Applied Mathematics and Mechanics

, Volume 33, Issue 10, pp 1329–1350 | Cite as

Lagrangian cell-centered conservative scheme

  • Quan-wen Ge (葛全文)Email author


This paper presents a Lagrangian cell-centered conservative gas dynamics scheme. The piecewise constant pressures of cells arising from the current time sub-cell densities and the current time isentropic speed of sound are introduced. Multipling the initial cell density by the initial sub-cell volumes obtains the sub-cell Lagrangian masses, and dividing the masses by the current time sub-cell volumes gets the current time subcell densities. By the current time piecewise constant pressures of cells, a scheme that conserves the momentum and total energy is constructed. The vertex velocities and the numerical fluxes through the cell interfaces are computed in a consistent manner due to an original solver located at the nodes. The numerical tests are presented, which are representative for compressible flows and demonstrate the robustness and accuracy of the Lagrangian cell-centered conservative scheme.

Key words

sub-cell force Lagrange cell-centered scheme Lagrangian cell-centered conservative gas dynamics scheme piecewise constant pressure of cell 

Chinese Library Classification


2010 Mathematics Subject Classification

76N15 65M06 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Maire, P. H., Abgrall, R., Breil, J., and Ovadia, J. A cell-centered Lagrangian scheme for compressible flow problems. SIAM J. Sci. Comput., 29(4), 1781–1824 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Von Neumann, J. and Richtmyer, R. D. A method for the numerical calculations of hydrodynamics shocks. J. Appl. Phys., 21, 232–238 (1950)MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Wilkins, M. L. Calculation of elastic plastic flow. Methods in Computational Physics (ed. Alder, B.), Vol. 3, Academic Press, New York (1964)Google Scholar
  4. [4]
    Caramana, E. J. and Shashkov, M. J. Elimination of artificial grid distorsion and hourglass-type motions by means of Lagrangian subzonal masses and pressures. J. Comput. Phys., 142, 521–561 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Caramana, E. J., Shashkov, M. J., and Whalen, P. P. Formulations of artificial viscosity for multidimensional shock wave computations. J. Comput. Phys., 144, 70–97 (1998)MathSciNetCrossRefGoogle Scholar
  6. [6]
    Campbell, J. C. and Shashov, J. C. A tensor artificial viscosity using a mimetic finite difference algorithm. J. Comput. Phys., 172, 739–765 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Caramana, E. J., Burton, D. E., Shashov, M. J., and Whalen, P. P. The construction of compatible hydrodynamics algorithms utilizing conservation of total energy. J. Comput. Phys., 146, 227–262 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Campbell, J. C. and Shashov, M. J. A compatible Lagrangian hydrodynamics algorithm for unstructured grids. Selcuk J. Appl. Math., 4(2), 53–70 (2003)zbMATHGoogle Scholar
  9. [9]
    Scovazzi, G., Christon, M. A., Hughes, T. J. R., and Shadid, J. N. Stabilized shock hydrodynamics: I. a Lagrangian method. Comput. Methods Appl. Mech. Engrg., 196, 923–966 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Scovazzi, G. Stabilized shock hydrodynamics: II. design and physical interpretation of the SUPG operator for Lagrangian computations. Comput. Methods Appl. Mech. Engrg., 196, 966–978 (2007)Google Scholar
  11. [11]
    Scovazzi, G., Love, E., and Shashkov, M. J. Multi-scale Lagrangian shock hydrodynamics on Q 1/P 0 finite elements: theoretical framework and two-dimensional computations. Comput. Methods Appl. Mech. Engrg., 197, 1056–1079 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Godunov, S. K., Zabrodine, A., Ivanov, M., Kraiko, A., and Prokopov, G. Résolution Numérique des Problèmes Multidimensionnels de la Dynamique des Gaz, Editions Mir, Moscow (1979)Google Scholar
  13. [13]
    Adessio, F. L., Carroll, D. E., Dukowicz, J. K., Johnson, J. N., Kashiwa, B. A., Maltrud, M. E., and Ruppel, H. M. Caveat: a Computer Code for Fluid Dynamics Problems with Large Distortion and Internal Slip, Technical Report LA-10613-MS, Los Alamos National Laboratory (1986)Google Scholar
  14. [14]
    Dukowicz, J. K. and Meltz, B. Vorticity errors in multidimensional Lagrangian codes. J. Comput. Phys., 99, 115–134 (1992)zbMATHCrossRefGoogle Scholar
  15. [15]
    Desp’res, B. and Mazeran, C. Lagrangian gas dynamics in two dimensions and Lagrangian systems. Arch. Rational Mech. Anal., 178, 327–372 (2005)MathSciNetCrossRefGoogle Scholar
  16. [16]
    Carré, G., Delpino, S., Desp’res, B., and Labourasse, E. A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension. J. Comput. Phys., 228, 5160–5183 (2009)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute of Applied Physics and Computational MathematicsBeijingP. R. China

Personalised recommendations