Inverse Lax–Wendroff Procedure for Numerical Boundary Conditions of Hyperbolic Equations: Survey and New Developments

  • Sirui Tan
  • Chi-Wang ShuEmail author
Part of the Fields Institute Communications book series (FIC, volume 66)


In this paper, we give a survey and discuss new developments and computational results for a high order accurate numerical boundary condition based on finite difference methods for solving hyperbolic equations on Cartesian grids, while the physical domain can be arbitrarily shaped. The challenges result from the wide stencil of the high order interior scheme and the fact that the physical boundary does not usually coincide with grid lines. There are two main ingredients of the method. The first one is an inverse Lax-Wendroff procedure for inflow boundary conditions and the other one is a robust and high order accurate extrapolation for outflow boundary conditions. The method is high order accurate, stable under standard CFL conditions determined by the interior schemes, and easy to implement. We show applications in simulating interactions between compressible inviscid flows and rigid (static or moving) boundaries.


Numerical boundary conditions Hyperbolic conservation laws Cartesian mesh Inverse Lax-Wendroff procedure Extrapolation 



Research is supported by AFOSR grant FA9550-09-1-0126 and NSF grant DMS-1112700.


  1. 1.
    Appelö, D., Petersson, N.A.: A fourth-order accurate embedded boundary method for the wave equation. Preprint.
  2. 2.
    Arienti, M., Hung, P., Morano, E., Shepherd, J.E.: A level set approach to Eulerian-Lagrangian coupling. J. Comput. Phys. 185, 213–251 (2003) zbMATHCrossRefGoogle Scholar
  3. 3.
    Bassi, F., Rebay, S.: High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys. 138, 251–285 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Berger, M.J., Helzel, C., LeVeque, R.J.: h-box methods for the approximation of hyperbolic conservation laws on irregular grids. SIAM J. Numer. Anal. 41, 893–918 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Carpenter, M.H., Gottlieb, D., Abarbanel, S., Don, W.-S.: The theoretical accuracy of Runge-Kutta time discretizations for the initial boundary value problem: a study of the boundary error. SIAM J. Sci. Comput. 16, 1241–1252 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    De Palma, P., de Tullio, M.D., Pascazio, G., Napolitano, M.: An immersed-boundary method for compressible viscous flows. Comput. Fluids 35, 693–702 (2006) zbMATHCrossRefGoogle Scholar
  7. 7.
    de Tullio, M.D., De Palma, P., Iaccarino, G., Pascazio, G., Napolitano, M.: An immersed boundary method for compressible flows using local grid refinement. J. Comput. Phys. 225, 2098–2117 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Falcovitz, J., Alfandary, G., Hanoch, G.: A two-dimensional conservation laws scheme for compressible flows with moving boundaries. J. Comput. Phys. 138, 83–102 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Forrer, H., Berger, M.: Flow simulations on Cartesian grids involving complex moving geometries. In: Jeltsch, R. (ed.) Proc. 7th Intl. Conf. on Hyperbolic Problems, pp. 315–324. Birkhäuser, Basel (1998) Google Scholar
  10. 10.
    Forrer, H., Jeltsh, R.: A high-order boundary treatment for Cartesian-grid methods. J. Comput. Phys. 140, 259–277 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Ghias, R., Mittal, R., Dong, H.: A sharp interface immersed boundary method for compressible viscous flows. J. Comput. Phys. 225, 528–553 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Goldberg, M., Tadmor, E.: Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. I. Math. Comput. 32, 1097–1107 (1978) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Goldberg, M., Tadmor, E.: Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. II. Math. Comput. 36, 603–626 (1981) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high order accurate essentially non-oscillatory schemes. III. J. Comput. Phys. 71, 231–303 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Helzel, C., Berger, M.J., LeVeque, R.J.: A high-resolution rotated grid method for conservation laws with embedded geometries. SIAM J. Sci. Comput. 26, 785–809 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Hu, X.Y., Khoo, B.C., Adams, N.A., Huang, F.L.: A conservative interface method for compressible flows. J. Comput. Phys. 219, 553–578 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Huang, L., Shu, C.-W., Zhang, M.: Numerical boundary conditions for the fast sweeping high order WENO methods for solving the Eikonal equation. J. Comput. Math. 26, 336–346 (2008) MathSciNetzbMATHGoogle Scholar
  18. 18.
    Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Kreiss, H.-O., Petersson, N.A.: A second order accurate embedded boundary method for the wave equation with Dirichlet data. SIAM J. Sci. Comput. 27, 1141–1167 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Kreiss, H.-O., Petersson, N.A., Yström, J.: Difference approximations for the second order wave equation. SIAM J. Numer. Anal. 40, 1940–1967 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Kreiss, H.-O., Petersson, N.A., Yström, J.: Difference approximations of the Neumann problem for the second order wave equation. SIAM J. Numer. Anal. 42, 1292–1323 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Krivodonova, L., Berger, M.: High-order accurate implementation of solid wall boundary conditions in curved geometries. J. Comput. Phys. 211, 492–512 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Lax, P.D., Wendroff, B.: Systems of conservation laws. Commun. Pure Appl. Math. 13, 217–237 (1960) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    LeVeque, R.J., Calhoun, D.: Cartesian grid methods for fluid flow in complex geometries. In: Fauci, L.J., Gueron, S. (eds.) Computational Modeling in Biological Fluid Dynamics, IMA Vol. Math. Appl., vol. 124, pp. 117–143. Springer, New York (2001) CrossRefGoogle Scholar
  25. 25.
    LeVeque, R.J., Li, Z.: The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31, 1019–1044 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    LeVeque, R.J., Li, Z.: Immersed interface methods for Stokes flow with elastic boundaries or surface tensions. SIAM J. Sci. Comput. 18, 709–735 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Lombard, B., Piraux, J., Gélis, C., Virieux, J.: Free and smooth boundaries in 2-D finite-difference schemes for transient elastic waves. Geophys. J. Int. 172, 252–261 (2008) CrossRefGoogle Scholar
  28. 28.
    Mittal, R., Iaccarino, G.: Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239–261 (2005) MathSciNetCrossRefGoogle Scholar
  29. 29.
    Peskin, C.S.: Flow patterns around the heart valves. J. Comput. Phys. 10, 252–271 (1972) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Shyue, K.-M.: A moving-boundary tracking algorithm for inviscid compressible flow. In: Benzoni-Gavage, S., Serre, D. (eds.) Hyperbolic Problems: Theory, Numerics, Applications, pp. 989–996. Springer, Berlin (2008) CrossRefGoogle Scholar
  32. 32.
    Sjögreen, B., Petersson, N.A.: A Cartesian embedded boundary method for hyperbolic conservation laws. Commun. Comput. Phys. 2, 1199–1219 (2007) MathSciNetzbMATHGoogle Scholar
  33. 33.
    Tan, S., Shu, C.-W.: Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws. J. Comput. Phys. 229, 8144–8166 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Tan, S., Shu, C.-W.: A high order moving boundary treatment for compressible inviscid flows. J. Comput. Phys. 230, 6023–6036 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Tan, S., Wang, C., Shu, C.-W., Ning, J.: Efficient implementation of high order inverse Lax-Wendroff boundary treatment for conservation laws. J. Comput. Phys. 231, 2510–2527 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Xiong, T., Zhang, M., Zhang, Y.-T., Shu, C.-W.: Fifth order fast sweeping WENO scheme for static Hamilton-Jacobi equations with accurate boundary treatment. J. Sci. Comput. 45, 514–536 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Zhang, Y.-T., Chen, S., Li, F., Zhao, H., Shu, C.-W.: Uniformly accurate discontinuous Galerkin fast sweeping methods for Eikonal equations. SIAM J. Sci. Comput. 33, 1873–1896 (2011) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA

Personalised recommendations