On a boundedness-preserving semi-linear discretization of a two-dimensional nonlinear diffusion–reaction model

Abstract

Departing from a method to approximate the solutions of a two-dimensional generalization of the well-known Fisher's equation from population dynamics, we extend this computational technique to calculate the solutions of a FitzHugh–Nagumo model and derive conditions under which its positive and bounded analytic solutions are estimated consistently by positive and bounded numerical approximations. The constraints are relatively flexible, and they are provided exclusively in terms of the model coefficients and the computational parameters. The proofs are established with the help of the theory of M-matrices, using the facts that such matrices are non-singular, and that the entries of their inverses are positive numbers. Some numerical experiments are performed in order to show that our method is capable of preserving the positivity and the boundedness of the numerical solutions. The simulations evince a good agreement between the numerical estimations and the corresponding exact solutions derived in this work.

Keywords:

• positivity
• boundedness
• two-dimensional FitzHugh–Nagumo equation
• semi-linear discretization
• finite-difference scheme

2010 AMS Subject Classifications::

• 65M06
• 65N06
• 65Q10
• 92B05

Acknowledgements

The authors wish to thank the anonymous reviewers for their invaluable comments and suggestions, all of which led to a substantial improvement of the final work. The present manuscript reports on results of the research project PIM12-1 at the Universidad Autónoma de Aguascalientes.