Time reversibility and stream correction in the CABARET scheme for the two-dimensional equation of convective transport


  • V.M. Goloviznin Nuclear Safety Institute (IBRAE) of RAS
  • D.Yu. Gorbachev Lomonosov Moscow State University
  • A.M. Kolokolnikov Lomonosov Moscow State University
  • P.A. Maiorov Lomonosov Moscow State University
  • B.A. Tlepsuk Lomonosov Moscow State University




CABARET scheme, shallow water equations, conservative schemes, time reversible schemes, numerical simulation


The CABARET (Compact Accurately Boundary Adjusting-REsolution Technique) difference scheme with five different modifications of nonlinear flow correction based on the maximum principle is considered for the two-dimensional linear convection equation. The computational efficiency of these different modifications is analyzed by the results of solving the Crowley’s problem on the cone rotation around the axis not coincident with the axis of the cone using condensing orthogonal grids. A number of recommendations are formulated to improve the computational efficiency of the entire class of CABARET schemes for the hyperbolic-type conservation laws and for the processes with dominant grid transfer.

Author Biographies

V.M. Goloviznin

D.Yu. Gorbachev

A.M. Kolokolnikov

P.A. Maiorov

B.A. Tlepsuk


  1. B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics (Nauka, Moscow, 1978; Amer. Math. Soc., Providence, 1982).
  2. A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Fizmatlit, Moscow, 2012; CRC Press, Boca Raton, 2001).
  3. 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].
  4. S. K. Godunov, “A Difference Method for Numerical Calculation of Discontinuous Solutions of the Equations of Hydrodynamics,” Mat. Sb. 47 (3), 271-306 (1959).
  5. A. A. Samarskii and Yu. P. Popov, Difference Schemes for Solving Gas Dynamics Problems (Nauka, Moscow, 1992) [in Russian].
  6. A. Harten, “High Resolution Schemes for Hyperbolic Conservation Laws,” J. Comput. Phys. 49 (3), 357-393 (1983).
  7. V. M. Goloviznin and S. A. Karabasov, “Nonlinear Correction of Cabaret Scheme,” Mat. Model. 10 (12), 107-123 (1998).
  8. V. M. Goloviznin, V. N. Semenov, I. A. Korotkin, and S. A. Karabasov, “A Novel Computational Method for Modelling Stochastic Advection in Heterogeneous Media,” Transp. Porous Med. 66 (3), 439-456 (2007).
  9. W. P. Crowley, “Numerical Advection Experiments,” Mon. Wea. Rev. 96 (1), 1-11 (1968).



How to Cite

Головизнин В.М., Горбачев Д.Ю., Колокольников А.М., Майоров П.А., Тлепсук Б.А. Time Reversibility and Stream Correction in the CABARET Scheme for the Two-Dimensional Equation of Convective Transport // Numerical methods and programming. 2016. 17. 393-401. doi 10.26089/NumMet.v17r436



Section 1. Numerical methods and applications