This article is cited in 6 papers

Parametric continuation of the solitary traveling pulse solution in the reaction-diffusion system using the Newton–Krylov method

A. G. Makeev, N. L. Semendiayeva

Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119992, Russia

Abstract: The matrix-free Newton–Krylov method that uses the GMRES algorithm (an iterative algorithm for solving systems of linear algebraic equations) is used for the parametric continuation of the solitary traveling pulse solution in a three-component reaction-diffusion system. Using the results of integration on a short time interval, we replace the original system of nonlinear algebraic equations by another system that has more convenient (from the viewpoint of the spectral properties of the GMRES algorithm) Jacobi matrix. The proposed parametric continuation proved to be efficient for large-scale problems, and it made it possible to thoroughly examine the dependence of localized solutions on a parameter of the model.

Key words: reaction-diffusion equations, localized structures, parametric continuation of self-similar solutions, Newton–Krylov method, GMRES algorithm, bifurcation analysis.

UDC: 519.633

Received: 18.04.2008
Revised: 02.07.2008

 English version: Computational Mathematics and Mathematical Physics, 2009, 49:4, 623–637

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