Geodesic
Automatic step size control for explicit Numerov type methods
Matematičeskoe modelirovanie, Tome 17 (2005) no. 10, pp. 39-46.

Voir la notice de l'article provenant de la source Math-Net.Ru

A new algorithm of automatic step size control is presented for two explicit Numerov type methods of 6th and 8th order. These methods are in use with ODEs of second order, when the first derivatives are absent. It is shown that automatic step size control let decrease the global precision of the methods and dramatically improves their performance. The simple way of getting started with the methods is also proposed. 
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     author = {Ya. Yu. Mikhailin},
     title = {Automatic step size control for explicit {Numerov} type methods},
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Ya. Yu. Mikhailin. Automatic step size control for explicit Numerov type methods. Matematičeskoe modelirovanie, Tome 17 (2005) no. 10, pp. 39-46. http://geodesic.mathdoc.fr/item/MM_2005_17_10_a3/

[1] V. M. Alekseev, Lektsii po nebesnoi mekhanike, Regulyarnaya i khaoticheskaya dinamika, Izhevsk, 1999, 160 pp. | MR

[2] L. D. Landau, E. M. Lifshits, Kvantovaya mekhanika, Nauka, M., 1972, 368 pp. | MR

[3] Ch. Tsitouras, “Dissipative high phase-lag order methods”, Appl. Math. and Comput., 117 (2001), 35–43 | DOI | MR | Zbl

[4] G. Papageorgiou, Ch. Tsitouras and I. Famelis, “Explicit Numerov type methods for second order IVPs with oscillating solutions”, Int. J. Mod. Phys., 12 (2001), 657–666 | DOI | MR

[5] Ch. Tsitouras, “Explicit two step methods for second order linear IVPs”, Comput. Math. Appl., 43 (2002), 943–949 | DOI | MR | Zbl

[6] Ch. Tsitouras, “Explicit Numerov type methods with reduced number of stages”, Comput. Math. Appl., 45:1–3 (2003), 37–42 | DOI | MR | Zbl

[7] Ch. Tsitouras and S. N. Papakostas, “Cheap Error Estimation for Runge–Kutta methods”, SIAM J. Sci. Comput., 20 (1999), 2067–2088 | DOI | MR | Zbl

[8] M. M. Chawla and P. S. Rao, “An explicit sixth-order method with phase-lag of order eight for $y''=f(t,y)$”, J. Comput. Appl. Math., 17 (1987), 365–368 | DOI | MR | Zbl

[9] S. N. Papakostas and Ch. Tsitouras, “High phase-lag order Runge–Kutta and Nystrom pairs”, SIAM J. Sci. Comput., 21 (2000), 747–763 | DOI | MR

[10] T. E. Simos, “Eight-order methods for elastic scattering phase shifts”, Int. J. Theor. Phys., 36 (1997), 663–672 | DOI | MR | Zbl

[11] T. E. Simos, “Explicit eight-order method for the numerical integration of initial value problem with periodic or oscillating solutions”, Comput. Phys. Commun., 119 (1999), 32–44 | DOI | MR | Zbl

[12] Ch. Tsitouras and T. E. Simos, “High order explicit methods for the integration of periodic initial value problems”, Appl. Math. and Comput., 95 (1998), 15–26 | DOI | MR | Zbl

[13] J. R. Dormand, M. E. El-Mikkawy and P. J. Prince, “Families of Runge–Kutta–Nystrom formulae”, IMA J. Numer. Anal., 7 (1987), 235–250 | DOI | MR | Zbl

[14] Z. Kalogiratou and Ch. Tsitouras, Intermediate reconsideration of Runge–Kutta step, Conference on Scientific Computing and Differential Equations, ed. B. Owren, Grado, 1997 | MR

[15] A. Gordon Emslie and I. W. Walker, “Studies in the application of reccurence relations to special perturbation methods”, Celestial Mechanics, 19 (1979), 147–162 | DOI | MR | Zbl

[16] T. E. Hull, W. H. Enright, B. M. Fellen and A. E. Sedgwick, “Comparing numerical methods for ordinary differential equations”, SIAM J. Numer. Anal., 9 (1972), 603–637 | DOI | MR | Zbl

[17] O. Montenbruk, T. Pfleger, Astronomiya na personalnom kompyutere, Piter, SPb., 2002, 320 pp. | MR

[18] A. Jorba and M. Zou, A software package for the numerical integration of ODE by means of high-order Taylor methods, http://www.maia.ub.es/~angel/taylor.pdf | MR

[19] J. L. M. Quiroz Gonzalez and D. Thompson, “Getting Started With Numerov's Method”, Comput. in Phys., 5 (1997), 514–515 | DOI | MR

[20] V. G. Demin, I. I. Kosenko, P. S. Krasilnikov, S. D. Furta, Izbrannye zadachi nebesnoi mekhaniki, ed. P. S. Krasilnikov, Izdatelskii dom “Udmurtskii universitet”, Izhevsk, 1999, 210 pp. | MR | Zbl

[21] Panayotis Avgoustos Skordos, Multistep methods for integrating the Solar system, Submitted in partial fulfilment of the requirements for the degree of Master of Science at the Massachusetts, Institute of Technology, 1988 | MR

[22] W. H. Enright and J. D. Pryce, “Two FORTRAN packages for assessing initial value methods”, ACM Trans. Math. Software, 13 (1987), 1–27 | DOI | MR | Zbl