STABILITY ANALYSIS OF IMPLICIT FINITE-DIFFERENCE SCHEMES FOR PARABOLIC PROBLEMS ON GRAPHS

We consider a parabolic problem on branched structures. The Hodgkin-Huxley reaction-diffusion 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 theta-implicit finite difference scheme which is based on the theta-method for ODEs. The stability and convergence of the discrete solution is proved in L-2, H-1, and L-infinity 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.