A numerical study of mixed parabolic-gradient systems
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.
scientific article; zbMATH DE number 1658296