Parametric continuation of the solitary traveling pulse solution in the reaction-diffusion system using the Newton-Krylov method. (Russian, English) Zbl 1199.65110

Makeev, A. G.; Semendyaeva, N. L.

Comput. Math., Math. Phys. 49, No. 4, 623-637 (2009); translation from Zh. Vychisl. Mat. Mat. Fiz. 49, No. 4, 646-661 (2009).

Summary: 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 a more convenient (from the viewpoint of the spectral properties of the GMRES algorithm) Jacobi matrix. The proposed parametric continuation proves to be efficient for large-scale problems, and it makes it possible to thoroughly examine the dependence of localized solutions on a parameter of the model.

MSC:

65F10	Iterative numerical methods for linear systems
35K57	Reaction-diffusion equations
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

Keywords:

reaction-diffusion equation; localized solutions; Newton-Krylov method; iterative algorithm