Decay of Unstable Strong Discontinuities in the Case of a Convex-Flux Scalar Conservation Law Approximated by the CABARET Scheme

  • N. A. ZyuzinaEmail author
  • V. V. OstapenkoEmail author


Monotonicity conditions for the CABARET scheme approximating a quasilinear scalar conservation law with a convex flux are obtained. It is shown that the monotonicity of the CABARET scheme for Courant numbers \(r \in (0.5,1]\) does not ensure the complete decay of unstable strong discontinuities. For the CABARET scheme, a difference analogue of an entropy inequality is derived and a method is proposed ensuring the complete decay of unstable strong discontinuities in the difference solution for any Courant number at which the CABARET scheme is stable. Test computations illustrating these properties of the CABARET scheme are presented.


monotone CABARET scheme scalar conservation law strong discontinuity difference analogue of entropy inequality. 



This work was supported in part by the Russian Foundation for Basic Research, project no. 16-01-00333.


  1. 1.
    V. M. Goloviznin and A. A. Samarskii, “Finite approximation of convective transport with a space splitting of time derivative,” Mat. Model. 10 (1), 86–100 (1998).MathSciNetzbMATHGoogle Scholar
  2. 2.
    V. M. Goloviznin and A. A. Samarskii, “Some properties of the difference scheme CABARET,” Mat. Model. 10 (1), 101–11 (1998).MathSciNetzbMATHGoogle Scholar
  3. 3.
    B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics (Nauka, Moscow, 1978; Am. Math. Soc., Providence, 1983).Google Scholar
  4. 4.
    A. G. Kulikovskii, N. V. Pogorelov, and F. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Fizmatlit, Moscow, 2001; Chapman and Hall/CRC, London, 2001).Google Scholar
  5. 5.
    V. M. Goloviznin, “Balanced characteristic method for systems of hyperbolic conservation laws,” Dokl. Math. 72 (1), 619–623 (2005).zbMATHGoogle Scholar
  6. 6.
    P. Woodward and P. Colella, “The numerical simulation of two-dimensional fluid flow with strong shocks,” J. Comput. Phys. 54 (1), 115–173 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    V. V. Ostapenko, “On the monotonicity of the balanced characteristic scheme,” Mat. Model. 21 (7), 29–42 (2009).MathSciNetzbMATHGoogle Scholar
  8. 8.
    V. V. Ostapenko, “On the strong monotonicity of the CABARET scheme,” Comput. Math. Math. Phys. 52 (3), 387–399 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    V. M. Goloviznin and A. A. Kanaev, “The principle of minimum of partial local variations for determining convective flows in the numerical solution of one-dimensional nonlinear scalar hyperbolic equations,” Comput. Math. Math. Phys. 51 (7), 824–839 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    V. M. Goloviznin, M. A. Zaitsev, S. A. Karabasov, and I. A. Korotkin, New CFD Algorithms for Multiprocessor Computer Systems (Mosk. Gos. Univ., Moscow, 2013) [in Russian].Google Scholar
  11. 11.
    S. A. Karabasov and V. M. Goloviznin, “New efficient high-resolution method for nonlinear problems in aeroacoustics,” AIAA J. 45 (12), 2861–2871 (2007).CrossRefGoogle Scholar
  12. 12.
    S. A. Karabasov, P. S. Berloff, and V. M. Goloviznin, “Cabaret in the ocean gyres,” Ocean Model. 30 (2), 155–168 (2009).CrossRefGoogle Scholar
  13. 13.
    V. M. Goloviznin and V. A. Isakov, “Balance-characteristic scheme as applied to the shallow water equations over a rough bottom,” Comput. Math. Math. Phys. 57 (7), 1140–1157 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    O. A. Kovyrkina and V. V. Ostapenko, “On monotonicity of two-layer in time cabaret scheme,” Math. Model. Comput. Simul. 5 (2), 180–189 (2013).MathSciNetCrossRefGoogle Scholar
  15. 15.
    O. A. Kovyrkina and V. V. Ostapenko, “Monotonicity of the CABARET scheme approximating a hyperbolic equation with a sign-changing characteristic field,” Comput. Math. Math. Phys. 56 (5), 783–801 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    O. A. Kovyrkina and V. V. Ostapenko, “On the monotonicity of the CABARET scheme in the multidimensional case,” Dokl. Math. 91 (3), 323–328 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    N. A. Zyuzina and V. V. Ostapenko, “On the monotonicity of the CABARET scheme approximating a scalar conservation law with a convex flux,” Dokl. Math. 93 (1), 69–73 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    N. A. Zyuzina and V. V. Ostapenko, “Monotone approximation of a scalar conservation law based on the CABARET scheme in the case of a sign-changing characteristic field,” Dokl. Math. 94 (2), 538–542 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (Soc. Ind. Appl. Math., Philadelphia, 1972).Google Scholar
  20. 20.
    V. V. Ostapenko, Hyperbolic Systems of Conservation Laws and Their Application to Shallow Water Theory (Novosib. Gos. Univ., Novosibirsk, 2014) [in Russian].Google Scholar
  21. 21.
    K. O. Friedrichs and P. D. Lax, “Systems of conservation equation with convex extension,” Proc. Natl. Acad. Sci. USA 68 (8), 1686–1688 (1971).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Yu. I. Shokin and N. N. Yanenko, Method of Differential Approximation (Nauka, Novosibirsk, 1985) [in Russian].zbMATHGoogle Scholar
  23. 23.
    P. Lax and B. Wendroff, “Systems of conservation laws,” Commun. Pure Appl. Math. 13, 217–237 (1960).CrossRefzbMATHGoogle Scholar
  24. 24.
    A. Harten, J. M. Hyman, and P. D. Lax, “On finite-difference approximations and entropy conditions for shocks,” Commun. Pure Appl. Math. 29, 297–322 (1976).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    A. Harten, “High resolution schemes for hyperbolic conservation laws,” J. Comput. Phys. 49, 357–393 (1983).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    A. Harten and S. Osher, “Uniformly high-order accurate nonoscillatory schemes,” SIAM J. Numer. Anal. 24 (2), 279–309 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    V. P. Pinchukov and C.-W. Shu, High-Order Numerical Methods for Problems in Aerodynamics (Sib. Otd. Ross. Akad. Nauk, Novosibirsk, 2000) [in Russian].zbMATHGoogle Scholar
  28. 28.
    V. V. Ostapenko, “Symmetric compact schemes with higher order conservative artificial viscosities,” Comput. Math. Math. Phys. 42 (7), 980–999 (2002).MathSciNetGoogle Scholar
  29. 29.
    T. C. Fisher and M. H. Carpenter, High-Order Entropy Stable Finite Difference Schemes for Nonlinear Conservation Laws: Finite Domains (Langley Research Center, Hampton, Virginia, 2013).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Lavrent’ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of SciencesNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

Personalised recommendations