Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension

Raimund Bürger; Ricardo Ruiz; Kai Schneider; Mauricio Sepúlveda

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 4, page 535-563
  • ISSN: 0764-583X

Abstract

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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 L1 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.

How to cite

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Bürger, Raimund, et al. "Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension." ESAIM: Mathematical Modelling and Numerical Analysis 42.4 (2008): 535-563. <http://eudml.org/doc/250404>.

@article{Bürger2008,
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, whose leaves are the non-uniform finite volumes on which the numerical divergence is eventually evaluated. Moreover using the L1 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. },
author = {Bürger, Raimund, Ruiz, Ricardo, Schneider, Kai, Sepúlveda, Mauricio},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Degenerate parabolic equation; adaptive multiresolution scheme; monotone scheme; upwind difference scheme; boundary conditions; entropy solution.; degenerate parabolic equation; adaptive multiresolution scheme; entropy solution; error bounds; adaptive mesh refinement},
language = {eng},
month = {5},
number = {4},
pages = {535-563},
publisher = {EDP Sciences},
title = {Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension},
url = {http://eudml.org/doc/250404},
volume = {42},
year = {2008},
}

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
DA - 2008/5//
PB - EDP Sciences
VL - 42
IS - 4
SP - 535
EP - 563
AB - 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 L1 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.
LA - eng
KW - Degenerate parabolic equation; adaptive multiresolution scheme; monotone scheme; upwind difference scheme; boundary conditions; entropy solution.; degenerate parabolic equation; adaptive multiresolution scheme; entropy solution; error bounds; adaptive mesh refinement
UR - http://eudml.org/doc/250404
ER -

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