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Fully adaptive multiresolution schemes forstrongly degenerate parabolic equations in one space dimension

Published online by Cambridge University Press:  27 May 2008

Raimund Bürger,
Ricardo Ruiz,
Kai Schneider and
Mauricio Sepúlveda
Raimund Bürger
Affiliation:
Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile. rburger@ing-mat.udec.cl; rruiz@ing-mat.udec.cl; mauricio@ing-mat.udec.cl
Ricardo Ruiz
Affiliation:
Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile. rburger@ing-mat.udec.cl; rruiz@ing-mat.udec.cl; mauricio@ing-mat.udec.cl
Kai Schneider
Affiliation:
Centre de Mathématiques et d'Informatique, Université de Provence, 39 rue Joliot-Curie, 13453 Marseille Cedex 13, France. kschneid@cmi.univ-mrs.fr
Mauricio Sepúlveda
Affiliation:
Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile. rburger@ing-mat.udec.cl; rruiz@ing-mat.udec.cl; mauricio@ing-mat.udec.cl
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Abstract

We present a fully adaptive multiresolution scheme for spatially one-dimensional quasilinear strongly degenerate parabolic equations with zero-flux and periodic boundary conditions. The numerical scheme is based on a finite volume discretization using the Engquist-Osher numerical flux and explicit time stepping. An adaptive multiresolution scheme based on cell averages is then used to speed up the CPU time and the memory requirements of the underlying finite volume scheme, whose first-order version is known to converge to an entropy solution of the problem. A particular feature of the method is the storage of the multiresolution representation of the solution in a graded tree, whoseleaves are the non-uniform finite volumes on which the numerical divergence is eventually evaluated. Moreover using the L 1 contraction of the discrete time evolution operator we derive the optimal choice of the threshold in the adaptive multiresolution method. Numerical examples illustrate thecomputational efficiency together with the convergence properties.

Keywords

Degenerate parabolic equationadaptive multiresolution schememonotone schemeupwind difference schemeboundary conditionsentropy solution.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 4 , July 2008, pp. 535 - 563
Copyright
© EDP Sciences, SMAI, 2008

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