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.
Keywords: biological transport networks, gradient flow, numerical schemes
CC-BY-NC-ND 4.0
@article{SMAI-JCM_2019__5__229_0,
author = {Di Fang and Shi Jin and Peter Markowich and Beno{\^\i}t Perthame},
title = {Implicit and {Semi-implicit} {Numerical} {Schemes} for the {Gradient} {Flow} of the {Formation} of {Biological} {Transport} {Networks}},
journal = {The SMAI Journal of computational mathematics},
pages = {229--249},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {5},
year = {2019},
doi = {10.5802/smai-jcm.59},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.59/}
}
TY - JOUR AU - Di Fang AU - Shi Jin AU - Peter Markowich AU - Benoît Perthame TI - Implicit and Semi-implicit Numerical Schemes for the Gradient Flow of the Formation of Biological Transport Networks JO - The SMAI Journal of computational mathematics PY - 2019 SP - 229 EP - 249 VL - 5 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.59/ DO - 10.5802/smai-jcm.59 LA - en ID - SMAI-JCM_2019__5__229_0 ER -
%0 Journal Article %A Di Fang %A Shi Jin %A Peter Markowich %A Benoît Perthame %T Implicit and Semi-implicit Numerical Schemes for the Gradient Flow of the Formation of Biological Transport Networks %J The SMAI Journal of computational mathematics %D 2019 %P 229-249 %V 5 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.59/ %R 10.5802/smai-jcm.59 %G en %F SMAI-JCM_2019__5__229_0
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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