A modification of the CABARET scheme for the computation of multicomponent gaseous flows





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


An explicit numerical algorithm for computing one-dimensional motion of multicomponent gaseous mixtures is proposed. A physical model and the equations of motion are presented in conservative and characteristic forms. 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 the Riemann problem with different gases on the different sides of the initial discontinuity. The resulting numerical solutions are compared with the analytical one and with those obtained by other numerical approaches. It is shown that the proposed algorithm is of high accuracy on the class of problems being considered.

Author Biographies

A.V. Danilin

A.V. Solovjev


  1. E. Johnsen and T. Colonius, “Implementation of WENO Schemes in Compressible Multicomponent Flow Problems,” J. Comput. Phys. 219 (2), 715-732 (2006).
  2. V. Coralic and T. Colonius, “Finite-Volume WENO Scheme for Viscous Compressible Multicomponent Flows,” J. Comput. Phys. 274. 95-121 (2014).
  3. D. Igra and K. Takayama, “A High Resolution Upwind Scheme for Multi-Component Flows,” Int. J. Numer. Meth. Fluids 38 (10), 985-1007 (2002).
  4. R. Abgrall and S. Karni, “Computations of Compressible Multifluids,” J. Comput. Phys. 169 (2), 594-623 (2001).
  5. J.-P. Cocchi and R. Saurel, “A Riemann Problem Based Method for the Resolution of Compressible Multimaterial Flows,” J. Comput. Phys. 137 (2), 265-298 (1997).
  6. J.-P. Cocchi, R. Saurel, and J. C. Loraud, “Treatment of Interface Problems with Godunov-Type Schemes,” Shock Waves 5 (6), 347-357 (1996).
  7. R. Saurel and R. Abgrall, “A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows,” J. Comput. Phys. 150 (2), 425-467 (1999).
  8. V. S. Surov and E. N. Stepanenko, “The Grid Method of Characteristics for Calculating Flows of a Single-Velocity Multicomponent Heat-Conducting Medium,” Vestn. Chelyabinsk Univ., Ser. Fiz., No. 8, 15-22 (2010).
  9. V. S. Surov, “Method of the Characteristics for Calculation of Currents of One-Speed Heterogeneous Mixtures in Lagrange Variables,” Mat. Model. 15 (5), 37-46 (2003).
  10. R. Abgrall, “How to Prevent Pressure Oscillations in Multicomponent Flow Calculations: A Quasi Conservative Approach,” J. Comput. Phys. 125 (1), 150-160 (1996).
  11. B. Larrouturou, “How to Preserve the Mass Fractions Positivity when Computing Compressible Multi-Component Flows,” J. Comput. Phys. 95 (1), 59-84 (1991).
  12. V. M. Goloviznin and A. A. Samarskii, “Some Characteristics of Finite Difference Scheme, “Cabaret’’,” Mat. Model. 10 (1), 101-116 (1998).
  13. V. M. Goloviznin and S. A. Karabasov, “Nonlinear Correction of Cabaret Scheme,” Mat. Model. 10 (12), 107-123 (1998).
  14. 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).
  15. V. M. Goloviznin, “Balanced Characteristic Method for 1D Systems of Hyperbolic Conservation Laws in Eulerian Representation,” Mat. Model. 18 (11), 14-30 (2006).
  16. 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).
  17. 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).



How to Cite

Данилин А.В., Соловьев А.В. A Modification of the CABARET Scheme for the Computation of Multicomponent Gaseous Flows // Numerical methods and programming. 2015. 16. 18-25. doi 10.26089/NumMet.v16r103



Section 1. Numerical methods and applications