Stability analysis of implicit finite-difference schemes for parabolic problems on graphs

Abstract

We consider a parabolic problem on branched structures. The Hodgkin–Huxley reactiondiffusion equation is a well-known example of such type models. The diffusion equations on edges of a graph are coupled by two types of conjugation conditions at branch points. The first one describes a conservation of the fluxes at vertexes, and the second conjugation condition defines the conservation of the current flowing at the soma in neuron models. The differential problem is approximated by a θ-implicit finite difference scheme which is based on the θ-method for ODEs. The stability and convergence of the discrete solution is proved in L2, H1, and L∞ norms. The main goal is to estimate the influence of the approximation errors introduced at the branch points of the first type. Results of numerical experiments are presented.