A numerical study of mixed parabolic-gradient systems
type: Item
instance of: scholarly article
title: A numerical study of mixed parabolic-gradient systems
zbMATH Open document ID: 1013.65093
DOI: 10.1016/S0377-0427(00)00605-1
author: J. G. Verwer
author: B. P. Sommeijer
published in: Journal of Computational and Applied Mathematics
publication date: 2003-06-26T00:00:00Z
reviewed by: S. Burys
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 92C20
zbMATH DE Number: 1658296
zbMATH Keywords: neurobiology
zbMATH Keywords: error bound
zbMATH Keywords: numerical examples
zbMATH Keywords: mixed parabolic-gradient system
zbMATH Keywords: nervous system
zbMATH Keywords: parabolic equations
zbMATH Keywords: fourth-order difference scheme
zbMATH Keywords: Hermite interpolation
zbMATH Keywords: stability
zbMATH Keywords: Runge-Kutta method
zbMATH Keywords: Runge-Kutta-Chebyshev method
describes a project that uses: RKC
MaRDI profile type: Publication
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full work available at URL: s0377-0427(00)00605-1
OpenAlex ID: W2134306279
Wikidata QID: Q128116354
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review text: The authors consider the following mixed parabolic-gradient system of differential equations NEWLINE\[NEWLINE\partial\rho_l/\partial t= d_l\Delta\rho_l- \delta_l\rho_l+ S_L,\quad 1\leq l\leq L,\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINEdr_n/dt= \sum_l \lambda_{n,l}\nabla\rho_l (r_n(t), t),\quad 1\leq n\leq N\quad\text{for }x\in [0,1]^m,\quad t>0.\tag{2}NEWLINE\]NEWLINE Here \(d_l\), \(\delta_l\), \(\lambda_{n,l}\) are positive constants and \(S_l\) are source fluxes couplin (1) and (2). Such systems model the formation of axonal connections in the nervous system, where \(\rho_l\) represents the concentration of a chemical \(l\) and \(r_n\) represents the position of the growth cone of axon \(n\).NEWLINENEWLINENEWLINEFor \(m=2\) the parabolic equations are discretized on a uniform space grid by a fourth-order difference scheme and the functions \(\rho_l\) in the ordinary differential equations are spatially approximated by means of piecewise cubic Hermite interpolation. Under a few simplifying assumptions a stability analysis to the semi-discrete system is performed and the power boundedness of a Runge-Kutta method is examined. One section of the paper is devoted to an application of the Runge-Kutta-Chebyshev method, a subclass of the Runge-Kutta one.NEWLINENEWLINENEWLINENumerical examples are given and possible future generalizations and reseachers are mentioned.
description: scientific article; zbMATH DE number 1658296