Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Fully adaptive multiresolution finite volume schemes for conservation laws

Authors: Albert Cohen, Sidi Mahmoud Kaber, Siegfried Müller and Marie Postel
Journal: Math. Comp. 72 (2003), 183-225
MSC (2000): Primary 41A58, 65M50, 65M12
Published electronically: December 5, 2001
MathSciNet review: 1933818
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by A. Harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation. In this approach the solution is still represented at each time step on the finest grid, resulting in an inherent limitation of the potential gain in memory space and computational time. The present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes, in which the solution is represented and computed in a dynamically evolved adaptive grid. A crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages. Several solutions to this problem are proposed, analyzed, and compared in terms of accuracy and complexity.

References [Enhancements On Off] (What's this?)

  • 1. Abgrall, R. (1997) Multiresolution analysis on unstructured meshes: applications to CFD, in Chetverushkin and al. eds. Experimentation, modeling and computation in flow, turbulence and combustion, vol.2, John Wiley & Sons.
  • 2. Arandiga, F. and R. Donat (1999) A class of nonlinear multiscale decomposition, to appear in Numerical Algorithms.
  • 3. Berger, M. J. and P. Collela (1989) Local adaptive mesh refinement for shock hydrodynamics, Journal of Computational Physics, 82, pp 64-84.
  • 4. Babuska, I. and B. Guo (1998) The h-p version of the finite element method for domains with curved boundaries, SIAM J. Numer. Anal. 25, 837-861. MR 89i:65111
  • 5. Bihari, B. and A. Harten (1997) Multiresolution schemes for the numerical solution of 2-D conservation laws, SIAM J. Sci. Comput. 18(2), 315-354. MR 98f:65092
  • 6. Carnicer, J.M., W. Dahmen and J.M. Peña (1996) Local decomposition of refinable spaces and wavelets Appl. Comput. Harmon. Anal. 3, 127-153. MR 97f:42050
  • 7. Cavaretta, A., W. Dahmen and C.A. Micchelli (1991), Stationary subdivision, Memoirs of AMS 453. MR 92h:65017
  • 8. Chiavassa, G. and R. Donat (1999) Numerical experiments with point value multiresolution for 2d compressible flows, Technical Report GrAN-99-4, University of Valencia.
  • 9. Cockburn, B., F. Coquel and P. Lefloch (1994) An error estimate for finite volume methods for multidimensional conservation laws, Math. Comp. 63, 77-103. MR 95d:65078
  • 10. Cohen, A. (2000) Wavelet methods in numerical analysis, Handbook of Numerical Analysis, vol. VII, P.G. Ciarlet and J.L. Lions, eds., North-Holland, Amsterdam, pp. 417-711. CMP 2001:08
  • 11. Cohen, A., W. Dahmen and R. DeVore (2001) Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. Comp. 70, 27-75. CMP 2001:06
  • 12. Cohen, A., I. Daubechies and J.-C. Feauveau (1992) Biorthogonal bases of compactly supported wavelets, Comm. Pure and Applied Math. 45, 485-560. MR 93e:42044
  • 13. Cohen, A., N. Dyn, S.M. Kaber and M. Postel (2000) Multiresolution schemes on triangles for scalar conservation laws, J. Comp. Phys. 161, 264-286. MR 2000m:65168
  • 14. Cohen, A., S.M. Kaber and M. Postel (1999) Multiresolution analysis on triangles: application to conservation laws, in Finite volumes for complex applications II, R. Vielsmeier, F. Benkhaldoun and D. Hänel eds., Hermes, Paris.
  • 15. Dahmen, W. (1997) Wavelet and multiscale methods for operator equations, Acta Numerica 6, 55-228. MR 98m:65102
  • 16. Dahmen, W., B. Gottschlich-Müller and S. Müller (2000) Multiresolution Schemes for Conservation Laws, Numerische Mathematik DOI 10.1007/s00210000222
  • 17. Daubechies, I. (1992) Ten Lectures on Wavelets, SIAM, Philadelphia. MR 93e:42045
  • 18. DeVore, R. (1998) Nonlinear Approximation, Acta Numerica, 51-151. MR 2001a:41034
  • 19. Dyn, N. (1992) Subdivision algorithms in computer-aided geometric design, in: Advances in Numerical Analysis II, W.A. Light ed., Clarendon Press, Oxford.
  • 20. Gottschlich-Müller, B. and S. Müller (1999) Adaptive finite volume schemes for conservation laws based on local multiresolution techniques, in M. Fey and R. Jeltsch eds., Hyperbolic Problems: Theory, Numerics, Applications, Birkhäuser, Basel, pp. 385-394. MR 2000f:65112
  • 21. Harten, A. (1994) Adaptive multiresolution schemes for shock computations, J. Comp. Phys. 115, 319-338. MR 96d:65175
  • 22. Harten, A. (1995) Multiresolution algorithms for the numerical solution of hyperbolic conservation laws, Comm. Pure and Applied Math. 48, 1305-1342. MR 97e:65094
  • 23. Harten, A., B. Engquist, S. Osher, and S.R. Chakravarthy (1987) Uniformly high order accurate essentially non-oscillatory schemes III, J. Comp. Phys. 71, 231-303. MR 90a:65199
  • 24. Jaffard, S. (1991) Pointwise smoothness, two-microlocalization and wavelet coefficients, Publicacions Matemàtiques 35, 155-168. MR 92j:46057
  • 25. Kröner, D. (1997), Numerical schemes for conservation laws, Advances in Numerical Mathematics. Wiley-Teubner. MR 98b:65003
  • 26. LeVeque, R.J. (1992) Numerical methods for conservation laws, Birkhäuser Verlag, Basel. MR 92m:65106
  • 27. Lucier, B. (1986) A moving mesh numerical method for hyperbolic conservation laws, Math. Comp. 46, 59-69. MR 87m:65141
  • 28. Meyer, Y. (1990) Ondelettes et Opérateurs, Hermann, Paris, english tranlation by D.H. Salinger (1992), Cambridge University Press, Cambridge. MR 93i:42002; MR 93i:42003; MR 93i:42004
  • 29. Sanders, R. (1983) On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40, 91-106. MR 84a:65075
  • 30. Sjögreen, B. (1995) Numerical experiments with the multiresolution scheme for the compressible Euler equations, J. Comp. Phys., 117, 251-261.
  • 31. Schröder-Pander, F. and T. Sonar (1995) Preliminary investigations on multiresolution analysis on unstructured grids, DLR Report IB 223-95 A 36, 1995, DLR Göttingen.
  • 32. L.J. Durlofsky, B. Engquist, and S. Osher (1992)
    Triangle based adaptive stencils for the solution of hyperbolic conservation laws.
    J. Comp. Phys., 98, 64-73.
  • 33. E. Godlewski and P-A. Raviart (1996)
    Numerical Approximation of Hyperbolic Systems of Conservation Laws, volume 118.
    Springer. MR 98d:65109
  • 34. S. M. Kaber and M. Postel (1999)
    Finite volume schemes on triangles coupled with multiresolution analysis.
    328, série I, 817,822. MR 2000a:65147
  • 35. A. Voss and S. Müller (1999)
    A manual for the template class library igpm_t_lib.
    Technical Report 197, IGPM, RWTH Aachen.
  • 36. J. Ballmann, F. Bramkamp, and S. Müller (2000)
    Development of a flow solver employing local adaptation based on multiscale analysis on B-spline grids.
    In Proceedings of 8th Annual Conf. of the CFD Society of Canada, June, 11 to 13, 2000 Montreal.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 41A58, 65M50, 65M12

Retrieve articles in all journals with MSC (2000): 41A58, 65M50, 65M12

Additional Information

Albert Cohen
Affiliation: Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, BC 187, 75252 Paris cedex 05, France

Sidi Mahmoud Kaber
Affiliation: Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, BC 187, 75252 Paris cedex 05, France

Siegfried Müller
Affiliation: Institut für Geometrie und Praktische Mathematik, RWTH, Templergraben 55, D-52056 Aachen, Germany

Marie Postel
Affiliation: Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, BC 187, 75252 Paris cedex 05, France

Keywords: Conservation laws, finite volume schemes, adaptivity, multiresolution, wavelets.
Received by editor(s): May 30, 2000
Received by editor(s) in revised form: February 6, 2001
Published electronically: December 5, 2001
Additional Notes: The work of S. Müller has been supported by the EU–TMR network “Wavelets in Numerical Simulations”.
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society