Stability Analysis of Implicit Finite-Difference Schemes for Parabolic Problems on Graphs

Abstract

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 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 L ₂, H ¹, 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.

Keywords:

• Convergence
• Error estimates
• Euler algorithm
• Finite difference method
• Graphs
• Parabolic problem
• Stability
• θ-method

Mathematics Subject Classification:

• 6M06
• 6M12
• 6M15

ACKNOWLEDGMENTS

The authors would like to thank the referees for their constructive criticism which helped to improve the clarity of this note.