A modification of the CABARET scheme for numerical simulation of multicomponent gaseous flows in two-dimensional domains


  • A.V. Danilin Nuclear Safety Institute (IBRAE) of RAS https://orcid.org/0000-0001-7349-0600
  • A.V. Solovjev Nuclear Safety Institute (IBRAE) of RAS
  • A.M. Zaitsev Nuclear Safety Institute (IBRAE) of RAS




one-velocity multicomponent medium, systems of hyperbolic equations, CABARET scheme, computational fluid dynamics (CFD), conservative methods, finite volume methods, turbulent mixing


An explicit numerical algorithm for calculation of two-dimensional motion of multicomponent gas mixtures is proposed. A physical model as well as conservative and characteristic forms of governing equations are given. The discretization of the governing equations is made in accordance with the CABARET (Compact Accurately Boundary Adjusting-REsolution Technique) approach. The proposed algorithm is tested on problems of air shock waves passing through dense and dilute volume inhomogeneities with initial conditions adopted from numerical and experimental studies of other authors. A good agreement between the results of these studies and those obtained by the CABARET approach is shown.

Author Biographies

A.V. Danilin

A.V. Solovjev

A.M. Zaitsev


  1. R. Abgrall, “How to Prevent Pressure Oscillations in Multicomponent Flow Calculations: A Quasi Conservative Approach,” J. Comput. Phys. 125 (1), 150-160 (1996).
  2. V. M. Goloviznin and A. A. Samarskii, “Some Characteristics of Finite Difference Scheme ’Cabaret’,” Mat. Model. 10 (1), 101-116 (1998).
  3. V. M. Goloviznin and S. A. Karabasov, “Nonlinear Correction of Cabaret Scheme,” Mat. Model. 10 (12), 107-123 (1998).
  4. 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).
  5. V. M. Goloviznin, “Balanced Characteristic Method for 1D Systems of Hyperbolic Conservation Laws in Eulerian Representation,” Mat. Model. 18 (11), 14-30 (2006).
  6. 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).
  7. V. G. Kondakov, A Generalization of the ’Cabaret’ Scheme to Multidimensional Equations of Gas Dynamics , Candidate’s Dissertation in Mathematics and Physics (Moscow State Univ., Moscow, 2014).
  8. J. J. Quirk and S. Karni, “On the Dynamics of a Shock-Bubble Interaction,” J. Fluid Mech. series 318}, 129-163 (1996).
  9. J. W. Jacobs, “The Dynamics of Shock Accelerated Light and Heavy Gas Cylinders,” Phys. Fluids A series 5} (9), 2239-2247 (1993).
  10. R. S. Lagumbay, Modeling and Simulation of Multiphase/Multicomponent Flows , PhD Thesis (University of Colorado, Boulder, 2006).
  11. K. R. Bates, N. Nikiforakis, and D. Holder, “Richtmyer-Meshkov Instability Induced by the Interaction of a Shock Wave with a Rectangular Block of {@@m SF}_6,” Phys. Fluids 19 (2007). doi 10.1063/1.2565486
  12. J.-F. Haas and B. Sturtevant, “Interaction of Weak Shock Waves with Cylindrical and Spherical Gas Inhomogeneities,” J. Fluid Mech. 181, 41-76 (1987).
  13. D. A. Holder, A. V. Smith, C. J. Barton, and D. L. Youngs, “Shock-Tube Experiments on Richtmyer-Meshkov Instability Growth Using an Enlarged Double-Bump Perturbation,” Laser Part. Beams 21 (3), 411-418 (2003).
  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. M. Latini, O. Schilling, W. S. Don, “Effects of WENO Flux Reconstruction Order and Spatial Resolution on Reshocked Two-Dimensional Richtmyer-Meshkov Instability,” J. Comput. Phys. 221 (2), 805-836 (2007).
  16. R. L. Holmes, J. W. Grove, and D. H. Sharp, “Numerical Investigation of Richtmyer-Meshkov Instability Using Front Tracking,” J. Fluid Mech. 301, 51-64 (1995).
  17. R. H. Cohen, W. P. Dannevik, A. M. Dimits, et al., “Three-Dimensional Simulation of a Richtmyer-Meshkov Instability with a Two-Scale Initial Perturbation,” Phys. Fluids 14, 3692-3709 (2002).
  18. B. D. Collins and J. W. Jacobs, “PLIF Flow Visualization and Measurements of the Richtmyer-Meshkov Instability of an Air/SF6 Interface,” J. Fluid Mech. 464, 113-136 (2002).



How to Cite

Данилин А.В., Соловьев А.В., Зайцев А.М. A Modification of the CABARET Scheme for Numerical Simulation of Multicomponent Gaseous Flows in Two-Dimensional Domains // Numerical methods and programming. 2015. 16. 436-445. doi 10.26089/NumMet.v16r341



Section 1. Numerical methods and applications