A modification of the CABARET scheme for resolving the sound points in gas flows





systems of hyperbolic equations, CABARET scheme, computational fluid dynamics (CFD), conservative methods, sound points, rarefaction shock waves


An explicit numerical algorithm for resolving the sound points in the framework of the CABARET scheme is proposed. The sound points are characterized by a sign change in at least one of characteristic velocities. The flows in mesh nodes corresponding to sound points are calculated by solving the Riemann problem. This approach is successfully tested on a problem with sound transitions on rarefaction waves, on a problem with two diverging supersonic flows, and on a problem with supersonic flow over a forward facing step.

Author Biographies

A.V. Danilin

A.V. Solovjev


  1. M. V. Abakumov, S. I. Mukhin, Yu. P. Popov, and D. V. Rogozhkin, Shock Waves of Rarefaction in Computational Gas Dynamics , Preprint No. 3 (Keldysh Institute of Applied Mathematics, Moscow, 2006).
  2. Yu. M. Davydov, “Application of the Differential Approximation Method to Study and Construction of Nonlinear Difference Schemes,” Numer. Methods Continuum Mech. 11 (4), 41-59 (1980).
  3. O. A. Kuznetsov, Numerical Testing of Roe-Einfeldt Scheme for Hydrodynamics , Preprint No. 043 (Keldysh Institute of Applied Mathematics, Moscow, 1998).
  4. V. G. Kondakov, A Generalization of the Cabaret Scheme to Multidimensional Gas Dynamics Equations , Candidate’s Dissertation in Mathematics and Physics (Moscow State Univ., Moscow, 2014).
  5. V. M. Goloviznin, A. V. Solovjev, and V. A. Isakov, “An Approximation Algorithm for the Treatment of Sound Points in the CABARET Scheme,” Vychisl. Metody Programm. 17, 166-176 (2016).
  6. Yu. P. Raizer, Introduction to Hydrogasdynamics and the Theory of Shock Waves for Physicists (Intellekt, Dolgoprudnyi, 2011) [in Russian].
  7. B. Einfeldt, C. D. Munz, P. L. Roe, and B. Sjögreen, “On Godunov-Type Methods near Low Densities,” J. Comput. Phys. 92 (2), 273-295 (1991).
  8. P. Woodward and P. Colella, “The Numerical Simulation of Two-Dimensional Fluid Flow with Strong Shocks,” J. Comput. Phys. 54 (1), 115-173 (1984).
  9. V. M. Goloviznin and A. A. Samarskii, “Some Characteristics of Finite Difference Scheme Cabaret,” Mat. Model. 10 (1), 101-116 (1998).
  10. V. M. Goloviznin and S. A. Karabasov, “Nonlinear Correction of Cabaret Scheme,” Mat. Model. 10 (12), 107-123 (1998).
  11. V. M. Goloviznin, S. A. Karabasov, and I. M. Kobrinskii, “Balance-Characteristic Schemes with Separated Conservative and Flux Variables,” Mat. Model. 15 (9), 29-48 (2003).
  12. V. M. Goloviznin, “Balanced Characteristic Method for 1D Systems of Hyperbolic Conservation Laws in Eulerian Representation,” Mat. Model. 18 (11), 14-30 (2006).
  13. S. A. Karabasov and V. M. Goloviznin, “Compact Accurately Boundary-Adjusting High-Resolution Technique for Fluid Dynamics,” J. Comput. Phys. 228 (19), 7426-7451 (2009).
  14. A. V. Danilin and A. V. Solovjev, “A Modification of the CABARET Scheme for the Computation of Multicomponent Gaseous Flows,” Vychisl. Metody Programm. 16, 18-25 (2015).
  15. A. V. Danilin, A. V. Solovjev, and A. M. Zaitsev, “A Modification of the CABARET Scheme for Numerical Simulation of Multicomponent Gaseous Flows in Two-Dimensional Domains,” Vychisl. Metody Programm. 16, 436-445 (2015).



How to Cite

Данилин А., Соловьев А. A Modification of the CABARET Scheme for Resolving the Sound Points in Gas Flows // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2020. 20. 481-488. doi 10.26089/NumMet.v20r442



Section 1. Numerical methods and applications

Most read articles by the same author(s)