Journal of Scientific Computing

, Volume 66, Issue 2, pp 761–791 | Cite as

High Order Boundary Extrapolation Technique for Finite Difference Methods on Complex Domains with Cartesian Meshes



The application of suitable numerical boundary conditions for hyperbolic conservation laws on domains with complex geometry has become a problem with certain difficulty that has been tackled in different ways according to the nature of the numerical methods and mesh type. In this paper we present a technique for the extrapolation of information from the interior of the computational domain to ghost cells designed for structured Cartesian meshes (which, as opposed to non-structured meshes, cannot be adapted to the morphology of the domain boundary). This technique is based on the application of Lagrange interpolation with a filter for the detection of discontinuities that permits a data dependent extrapolation, with higher order at smooth regions and essentially non oscillatory properties near discontinuities.


Finite difference WENO schemes Cartesian grids Extrapolation 


  1. 1.
    Aràndiga, F., Baeza, A., Belda, A.M., Mulet, P.: Analysis of WENO schemes for full and global accuracy. SIAM J. Numer. Anal. 49(2), 893–915 (2011)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Baeza, A., Mulet, P.: Adaptive mesh refinement techniques for high-order shock capturing schemes for multi-dimensional hydrodynamic simulations. Int. J. Numer. Methods Fluids 52, 455–471 (2006)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Berger, M.J., Colella, P.: Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 82, 64–84 (1989)CrossRefMATHGoogle Scholar
  4. 4.
    Boiron, O., Chiavassa, G., Donat, R.: A high-resolution penalization method for large Mach number flows in the presence of obstacles. Comput. Fluids 38, 703–714 (2009). doi: 10.1016/j.compfluid.2008.07.003 CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Faà di Bruno, C.F.: Note sur un nouvelle formule de calcul différentiel. Q. J. Math. 1, 359–360 (1857)Google Scholar
  6. 6.
    Carpenter, M., 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)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Donat, R., Marquina, A.: Capturing shock reflections: an improved flux formula. J. Comput. Phys. 125, 42–58 (1996)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Godlewski, E., Raviart, P.A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, vol. 118. Springer, New York (1996). doi: 10.1007/978-1-4612-0713-9 CrossRefGoogle Scholar
  9. 9.
    Gustafsson, B., Kreiss, H.O., Oliger, J.: Time Dependent Problems and Difference Methods. Pure and Applied Mathematics. Wiley, New York (1995). A Wiley-Interscience PublicationGoogle Scholar
  10. 10.
    Gustafsson, B., Kreiss, H.O., Sundström, A.: Stability theory of difference approximations for mixed initial boundary value problems. II. Math. Comput. 26, 649–686 (1972)CrossRefMATHGoogle Scholar
  11. 11.
    Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high order accurate essentially non-oscillatory schemes. III. J. Comput. Phys. 71(2), 231–303 (1987)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    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)MathSciNetMATHGoogle Scholar
  13. 13.
    Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Marquina, A., Mulet, P.: A flux-split algorithm applied to conservative models for multicomponent compressible flows. J. Comput. Phys. 185, 120–138 (2003)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Oliger, J., Sundström, A.: Theoretical and practical aspects of some initial boundary value problems in fluid dynamics. SIAM J. Appl. Math. 35(3), 419–446 (1978)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Pletcher, R.H., Tannehill, J.C., Anderson, D.: Computational Fluid Mechanics and Heat Transfer, 3rd edn. CRC Press, Boca Raton (2012)Google Scholar
  17. 17.
    Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. II. J. Comput. Phys. 83(1), 32–78 (1989)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Sjogreen, B., Petersson, N.: A Cartesian embedded boundary method for hyperbolic conservation laws. Commun. Comput. Phys. 2, 1199–1219 (2007)MathSciNetGoogle Scholar
  20. 20.
    Tan, S., Shu, C.W.: Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws. J. Comput. Phys. 229, 8144–8166 (2010)CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    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(6), 2510–2527 (2012)CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115–173 (1984)CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Xiong, T., Zhang, M., Zhang, Y.T., Shu, C.W.: Fast sweeping fifth order WENO scheme for static Hamilton-Jacobi equations with accurate boundary treatment. J. Sci. Comput. 45, 514–536 (2010)CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Yee, H.C.: Numerical approximation of boundary conditions with applications to inviscid equations of gas dynamics. Technical report N81–19834, NASA (1981)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Departament de Matemàtica AplicadaUniversitat de ValènciaValenciaSpain

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