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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
DOI: https://doi.org/10.1090/S0025-5718-01-01391-6
Published electronically: December 5, 2001
MathSciNet review: 1933818
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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.


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Additional Information

Albert Cohen
Affiliation: Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, BC 187, 75252 Paris cedex 05, France
Email: cohen@ann.jussieu.fr

Sidi Mahmoud Kaber
Affiliation: Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, BC 187, 75252 Paris cedex 05, France
Email: kaber@ann.jussieu.fr

Siegfried Müller
Affiliation: Institut für Geometrie und Praktische Mathematik, RWTH, Templergraben 55, D-52056 Aachen, Germany
Email: mueller@igpm.rwth-aachen.de

Marie Postel
Affiliation: Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, BC 187, 75252 Paris cedex 05, France
Email: postel@ann.jussieu.fr

DOI: https://doi.org/10.1090/S0025-5718-01-01391-6
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