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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, whose leaves are the non-uniform finite volumes on which the numerical divergence is eventually evaluated. Moreover using the contraction of the discrete time evolution operator we derive the optimal choice of the threshold in the adaptive multiresolution method. Numerical examples illustrate the computational efficiency together with the convergence properties.
Bürger, Raimund ; Ruiz, Ricardo ; Schneider, Kai 1 ; Sepúlveda, Mauricio
@article{M2AN_2008__42_4_535_0,
author = {B\"urger, Raimund and Ruiz, Ricardo and Schneider, Kai and Sep\'ulveda, Mauricio},
title = {Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {535--563},
publisher = {EDP-Sciences},
volume = {42},
number = {4},
year = {2008},
doi = {10.1051/m2an:2008016},
mrnumber = {2437773},
zbl = {1147.65066},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an:2008016/}
}
TY - JOUR AU - Bürger, Raimund AU - Ruiz, Ricardo AU - Schneider, Kai AU - Sepúlveda, Mauricio TI - Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2008 SP - 535 EP - 563 VL - 42 IS - 4 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an:2008016/ DO - 10.1051/m2an:2008016 LA - en ID - M2AN_2008__42_4_535_0 ER -
%0 Journal Article %A Bürger, Raimund %A Ruiz, Ricardo %A Schneider, Kai %A Sepúlveda, Mauricio %T Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2008 %P 535-563 %V 42 %N 4 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an:2008016/ %R 10.1051/m2an:2008016 %G en %F M2AN_2008__42_4_535_0
Bürger, Raimund; Ruiz, Ricardo; Schneider, Kai; Sepúlveda, Mauricio. Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 42 (2008) no. 4, pp. 535-563. doi : 10.1051/m2an:2008016. http://geodesic.mathdoc.fr/articles/10.1051/m2an:2008016/
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