Adaptive mesh refinement strategy for a non conservative transport problem
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014) no. 5, pp. 1381-1412.

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.

Classification : 49M,  35L
Mots clés : adaptive finite volumes, non conservative PDE, discontinuous flux
     author = {Aymard, Benjamin and Cl\'ement, Fr\'ed\'erique and Postel, Marie},
     title = {Adaptive mesh refinement strategy for a non conservative transport problem},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     pages = {1381--1412},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {5},
     year = {2014},
     doi = {10.1051/m2an/2014014},
     mrnumber = {3264358},
     language = {en},
     url = {}
Aymard, Benjamin; Clément, Frédérique; Postel, Marie. Adaptive mesh refinement strategy for a non conservative transport problem. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014) no. 5, pp. 1381-1412. doi : 10.1051/m2an/2014014.

[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. | MR 3006538

[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. | MR 3123827 | Zbl 1285.65057

[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. | MR 2396490 | Zbl 1137.65393

[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. | MR 1933818 | Zbl 1010.65035

[6] A. Cohen, S.M. Kaber and M. Postel, Adaptive multiresolution for finite volume solutions of gas dynamics. Comput. Fluids 32 (2003) 31-38. | Zbl 1009.76520

[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. | MR 2766462

[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. | MR 2581033 | Zbl 1204.35017

[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. | MR 2991834 | Zbl 1273.65119

[10] N. Echenim, F. Clément and M. Sorine, Multiscale modeling of follicular ovulation as a reachability problem. Multiscale Model. Simul. 6 (2007) 895-912. | MR 2368972 | Zbl 1149.35388

[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. | MR 2187068 | Zbl 1076.92017

[12] A. Harten, Multiresolution algorithms for the numerical solutions of hyperbolic conservation laws. Commun. Pure Appl. Math. 48 (1995) 1305-1342. | MR 1369391 | Zbl 0860.65078

[13] Summer school on multiresolution and adaptive mesh refinement methods, edited by V. Louvet and M. Massot, vol. 34. ESAIM: Proc. (2011) 1-290. | MR 2905893

[14] P. Michel, Multiscale modeling of follicular ovulation as a mass and maturity dynamical system. Multiscale Model. Simul. 9 (2011) 282-313. | MR 2801206 | Zbl 1219.35333

[15] S. Müller, Adaptive multiscale schemes for conservation laws, vol. 27 of Lect. Notes Comput. Sci. Eng. Springer-Verlag, Berlin (2003). | MR 1952371 | Zbl 1016.76004

[16] B. Perthame,Transport Equations In Biology. Front. Math. Birkhauser Verlag, Basel (2007). | MR 2270822 | Zbl 1185.92006

[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. Cell 132 (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. | MR 3018037

[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.