Commun. Comput. Phys.,
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Volume 3.

Spectral Methods for Resolving Spike Dynamics in the Geirer-Meinhardt Model

Michael McCourt 1*, Nicholas Dovidio 2, Michael Gilbert 3

1 Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA.
2 Department of Mathematics, Davidson College, North Carolina, USA.
3 Department of Mathematics and Physics, Colorado State University-Pueblo, Colorado, USA.

Received 23 December 2006; Accepted (in revised version) 5 July 2007
Available online 30 October 2007


The Gierer-Meinhardt reaction-diffusion model is analyzed using a spectral collocation method. This reaction-diffusion system is governed by activator and inhibitor concentrations. Initially, the system is considered in one dimension and then in two dimensions; numerical results are presented for both cases. The algorithmic complexity and accuracy are compared to those of a moving finite element method. Finally, observations are made concerning when to use the proposed spectral method as opposed to the established moving mesh method.

AMS subject classifications: 65M70

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Key words: Spike dynamics, Gierer-Meinhardt model, spectral collocation.

*Corresponding author.
Email: (M. McCourt), (N. Dovidio), (M. Gilbert)

The Global Science Journal