A pseudospectral method of solution of Fisher's equation

In this paper, we develop an accurate and efficient pseudospectral solution of Fisher's equation, a prototypical reaction–diffusion equation. The solutions of Fisher's equation are characterized by propagating fronts that can be very steep for large values of the reaction rate coefficient. There is an ongoing effort to better adapt pseudospectral methods to the solution of differential equations with solutions that resemble shock waves or fronts typical of hyperbolic partial differential equations. The collocation method employed is based on Chebyshev–Gauss–Lobatto quadrature points. We 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.