Numerical Algorithms

, Volume 39, Issue 1–3, pp 143–154 | Cite as

Wavelets and adaptive grids for the discontinuous Galerkin method

  • Jorge L. Díaz Calle
  • Philippe R. B. Devloo
  • Sônia M. Gomes


In this paper, space adaptivity is introduced to control the error in the numerical solution of hyperbolic systems of conservation laws. The reference numerical scheme is a new version of the discontinuous Galerkin method, which uses an implicit diffusive term in the direction of the streamlines, for stability purposes. The decision whether to refine or to unrefine the grid in a certain location is taken according to the magnitude of wavelet coefficients, which are indicators of local smoothness of the numerical solution. Numerical solutions of the nonlinear Euler equations illustrate the efficiency of the method.


conservation laws discontinuous Galerkin implicit time integration upwind artificial diffusion adaptivity wavelets 


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  1. [1]
    R. Abgrall and A. Harten, Multiresolution representation in unstructured meshes, SIAM J. Numer. Anal. (1998). Google Scholar
  2. [2]
    B.L. Bihari and A. Harten, Multiresolution schemes for the numerical solution of 2-D conservation laws I, SIAM J. Sci. Comput. 18(2) (1997). Google Scholar
  3. [3]
    D.L. Bonhaus, A higher order accurate finite element method for viscous compressible flows, Ph.D. thesis, Virginia Polytechnics Institute and State University (November 1998). Google Scholar
  4. [4]
    A. Brooks and T. Hughes, Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg. 32 (1982). Google Scholar
  5. [5]
    G. Chiavassa and R. Donat, Numerical experiments with multilevel schemes for conservation laws, in: Godunov’s Methods: Theory and Applications, ed. Toro (Kluwer Academic/Plenum, Dordrecht, 1999). Google Scholar
  6. [6]
    B. Cockburn and C.-W. Shu, Runge–Kutta discontinuous Galerkin method for convection-dominated problems, J. Sci. Comput. 16 (2001). Google Scholar
  7. [7]
    A. Cohen, S. Muller, M. Postel and S.M. Ould-Kabe, Fully adaptive multiresolution finite volume schemes for conservation laws, Math. Comp. 72 (2002). Google Scholar
  8. [8]
    W. Dahmen, B. Gottschlich-Müller and S. Müller, Multiresolution schemes for conservation laws, Numer. Math. 88 (1998). Google Scholar
  9. [9]
    J.L. Díaz Calle, P.R.B. Devloo and S.M. Gomes, Stabilized discontinuous Galerkin method for hyperbolic equations, Comput. Methods Appl. Mech. Engrg., to appear. Google Scholar
  10. [10]
    M.O. Domingues, S.M. Gomes and L.A. Diaz, Adaptive wavelet representation and differentiation on block-structured grids, Appl. Numer. Math. 8(3/4) (2003). Google Scholar
  11. [11]
    A. Harten, Adaptive multiresolution schemes for shock computations, J. Comput. Phys. 115 (1994). Google Scholar
  12. [12]
    A. Harten, Multiresolution representation of data: A general framework, SIAM J. Numer. Anal. 33 (1996). Google Scholar
  13. [13]
    M. Holmström, Wavelet based methods for time dependent PDE, Ph.D. thesis, Uppsala University, Sweden (1997). Google Scholar
  14. [14]
    M.K. Kaibara and S.M. Gomes, Fully adaptive multiresolution scheme for shock computations, in: Godunov’s Methods: Theory and Applications, ed. Toro (Kluwer Academic/Plenum, Dordrecht, 1999). Google Scholar
  15. [15]
    B. Sjögreen, Numerical experiments with the multiresolution schemes for the compressible Euler equations, J. Comput. Phys. 117 (1995). Google Scholar
  16. [16]
    O.V. Vasilyev and C. Bowman, Second generation wavelet collocation method for the solution of partial differential equations, J. Comput. Phys. 165 (2000). Google Scholar
  17. [17]
    J. Waldén, Filter bank methods for hyperbolic PDEs, SIAM J. Numer. Anal. 36 (1999). Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Jorge L. Díaz Calle
    • 1
  • Philippe R. B. Devloo
    • 2
  • Sônia M. Gomes
    • 3
  1. 1.Commodity SystemsSão Paulo SPBrasil
  2. 2.FECUniversidade Estadual de CampinasCampinas SPBrasil
  3. 3.IMECCUniversidade Estadual de CampinasCampinas SPBrasil

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