CABARET scheme on moving grids for two-dimensional equations of gas dynamics and dynamic elasticity




conservative-characteristic methods, CABARET scheme, arbitrary Lagrangian-Eulerian (ALE) variables, hyperbolic type equations, free boundary


The conservative-characteristic CABARET scheme is widely used in solving many problems for systems of differential equations of hyperbolic type in Euler variables. The increasing urgency of the problems of interaction of deformable bodies with liquid and gas flows requires the adaptation of this method to Lagrangian and arbitrary Lagrangian-Eulerian variables. Earlier, the CABARET scheme was constructed for one-dimensional equations of gas dynamics in mass Lagrangian variables, as well as for three-dimensional equations of dynamic elasticity. In the first case, the constructed scheme could not be generalized to multidimensional problems, and in the second, a time-irreversible grid movement algorithm was used. This paper presents a generalization of the CABARET method to two-dimensional equations of gas dynamics and dynamic elasticity in arbitrary Lagrangian-Eulerian and Lagrangian variables. The constructed method is explicit, easily scalable, and has the property of temporal reversibility. The method is tested on various one-dimensional and two-dimensional problems for both systems of equations (collision of elastic bodies, transverse vibrations of an elastic beam, motion of the free boundary of an ideal gas).

Author Biographies

Nikita A. Afanasiev

Petr A. Maiorov


  1. Y. Bazilevs, M.-C. Hsu, D. J. Benson, et al., “Computational Fluid–Structure Interaction: Methods and Application to a Total Cavopulmonary Connection,” Comput. Mech. 45 (1), 77-89 (2009).
  2. K. Takizawa, D. Montes, M. Fritze, et al., “Methods for FSI Modeling of Spacecraft Parachute Dynamics and Cover Separation,” Math. Models Methods Appl. Sci. 23 (2), 307-338 (2013).
  3. A. Korobenko, M.-C. Hsu, I. Akkerman, et al., “Structural Mechanics Modeling and FSI Simulation of Wind Turbines,” Math. Models Methods Appl. Sci. 23 (2), 249-272 (2013).
  4. J.-F. Sigrist, D. Broc, and C. Lainé, “Dynamic Analysis of a Nuclear Reactor with Fluid–Structure Interaction: Part I: Seismic Loading, Fluid Added Mass and Added Stiffness Effects,” Nucl. Eng. Des. 236 (23), 2431-2443 (2006).
  5. J.-F. Sigrist, D. Broc, and C. Lainé, “Dynamic Analysis of a Nuclear Reactor with Fluid–Structure Interaction: Part II: Shock Loading, Influence of Fluid Compressibility,” Nucl. Eng. Des. 237 (3), 289-299 (2007).
  6. C. Michler, S. J. Hulshoff, E. H. van Brummelen, and R. de Borst, “A Monolithic Approach to Fluid–Structure Interaction,” Comput. Fluids 33 (5-6), 839-848 (2004).
  7. W. G. Dettmer and D. Peric, “On the Coupling between Fluid Flow and Mesh Motion in the Modeling of Fluid–Structure Interaction,” Comput. Mech. 43 (1), 81-90 (2008).
  8. O. O. Bendiksen, “Modern Developments in Computational Aeroelasticity,” Proc. Inst. Mech. Eng. Part G: J. Aerosp. Eng. 218 (3), 157-177 (2004).
  9. V. M. Goloviznin, M. A. Zaitsev, S. A. Karabasov, and I. A. Korotkin, Novel Algorithms of Computational Hydrodynamics for Multicore Computing (Mosk. Gos. Univ., Moscow, 2013) [in Russian].
  10. V. M. Goloviznin and B. N. Chetverushkin, “New Generation Algorithms for Computational Fluid Dynamics,” Zh. Vychisl. Mat. Mat. Fiz. 58 (8), 20-29 (2018) [Comput. Math. Math. Phys. 58 (8), 1217-1225 (2018)].
  11. 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).
  12. M. A. Zaitsev and S. A. Karabasov, “Cabaret Scheme for Computational Modelling of Linear Elastic Deformation Problems,” Mat. Model. 29 (11), 53-70 (2017).
  13. V. M. Goloviznin and S. A. Karabasov, “Nonlinear Correction of Cabaret Scheme,” Mat. Model. 10 (12), 107-123 (1998).
  14. N. Afanasiev and V. Goloviznin, “A Locally Implicit Time-Reversible Sonic Point Processing Algorithm for One-Dimensional Shallow-Water Equations,” J. Comput. Phys. 434 (2021). doi{10.1016/}.
  15. V. M. Goloviznin and N. A. Afanasiev, “Monolithic balance-characteristic method for solving problems of interaction of liquid and gas with deformable objects,” Mat. Model. 33 (10), 65-82 (2021).
  16. N. A. Afanasiev, V. M. Goloviznin, and A. V. Solovjev, “CABARET Scheme with Improved Dispersion Properties for Systems of Linear Hyperbolic-Type Differential Equations,” Vychisl. Metody Programm. 22 (1), 67-76 (2021).



How to Cite

Афанасьев Н.А., Майоров П.А. CABARET Scheme on Moving Grids for Two-Dimensional Equations of Gas Dynamics and Dynamic Elasticity // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2021. 22. 306-321. doi 10.26089/NumMet.v22r420



Methods and algorithms of computational mathematics and their applications