Simulación numérica de la ecuación de Nagumo por el método de diferencias finitas

Abstract

The aim of this work was to describe the Nagumo equation, its theoretical foundations and its numerical resolution through finite difference schemes that were implemented in Python to identify the method that best approximates the solution. The documentary research was of a theoretical-practical nature because an analysis was carried out on: the existence and uniqueness of the solutions of the Nagumo equation, it was presented two real solutions; the implicit, explicit, theta-scheme, Cranck-Nicolson schemes, their consistency, stability and convergence; at the same time the numerical simulation in Python was carried out, which allowed the comparison of the real solutions with the numerical solutions. It was obtained as results that the implicit, explicit and theta schemes have a linear order of convergence in time and quadratic in space, while the Cranck-Nicolson scheme is quadratic in time and space. Thanks to the study, it is concluded that the solutions of the Nagumo Equation exist and are unique, and from the simulations it is known that when the variable “α” approaches 0 the scheme that is closest to the real solution is that of Cranck - Nicolson, but when “α” approaches 0.5 it cannot be assured that the same scheme is the one that converges to the exact solution, in any temporal or spatial interval. It is recommended to read Chapter II in its entirety, since it will lay the foundations for the main topic of finite difference methods.