Comparison of Two Aspects of a PDE Model for Biological Network Formation
Type
ArticleKAUST Department
Applied Mathematics and Computational Sciences Department, King Abdullah University of Science and Technology (KAUST), Thuwal 4700, Saudi ArabiaComputer, Electrical and Mathematical Science and Engineering (CEMSE) Division
Applied Mathematics and Computational Science Program
Date
2022-10-17Permanent link to this record
http://hdl.handle.net/10754/681619
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Show full item recordAbstract
We compare the solutions of two systems of partial differential equations (PDEs), seen as two different interpretations of the same model which describes the formation of complex biological networks. Both approaches take into account the time evolution of the medium flowing through the network, and we compute the solution of an elliptic–parabolic PDE system for the conductivity vector m, the conductivity tensor C and the pressure p. We use finite differences schemes in a uniform Cartesian grid in a spatially two-dimensional setting to solve the two systems, where the parabolic equation is solved using a semi-implicit scheme in time. Since the conductivity vector and tensor also appear in the Poisson equation for the pressure p, the elliptic equation depends implicitly on time. For this reason, we compute the solution of three linear systems in the case of the conductivity vector m∈R2 and four linear systems in the case of the symmetric conductivity tensor C∈R2×2 at each time step. To accelerate the simulations, we make use of the Alternating Direction Implicit (ADI) method. The role of the parameters is important for obtaining detailed solutions. We provide numerous tests with various values of the parameters involved to determine the differences in the solutions of the two systems.Citation
Astuto, C., Boffi, D., Haskovec, J., Markowich, P., & Russo, G. (2022). Comparison of Two Aspects of a PDE Model for Biological Network Formation. Mathematical and Computational Applications, 27(5), 87. https://doi.org/10.3390/mca27050087Sponsors
G.R. acknowledges support from ITN-ETN Horizon 2020 Project ModCompShock, Modeling and Computation on Shocks and Interfaces, Project Reference 642768, from the Italian Ministry of Instruction, University and Research (MIUR), PRIN Project 2017 (No.2017KKJP4X entitled “Innovative numerical methods for evolutionary partial differential equations and applications”), and University of Catania, project ”Piano della Ricerca 2020/2022, Linea d’intervento 2, MOSCOVID”.Publisher
MDPI AGarXiv
2209.08292Additional Links
https://www.mdpi.com/2297-8747/27/5/87ae974a485f413a2113503eed53cd6c53
10.3390/mca27050087
Scopus Count
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