Geodesic
Parametric continuation of the solitary traveling pulse solution in the reaction-diffusion system using the Newton–Krylov method
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 4, pp. 646-661 
A. G. Makeev; N. L. Semendiayeva. Parametric continuation of the solitary traveling pulse solution in the reaction-diffusion system using the Newton–Krylov method. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 4, pp. 646-661. http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_4_a6/
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     author = {A. G. Makeev and N. L. Semendiayeva},
     title = {Parametric continuation of the solitary traveling pulse solution in the reaction-diffusion system using the {Newton{\textendash}Krylov} method},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {646--661},
     year = {2009},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_4_a6/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Dhooge A., Govaerts W., Kuznetsov Yu. A., “MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs”, ACM Trans. Math. Software, 29:2 (2003), 141–164 | DOI | MR | Zbl

[2] Saad Y., Schultz M. H., GMRES: A generalized minimal residual algorithm for solving non-symetric linear systems, Res. Rep. YALEU/DCS/RR254, August 24, 1983; SIAM J. Sci. Stat. Comput., 7:3 (1986), 856–869 | DOI | MR | Zbl

[3] Gear C. W., Saad Y., “Iterative solution of linear equations in ode codes”, SIAM J. Sci. Stat. Comput., 4:4 (1983), 583–601 | DOI | MR | Zbl

[4] Wigton L. B., Yu N. J., Young D. P., “GMRES acceleration of computational fluid dynamics codes”, Proc. 7th AIAA CFD Conf., 1985, 67–74

[5] Brown P. N., Hindmarsh A. C., “Matrix-free methods for stiff systems of ode's”, SIAM J. Numer. Analys, 2:3 (1986), 610–638 | DOI | MR

[6] Brown P. N., Saad Y., “Hybrid Krylov methods for nonlinear systems of equations”, SIAM J. Sci. Stat. Comput., 11:3 (1990), 450–481 | DOI | MR | Zbl

[7] Knoll D. A., Keyes D. E., “Jacobian-free Newton–Krylov methods: a survey of approaches and applications”, J. Comput. Phys., 193:2 (2004), 357–397 | DOI | MR | Zbl

[8] Tuckerman L., Barkley D., “Bifurcation analysis for timesteppers”, Numer. Meth. for Bifurcation Problems and Large-Scale Dynamical Systems, Springer, New York, 2000, 453–566 | MR

[9] Kelley C. T., Kevrekidis I. G., Qiao L., Newton–Krylov solvers for timesteppers, arxiv.org/pdf/math.DS/0404374

[10] Kelley C. T., Solving nonlinear equations with Newton's method, SIAM, Philadelphia, 2003 | MR

[11] Kurkina E. S., Semendyaeva N. L., “Fluctuation-induced transitions and oscillations in catalytic CO oxidation: Monte Carlo simulations”, Surface. Sci., 558:1–3 (2004), 122–134 | DOI

[12] Kurkina E. C., Semendyaeva H. L., “Issledovanie kolebatelnykh rezhimov v stokhasticheskoi modeli v geterogennoi kataliticheskoi reaktsii”, Zh. vychisl. matem. i matem. fiz., 44:10 (2004), 1808–1823 | MR | Zbl

[13] Kness M., Tuckerman L. S., Barkley D., “Symmetry-breaking bifurcations in one-dimensional excitable media”, Phys. Rev. A, 46:8 (1992), 5054–5062 | DOI

[14] Keller H. B., “Numerical solution of bifurcation and nonlinear eigenvalue problems”, Applic. Bifurcation Theory, Acad. Press, New York, 1977, 359–384 | MR

[15] Seydel R., “Tutorial on continuation”, Internat. J. Bifurcation. Chaos, 1:1 (1991), 3–11 | DOI | MR | Zbl

[16] Dembo R. S., Eisenstat S. C., Steihaug T., “Inexact Newton methods”, SIAM J. Numer. Analis, 19:2 (1982), 400–408 | DOI | MR | Zbl

[17] Eisenstat S. C., Walker H. E., “Choosing the forcing terms in an inexact newton method”, SIAM J. Sci. Comput., 17:1 (1996), 16–32 | DOI | MR | Zbl

[18] Makeev A. G., Maraudas D., Kevrekidis l.G., ““Coarse” stability and bifurcation analysis using stochastic simulators: kinetic Monte Carlo examples”, J. Chem. Phys., 116:23 (2002), 10083–10091 | DOI

[19] Makeev A. G., Maraudas D., Panagiatapaulas A. Z., Kevrekidis I. G., “Coarse bifurcation analysis of kinetic Monte Carlo simulations: a lattice-gas model with lateral interactions”, J. Chem. Phys., 117:18 (2002), 8229–8240 | DOI

[20] Or-Guil M., Kevrekidis I. G., Bär M., “Stable bound states of pulses in an excitable medium”, Physica D, 135:1–2 (2000), 154–174 | DOI | MR | Zbl