Implicit and Semi-implicit Numerical Schemes for the Gradient Flow of the Formation of Biological Transport Networks
The SMAI Journal of computational mathematics, Volume 5 (2019), pp. 229-249.

Implicit and semi-implicit time discretizations are developed for the Cai–Hu model describing the formation of biological transport networks. The model couples a nonlinear elliptic equation for the pressure with a nonlinear reaction-diffusion equation for the network conductance vector. Numerical challenges include the nonlinearity and the stiffness, thus an explicit discretization puts severe constraints on the time step. We propose an implicit and a semi-implicit discretizations, which decays the energy unconditionally or under a condition independent of the mesh size respectively, as will be proven in 1D and verified numerically in 2D.

Published online:
DOI: 10.5802/smai-jcm.59
Classification: 65M06, 92B99
Keywords: biological transport networks, gradient flow, numerical schemes
Di Fang 1; Shi Jin 2; Peter Markowich 3; Benoît Perthame 4

1 Department of Mathematics, University of Wisconsin-Madison, Madison, WI, USA
2 School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, China
3 Mathematical and Computer Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Kingdom of Saudi Arabia; Faculty of Mathematics, University of Vienna, Oskar-MorgensternPlatz 1, 1090 Vienna, Austria
4 Sorbonne Université, Université Paris-Diderot SPC, CNRS, INRIA, Laboratoire Jacques-Louis Lions, F-75005 Paris, France
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Di Fang; Shi Jin; Peter Markowich; Benoît Perthame. Implicit and Semi-implicit Numerical Schemes for the Gradient Flow of the Formation of Biological Transport Networks. The SMAI Journal of computational mathematics, Volume 5 (2019), pp. 229-249. doi : 10.5802/smai-jcm.59. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.59/

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