.

13. Sod G.A. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. Journal of Computational Physics, 1978, vol. 27, no. 1, pp. 1–31. doi: 10.1016/0021-9991(78)90023-2