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Implicit and Semi-implicit Numerical Schemes for the Gradient Flow of the Formation of Biological Transport Networks

Abstract : 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.
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https://hal.sorbonne-universite.fr/hal-01984371
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Submitted on : Wednesday, January 16, 2019 - 11:12:27 PM
Last modification on : Saturday, April 11, 2020 - 1:57:26 AM

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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. SMAI Journal of Computational Mathematics, Société de Mathématiques Appliquées et Industrielles (SMAI), 2019, 5, pp.229-249. ⟨10.5802/smai-jcm.59⟩. ⟨hal-01984371⟩

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