Skip to main content
Log in

Wavelets and adaptive grids for the discontinuous Galerkin method

  • Published:
Numerical Algorithms Aims and scope Submit manuscript


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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. R. Abgrall and A. Harten, Multiresolution representation in unstructured meshes, SIAM J. Numer. Anal. (1998).

  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).

  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).

  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).

  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. B. Cockburn and C.-W. Shu, Runge–Kutta discontinuous Galerkin method for convection-dominated problems, J. Sci. Comput. 16 (2001).

  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).

  8. W. Dahmen, B. Gottschlich-Müller and S. Müller, Multiresolution schemes for conservation laws, Numer. Math. 88 (1998).

  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.

  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).

  11. A. Harten, Adaptive multiresolution schemes for shock computations, J. Comput. Phys. 115 (1994).

  12. A. Harten, Multiresolution representation of data: A general framework, SIAM J. Numer. Anal. 33 (1996).

  13. M. Holmström, Wavelet based methods for time dependent PDE, Ph.D. thesis, Uppsala University, Sweden (1997).

  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. B. Sjögreen, Numerical experiments with the multiresolution schemes for the compressible Euler equations, J. Comput. Phys. 117 (1995).

  16. O.V. Vasilyev and C. Bowman, Second generation wavelet collocation method for the solution of partial differential equations, J. Comput. Phys. 165 (2000).

  17. J. Waldén, Filter bank methods for hyperbolic PDEs, SIAM J. Numer. Anal. 36 (1999).

Download references

Author information

Authors and Affiliations


Rights and permissions

Reprints and permissions

About this article

Cite this article

Díaz Calle, J.L., Devloo, P.R.B. & Gomes, S.M. Wavelets and adaptive grids for the discontinuous Galerkin method. Numer Algor 39, 143–154 (2005).

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: