# Adaptive mesh refinement strategy for a non conservative transport problem

Benjamin Aymard; Frédérique Clément; Marie Postel

- Volume: 48, Issue: 5, page 1381-1412
- ISSN: 0764-583X

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topAymard, Benjamin, Clément, Frédérique, and Postel, Marie. "Adaptive mesh refinement strategy for a non conservative transport problem." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.5 (2014): 1381-1412. <http://eudml.org/doc/273135>.

@article{Aymard2014,

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.},

author = {Aymard, Benjamin, Clément, Frédérique, Postel, Marie},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {adaptive finite volumes; non conservative PDE; discontinuous flux; non-conservative PDE; transport equation; multiresolution method; numerical test},

language = {eng},

number = {5},

pages = {1381-1412},

publisher = {EDP-Sciences},

title = {Adaptive mesh refinement strategy for a non conservative transport problem},

url = {http://eudml.org/doc/273135},

volume = {48},

year = {2014},

}

TY - JOUR

AU - Aymard, Benjamin

AU - Clément, Frédérique

AU - Postel, Marie

TI - Adaptive mesh refinement strategy for a non conservative transport problem

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2014

PB - EDP-Sciences

VL - 48

IS - 5

SP - 1381

EP - 1412

AB - 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.

LA - eng

KW - adaptive finite volumes; non conservative PDE; discontinuous flux; non-conservative PDE; transport equation; multiresolution method; numerical test

UR - http://eudml.org/doc/273135

ER -

## References

top- [1] B. Aymard, F. Clément, F. Coquel and M. Postel, Numerical simulation of the selection process of the ovarian follicles. ESAIM: Proc. 38 (2012) 99–117. Zbl1336.92020MR3006538
- [2] B. Aymard, F. Clément, F. Coquel and M. Postel, A numerical method for transport equations with discontinuous flux functions: Application to mathematical modeling of cell dynamics. SIAM J. Sci. Comput.35 (2013) 2442–2468. Zbl1285.65057MR3123827
- [3] R. Bürger, R. Ruiz, K. Schneider and M.A. Sepúlveda, Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux. J. Engrg. Math.60 (2008) 365–385. Zbl1137.65393MR2396490
- [4] F. Clément and D. Monniaux, Multiscale modelling of ovarian follicular selection. Prog. Biophys. Mol. Biol.113 (2013) 398–408.
- [5] A. Cohen, S.M. Kaber, S. Müller and M. Postel, Fully adaptive multiresolution finite volume schemes for conservation laws. Math. Comput.72 (2003) 183–225. Zbl1010.65035MR1933818
- [6] A. Cohen, S.M. Kaber and M. Postel, Adaptive multiresolution for finite volume solutions of gas dynamics. Comput. Fluids32 (2003) 31–38. Zbl1009.76520
- [7] Multiresolution and Adaptive Methods for Convection-Dominated Problems, edited by F. Coquel, Y. Maday, S. Müller, M. Postel and Q. Tran, vol. 29. ESAIM: Proc. (2009) 1–108. Zbl05656310MR2766462
- [8] F. Coquel, Q. Nguyen, M. Postel and Q. Tran, Local time stepping applied to implicit-explicit methods for hyperbolic systems. Multiscale Model. Simul.8 (2010) 540–570. Zbl1204.35017MR2581033
- [9] F. Coquel, M. Postel and Q. Tran, Convergence of time–space adaptive algorithms for nonlinear conservation laws. IMA J. Numer. Anal.32 (2012) 1440–1483. Zbl1273.65119MR2991834
- [10] N. Echenim, F. Clément and M. Sorine, Multiscale modeling of follicular ovulation as a reachability problem. Multiscale Model. Simul.6 (2007) 895–912. Zbl1149.35388MR2368972
- [11] N. Echenim, D. Monniaux, M. Sorine and F. Clément, Multi-scale modeling of the follicle selection process in the ovary. Math. Biosci.198 (2005) 57–79. Zbl1076.92017MR2187068
- [12] A. Harten, Multiresolution algorithms for the numerical solutions of hyperbolic conservation laws. Commun. Pure Appl. Math.48 (1995) 1305–1342. Zbl0860.65078MR1369391
- [13] Summer school on multiresolution and adaptive mesh refinement methods, edited by V. Louvet and M. Massot, vol. 34. ESAIM: Proc. (2011) 1–290. Zbl1302.65211MR2905893
- [14] P. Michel, Multiscale modeling of follicular ovulation as a mass and maturity dynamical system. Multiscale Model. Simul.9 (2011) 282–313. Zbl1219.35333MR2801206
- [15] S. Müller, Adaptive multiscale schemes for conservation laws, vol. 27 of Lect. Notes Comput. Sci. Eng. Springer-Verlag, Berlin (2003). Zbl1016.76004MR1952371
- [16] B. Perthame,Transport Equations In Biology. Front. Math. Birkhauser Verlag, Basel (2007). Zbl1185.92006MR2270822
- [17] A. Sakaue-Sawano, H. Kurokawa, T. Morimura, A. Hanyu, H. Hama, H. Osawa, S. Kashiwagi, K. Fukami, T. Miyata, H. Miyoshi, T. Imamura, M. Ogawa, H. Masai and A. Miyawaki, Visualizing spatiotemporal dynamics of multicellular cell-cycle progression. Cell132 (2008) 487–498.
- [18] P. Shang, Cauchy problem for multiscale conservation laws: Applications to structured cell populations. J. Math. Anal. Appl.401 (2013) 896–920. Zbl1307.35164MR3018037
- [19] M. Tomura, A. Sakaue-Sawano, Y. Mori, M. Takase-Utsugi, A. Hata, K. Ohtawa, O. Kanagawa and A. Miyawaki, Contrasting quiescent G0 phase with mitotic cell cycling in the mouse immune system. PLoS ONE 8 (2013) e73801.