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Adaptive mesh refinement strategy for a non conservativetransport problem∗

Published online by Cambridge University Press:  13 August 2014

Benjamin Aymard ,
Frédérique Clément  and
Marie Postel
Benjamin Aymard
Affiliation:
Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France. CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.. [email protected] INRIA Paris-Rocquencourt, EPI Mycenae, Domaine de Voluceau, BP105, 78153 Le Chesnay cedex, France.
Frédérique Clément
Affiliation:
INRIA Paris-Rocquencourt, EPI Mycenae, Domaine de Voluceau, BP105, 78153 Le Chesnay cedex, France.
Marie Postel
Affiliation:
Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France. CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.. [email protected] INRIA Paris-Rocquencourt, EPI Mycenae, Domaine de Voluceau, BP105, 78153 Le Chesnay cedex, France.
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Abstract

Long time simulations of transport equations raise computational challenges since theyrequire both a large domain of calculation and sufficient accuracy. It is thereforeadvantageous, in terms of computational costs, to use a time varying adaptive mesh, withsmall cells in the region of interest and coarser cells where the solution is smooth.Biological models involving cell dynamics fall for instance within this framework and areoften non conservative to account for cell division. In that case the thresholdcontrolling the spatial adaptivity may have to be time-dependent in order to keep up withthe progression of the solution. In this article we tackle the difficulties arising whenapplying a Multiresolution method to a transport equation with discontinuous fluxesmodeling localized mitosis. The analysis of the numerical method is performed on asimplified model and numerical scheme. An original threshold strategy is proposed andvalidated thanks to extensive numerical tests. It is then applied to a biological model inboth cases of distributed and localized mitosis.

Keywords

Adaptive finite volumesnon conservative PDEdiscontinuous flux
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 5 , September 2014 , pp. 1381 - 1412
Copyright
© EDP Sciences, SMAI 2014

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