An introduction to boundary conditions

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


The aim of this chapter is to introduce the unfamiliar reader to the topic of boundary conditions: we just want to give some insight into this question and do not pretend to give an exhaustive study. We recall first the main features of the initial boundary value problem (I.B.V.P.) before we present the numerical treatment of the question.


Boundary Data Riemann Problem Positive Eigenvalue Incoming Wave Absorb Boundary Condition 
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. Higdon, R.L. Initial-boundary value problems for linear hyperbolic systems, SIAM Rev. 28 (1986), 177–217.MathSciNetCrossRefMATHGoogle Scholar
  2. Yee, H.C. Construction of explicit and implicit symmetric TVD schemes and their applications, J. Comp. Phys. 68 (1987), 151–179.MathSciNetCrossRefMATHGoogle Scholar
  3. Gustafsson, B., H.-O. Kreiss, and A. Sundström. Stability theory of difference approximations for mixed initial boundary value problems, II, Math. Comp., 26 (1972), 649–686.Google Scholar
  4. Engquist, B. and A. Majda. Absorbing boundary conditions for the numerical simulation of waves, Math. Comp. 31 (1977), 629–651.MathSciNetCrossRefMATHGoogle Scholar
  5. Hedstrom, G.W. Nonreflecting boundary conditions for nonlinear hyperbolic systems, J. Comp. Phys. 30 (1979), 222–237.MathSciNetCrossRefMATHGoogle Scholar
  6. Thompson, K.W. Time dependent boundary conditions for hyperbolic systems, II, J. Comp. Phys. 89 (1990), 439–461.Google Scholar
  7. Le Floch, Ph. Entropy weak solutions to nonlinear hyperbolic systems in nonconservation form, in Nonlinear Hyperbolic Equations - Theory, Computation Methods, and Applications (Proceedings of the Second International Conference on Nonlinear Hyperbolic Problems, Aachen 1988), J. Ballmann and R. Jeltsch (Eds.), Notes on Numerical Fluid Mechanics 24, Vieweg, Braunschweig (1989), 362–373, and Comm. Part. Diff. Equations 13 (6) (1988), 669–127.Google Scholar
  8. Gisclon, M. Étude des conditions aux limites pour des système strictement hyperboliques, via l’approximation parabolique, Thesis Université Claude Bernard - Lyon I (France) (1994).Google Scholar
  9. Karni, S. Far field boundaries and their numerical treatment: an unconventional approach, Numerical methods for fluid dynamics III, Proceedings of the conference on Numerical methods for fluid dynamics (held in Oxford, March 1988 ), K.W. Morton and M.J. Baines (Eds.); IMA Conference Series 17, Clarendon Press, Oxford, 1988, 307–512.Google Scholar
  10. Sod, G.A. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comp. Phys. 27 (1978), 1–31.MathSciNetCrossRefMATHGoogle Scholar
  11. Sod, G.A. Numerical Methods in fluid dynamics: Initial and boundary value problems, Cambridge University Press, Cambridge 1987.Google Scholar
  12. Song, Y. and T. Tang. Dispersion and group velocity in numerical schemes for three-dimensional hydrodynamic equations, J. Comp. Phys. 105 (1993), 72–82.CrossRefMATHGoogle Scholar
  13. Spekreijse, S. Multigrid solution of monotone second-order discretization of hyperbolic conservation laws, Math. Comp. 49 (1987), 135–156.MathSciNetCrossRefMATHGoogle Scholar
  14. Steger, J. and R.F. Warming. Flux vector splitting of the inviscid gas dynamics equation with application to finite difference methods, J. Comp. Phys. 40 (1981), 263–293.MathSciNetCrossRefMATHGoogle Scholar
  15. Stewart, H.B. and B. Wendroff. Two phase flow: models and methods, J. Comp. Phys. 56 (1984), 363–409.MathSciNetCrossRefMATHGoogle Scholar
  16. Stoufflet, B. Implicit finite element methods for the Euler equations, in Numerical methods for the Euler equations of fluid dynamics, Proceedings of the IN-RIA workshop, Rocquencourt, France (1983), F. Angrand et al. (Eds.), SIAM Philadelphia (1985), 409–434.Google Scholar
  17. Sun, M.T., S.T. Wu, and M. Dryer. On the time-dependent numerical boundary conditions of magnetohydrodynamic flows, J. Comp. Phys. 116 (1995), 330–342.CrossRefMATHGoogle 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