Stable and monotonic numerical schemes for discontinuous flow modeling based on conservative characteristic form of conservation laws

Mathematics. Physics. Mechanics


Grudnitsky V. G.*, Mendel M. A.

Moscow Institute of Physics and Technology, 9, Institutskiy per., Dolgoprudny, Moscow region, 141701, Russia



The article is concerned with numerical schemes for gas-dynamics problems.
The goal of research was to develop effective numerical algorithms using conservative characteristic form of conservation laws.
Conservative characteristic form was established from divergence form of conservation laws with the aid of identity flows transformation. No linearization of the flow’s vector-function (which requires continuity of the function) was made as opposed to Euler equations. That’s why final equations are conservative and also admit discontinuous solutions. Quasi-linear conservative representation of conservation laws allows developing conservative characteristic numerical schemes and formulating necessary and sufficient conditions of stability and monotonicity.
There were made two numerical schemes which are presented in this article. First one is appropriate for one-dimensional non-static problems. A number of tests (propagation of a shock-wave through high and low density area) were made. The results appeared to be extremely accurate next to discontinuities. Shock-wave and tangential discontinuity fronts are propagated without numerical diffusion influence. Two-dimensional supersonic and hypersonic flows modeling are first extension of the method. The fact that fluid velocity is greater than sound velocity allows us to solve time-independent equations. Static computations of supersonic flow in a flat duct with a wedge were made. The scheme can be simply modified to solve cylindrical-cemetery problems.
Numerical schemes for three-dimensional models call for further investigations. Present study demonstrates wide range of advantages which conservative form of conservation laws provides for construction of numerical schemes for gas dynamics. Monotonicity, stability, conservative and characteristic type make them work for discontinuous flows modeling and accounts for high-precisely results.


conservative characteristic scheme, shock-capturing, static supersonic flows, necessary and sufficient condition of stability and monotonicity


  1. Godunov S.K. Uspekhi matematicheskikh nauk, 1957, vol. 12, no. 1, pp. 176-177.
  2. Grudnitskii V.G. Algoritmy dlya chislennogo issledovaniya razryvnykh techenii, Sbornik statei, 1993, pp. 191-203.
  3. Grudnitskii V.G. Mat. modelirovanie, 1997, vol. 9, no. 12, pp. 121-125.
  4. Grudnitskii V.G. Dokladi Akademii nauk, 1998, vol. 362, no. 3, pp. 298-299.
  5. Grudnitskii V.G., Plotnikov., P.V. Novoe v chislennom modelirovanii, Sbornik statei, 2000, pp.148-164.
  6. Grudnitsky V.G. Computational Fluid Dynamics Journal, 2001, vol. 10, no.2, pp. 334-337.
  7. Grudnitskii V.G. Mat. modelirovanie, 2004, vol. 16, no.1, pp. 90-96.
  8. Grudnitskii. V.G. Mat. modelirovanie, 2005, vol. 17, no.12, pp. 119-128.
  9. Grudnitskii V.G. Mat. modelirovanie, 2006, vol. 18, no. 10, pp. 76-80.
  10. Grudnitskii V.G. Mat. modelirovanie, 2008, vol. 20, no. 2, pp. 93-104.
  11. Grudnitsky V.G. Materialy 27-th international symposium on shock wawes, 2009, pp. 184-185.
  12. Grudnitskii V.G. Nelineinye problemy zakonov sokhraneniya sploshnoi sredy (Nonlinear problems of conservation laws of continuous medium), Sputnik+, Moscow, 2009, pp. 1-84.
  13. Grudnitskii V.G. Obozrenie prikladnoi i promyshlennoi matematiki, 2011, vol. 18, no. 1, pp.725-726.
  14. Grudnitskii V.G. Materialy 9 mezhdunarodnoi konferentsii po neravnovesnym protsessam v soplakh i struyakh, 2012, pp. 552-554.
  15. Mendel M.A. Materialy 9 mezhdunarodnoi konferentsii po neravnovesnym protsessam v soplakh i struyakh, 2012, pp. 564-565.

Download — informational site MAI

Copyright © 2000-2020 by MAI