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Adaptive mesh refinement strategy for a non conservative transport 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.. marie.postel@upmc.fr 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.. marie.postel@upmc.fr 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 they require both a large domain of calculation and sufficient accuracy. It is therefore advantageous, in terms of computational costs, to use a time varying adaptive mesh, with small 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 are often non conservative to account for cell division. In that case the threshold controlling the spatial adaptivity may have to be time-dependent in order to keep up with the progression of the solution. In this article we tackle the difficulties arising when applying a Multiresolution method to a transport equation with discontinuous fluxes modeling localized mitosis. The analysis of the numerical method is performed on a simplified model and numerical scheme. An original threshold strategy is proposed and validated thanks to extensive numerical tests. It is then applied to a biological model in both 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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