The case of multidimensional systems

  • Edwige Godlewski
  • Pierre-Arnaud Raviart
Part of the Applied Mathematical Sciences book series (AMS, volume 118)


We shall mainly consider in this section a two-dimensional p × p hyperbolic system


Riemann Problem Contact Discontinuity Young Measure Multidimensional System Numerical Flux 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Engquist, B. and A. Majda. Absorbing boundary conditions for the numerical simulation of waves, Math. Comp. 31 (1984), 629–651.MathSciNetCrossRefGoogle Scholar
  2. Jeffrey, A. Quasilinear hyperbolic systems and waves, Research Notes in Mathematics 5, Pitman, London (1976).MATHGoogle Scholar
  3. Hirsch, C. Numerical computation of internal and external flows, vol 2: Computational methods for inviscid and viscous flows, John Wiley Sons, Chichester, 1990, reprinted 1995.Google Scholar
  4. Richtmyer, R.D. and K.W. Morton. Difference Methods for Initial-Value Problems, Interscience, New York (1967).MATHGoogle Scholar
  5. Chang, T. and L. Hsiao: The Riemann problem and interaction of waves in gas dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics 41, Longman Scientific and Technical, Harlow, UK. 1989.Google Scholar
  6. Charrier, P., B. Dubroca, and L. Flandrin: Un solveur de Riemann approché pour l’étude d’écoulements hypersoniques bidimensionnels, C. R. Acad. Sci. Paris 317, Sér. I, (1993), 1083–1086.Google Scholar
  7. Charrier, P. and B. Tessieras. On front-tracking methods applied to hyperbolic systems of nonlinear conservation laws, SIAM J. Numer. Anal. 23 (1986), 461–472.Google Scholar
  8. Chen, G.-Q. The method of quasi-decoupling for discontinuous solutions of conservation laws, Arch. Rat. Mech. Anal. 121 (1992), 131–185.CrossRefMATHGoogle Scholar
  9. Chen, G.-Q. and J. Glimm. Shock capturing and global solutions to the compressible Euler equations with geometrical structure, in Hyperbolic problems: theory, numerics, applications, Proceedings of the Fifth International Conference on Hyperbolic Problems, StonyBrook 1994, J. Glimm et al. (Eds.), World Scientific, Singapore (1996), 101–109.Google Scholar
  10. Liu, T.P. Solutions in the large for the equations of nonisentropic gas dynamics, Indiana Univ. Math. J. 26 (1977), 147–177.MathSciNetCrossRefMATHGoogle Scholar
  11. Liu, T.P. Admissible solutions of hyperbolic conservation laws, A.M.S. Memoirs 30 (240) (1981).Google Scholar
  12. Liu, T.-P. and Z. Xin. Overcompressive shock wave, in Nonlinear Evolution Equations that Change Type, IMA Volumes in Mathematics and its Applications 27, B.L. Keyfitz and M. Shearer (Eds.), Springer-Verlag, New York (1991), 139–145.Google Scholar
  13. Liu, T.-P. and L.-A. Ying. Nonlinear stability of strong detonations for a viscous combustion model, SIAM J. Math. Anal. 26 (3) (1995), 519–528.MathSciNetCrossRefMATHGoogle Scholar
  14. Liu, X.B. A maximum principle satisfying modification of triangle based adaptive stencils for the solution of scalar hyperbolic conservation laws, SIAM J. Numer. Anal. 30 (1993), 701–716.MathSciNetCrossRefMATHGoogle Scholar
  15. Liu X.D., S. Osher, and T. Chan. Weighted essentially non-oscillatory schemes, J. Comp. Phys. 115 (1994), 200–212.MathSciNetCrossRefMATHGoogle Scholar
  16. Liu, Y. and M. Vinokur. Nonequilibrium flow computations. An analysis of numerical formulations of conservation laws, J. Comp. Phys. 83 (1989), 373–397.Google Scholar
  17. Loh, C.Y. and W.H. Hui. A new Lagrangian method for steady supersonic flow computation I. Godunov scheme, J. Comp. Phys. 89 (1990), 207–240.MathSciNetCrossRefMATHGoogle Scholar
  18. Cargo, P. and G. Gallice: Un solveur de Roe pour les équations de la magnétohydrodynamique, C. R. Acad. Sci. Paris 320, Série I (1995), 1269–1272.Google Scholar
  19. Kreiss, H.-O. Stability theory for difference approximations of mixed initial boundary value problems. I, Math. Comp. 22 (1968), 703–714.Google Scholar
  20. Kröner, D. Absorbing boundary conditions for the linearized Euler equations in 2-D, Math. Comp. 57 (195) (1991), 153–167.MathSciNetCrossRefMATHGoogle Scholar
  21. Kröner, D. and M. Rokyta. Convergence of upwind finite volume schemes for scalar conservation laws in two dimensions, SIAM J. Numer. Anal. 31, 2 (1994), 324–343.MathSciNetCrossRefMATHGoogle Scholar
  22. Kruzkov, S. First-order quasilinear equations in several independent variables, Mat. Sb. 123 (1970), 228–255Google Scholar
  23. Kruzkov, S. English translation in Math. USSR. Sb. 10 (1970), 217–243.CrossRefGoogle Scholar
  24. Kumbaro, A. Modélisation, analyse mathématique et numérique des modèles bi-fluides d’écoulement diphasique, Doctoral dissertation, Thesis Université Paris-Sud, Orsay, France (1992).Google Scholar
  25. Kunhardt E.E. and C. Wu. Towards a more accurate Flux Corrected Transport algorithm, J. Comp. Phys. 68 (1987), 127–150.CrossRefMATHGoogle Scholar
  26. Lallemand, M.H. Dissipative properties of Runge—Kutta schemes with upwind spatial approximation for the Euler equations, INRIA Research Report 1173 (1990), INRIA Rocquencourt, 78153 Le Chesnay, France.Google Scholar
  27. Lallemand, M.H., F. Fezoui, and E. Perez. Un schéma multigrille en éléments finis décentré pour les équations d’Euler, INRIA Research Report 602 (1987), INRIA Rocquencourt, 78153 Le Chesnay, France.Google Scholar
  28. Larrouturou, B. Recent progress in reactive flow computations, in Computing methods in applied sciences and engineering (Proceedings of the ninth conference on computing methods in applied sciences and engineering, INRIA, Paris 1990), R. Glowinski and A. Lichnewsky (Eds.), SIAM, Philadelphia (1990), 249–272.Google Scholar
  29. Larrouturou, B. How to preserve the mass fractions positivitiy when computing compressible multicomponent flows, J. Comp. Phys. 92 (1991), 273–295.MathSciNetCrossRefGoogle Scholar
  30. Larrouturou, B. On upwind approximations of multi-dimensional multi-species flows, in Proceedings of the first European CFD Conference, Ch. Hirsh, J. Pénaux and E. Onate (Eds.), Elsevier, Amsterdam (1992), 117–126.Google Scholar
  31. Garcia-Navarro, P., M.E. Hubbard, and A. Priestley. Genuinely multidimensional upwinding for the 2d shallow water equations, J. Comp. Phys. 121 (1993), 79–93.MathSciNetCrossRefGoogle Scholar
  32. Glaister, P. An approximate linearized Riemann solver for the Euler equations for real gases, J. Comp. Phys. 74 (1994), 382–408.CrossRefGoogle Scholar
  33. Blunt, M. and B. Rubin: Implicit flux limiting schemes for petroleum reservoir simulation, J. Comp. Phys. 102 (1992), 194–210.MathSciNetCrossRefMATHGoogle Scholar
  34. Raviart, P.A. and L. Sainsaulieu. Mathematical and numerical modelling of two-phase flows, in Computing Methods in Applied Sciences and Engineering, Proceedings of the tenth International Conference on Computing Methods in Applied Sciences and Engineering, France (1995), R. Glowinski (Ed.), Nova Science Publishers, Inc., New York, 119–132.Google Scholar
  35. Durlofsky, L., B. Engquist, and S. Osher. Triangle based adaptive stencils for the solution of hyperbolic conservation laws, J. Comp. Phys. 98 (1993), 64–73.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • Edwige Godlewski
    • 1
  • Pierre-Arnaud Raviart
    • 2
  1. 1.Laboratoire d’Analyse NumériqueUniversité Pierre et Marie CurieParis Cedex 05France
  2. 2.Centre de Mathématiques AppliquéesEcole PolytechniquePalaiseau CedexFrance

Personalised recommendations