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Pseudospectral solutions of reaction-diffusion equations that model excitable media : convergence of solutions and applications Liceaga, Daniel Olmos

Abstract

In this thesis, I develop accurate and efficient pseudospectral methods to solve Fisher's, the Fitzhugh-Nagumo and the Beeler-Reuter equations. Based on these methods, I present a study of spiral waves and their interaction with a boundary. The solutions of Fisher's equation are characterized by propagating fronts with a, shock-like wave behavior when large values of the reaction rate coefficient is taken. The pseudospectral method employed for its solution is based on Chebyshev-Gauss-Lobatto quadrature points. I compare results for a single domain as well as for a subdivision of the main domain into subintervals. Instabilities that occur in the numerical solution for a single domain, analogous to those found by others, are attributed to round-off errors arising from numerical features of the discrete second derivative matrix operator. However, accurate stable solutions of Fisher's equation are obtained with a multidomain pseudospectral method. A detailed comparison of the present approach with the use of the sinc interpolation is also carried out. Also, I present a study of the convergence of different numerical schemes in the solution of the Fitzhugh-Nagumo equations. These equations, have spatial and temporal dynamics in two different scales and the solutions exhibit shock-like waves. The numerical schemes employed are Chebyshev multidomain, Fourier pseudospectral, finite difference methods and in particular a method developed by Barkley. I consider two different models of the local dynamics. I present results for plane wave propagation in one dimension and spiral waves for two dimensions. I use an operator splitting method with the Chebyshev multidomain approach in order to reduce the computational time. I conclude this thesis by presenting a study of the interaction of a meandering spiral wave with a boundary, where the Beeler-Reuter model is considered. The phenomenon of annihilation or reflection of a spiral at the boundaries of the domain is studied, when the trajectory of the tip of a spiral wave is essentially linear. This phenomenon is analyzed in terms of the variable j, which controls the reactivation of the sodium channel in the Beeler-Reuter model.

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