Finite-difference schemes for parabolic problems on graphs |
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Authors: | R Čiegis N Tumanova |
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Institution: | 1.Vilnius Gediminas Technical University,Vilnius,Lithuania |
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Abstract: | We consider a reaction–diffusion parabolic problem on branched structures. The Hodgkin–Huxley reaction–diffusion equations
are formulated on each edge of the graph. The problems are coupled by some conjugation conditions at branch points. It is
important to note that two different types of the flux conservation equations are considered. The first one describes a conservation
of the axial currents at branch points, and the second equation defines the conservation of the current flowing at the soma
in neuron models. We study three different types of finite-difference schemes. The fully implicit scheme is based on the backward
Euler algorithm. The stability and convergence of the discrete solution is proved in the maximum norm, and the analysis is
done by using the maximum principle method. In order to decouple computations at each edge of the graph, we consider two modified
schemes. In the predictor algorithm, the values of the solution at branch points are computed by using an explicit approximation
of the conservation equations. The stability analysis is done using the maximum principle method. In the predictor–corrector
method, in addition to the previous algorithm, the values of the solution at the branch points are recomputed by an implicit
algorithm, when the discrete solution is obtained on each subdomain. The stability of this algorithm is investigated numerically.
The results of computational experiments are presented. |
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