@article{ZVMMF_2004_44_10_a6,
author = {G. Yu. Kulikov and A. I. Merkulov},
title = {On one-step collocation methods with higher derivatives for solving ordinary differential equations},
journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
pages = {1782--1807},
year = {2004},
volume = {44},
number = {10},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZVMMF_2004_44_10_a6/}
}
TY - JOUR AU - G. Yu. Kulikov AU - A. I. Merkulov TI - On one-step collocation methods with higher derivatives for solving ordinary differential equations JO - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki PY - 2004 SP - 1782 EP - 1807 VL - 44 IS - 10 UR - http://geodesic.mathdoc.fr/item/ZVMMF_2004_44_10_a6/ LA - ru ID - ZVMMF_2004_44_10_a6 ER -
%0 Journal Article %A G. Yu. Kulikov %A A. I. Merkulov %T On one-step collocation methods with higher derivatives for solving ordinary differential equations %J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki %D 2004 %P 1782-1807 %V 44 %N 10 %U http://geodesic.mathdoc.fr/item/ZVMMF_2004_44_10_a6/ %G ru %F ZVMMF_2004_44_10_a6
G. Yu. Kulikov; A. I. Merkulov. On one-step collocation methods with higher derivatives for solving ordinary differential equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 44 (2004) no. 10, pp. 1782-1807. http://geodesic.mathdoc.fr/item/ZVMMF_2004_44_10_a6/
[1] Kollatts L., Chislennye metody resheniya differentsialnykh uravnenii, Izd-vo inostr. lit., M., 1953
[2] Henrici P., Discrete variable methods in ordinary differential equations, John Wiley and Sons, New York–London, 1962 | MR | Zbl
[3] Bakhvalov N. S., Chislennye metody, Nauka, M., 1975
[4] Kalitkin H. H., Chislennye metody, Nauka, M., 1978 | MR
[5] Ortega Dzh., Pul U., Vvedenie v chislennye metody resheniya differentsialnykh uravnenii, Nauka, M., 1986 | MR | Zbl
[6] Dekker K., Verver Ya., Ustoichivost metodov Runge–Kutty dlya zhestkikh nelineinykh differentsialnykh uravnenii, Mir, M., 1988 | MR
[7] Marchuk G. I., Metody vychislitelnoi matematiki, Nauka, M., 1989 | MR
[8] Samarskii A. A., Gulin A. V., Chislennye metody, Nauka, M., 1989 | MR
[9] Arushanyan O. B., Zaletkin S. F., Chislennoe reshenie obyknovennykh differentsialnykh uravnenii na Fortrane, Izd-vo MGU, M., 1990 | MR | Zbl
[10] Khairer E., Nersett S., Vanner G., Reshenie obyknovennykh differentsialnykh uravnenii. Nezhestkie zadachi, Mir, M., 1990 | MR
[11] Khairer E., Vanner G., Reshenie obyknovennykh differentsialnykh uravnenii. Zhestkie i differentsialno-algebraicheskie zadachi, Mir, M., 1999
[12] Butcher J. C., Numerical methods for ordinary differential equations, John Wiley and Sons, Chichester, 2003 | MR
[13] Gaiton A., Fiziologiya krovoobrascheniya: Minutnyi ob'em serdtsa i ego regulyatsiya, Meditsina, M., 1969
[14] Grodinz F., Teoriya regulirovaniya i biologicheskie sistemy, Mir, M., 1966
[15] Marchuk G. I., Matematicheskie metody v immunologii. Vychislitelnye metody i eksperimenty, Nauka, M., 1991 | MR
[16] Nerreter V., Raschet elektricheskikh tsepei na personalnoi EVM, Energoatomizdat, M., 1991
[17] Fehlberg E., “Eine methode zur fehlerverkleinerung bein Runge–Kutta verfahren”, Z. angen. Math. und Mech., 38 (1958), 421–426 | DOI | MR | Zbl
[18] Fehlberg E., “New high-order Runge–Kutta formulas with step size control for systems of first and second order differential equations”, Z. angen. Math. und Mech., 44, Sonderheft (1964), T17–T19 | MR
[19] Kastlunger K. H., Wanner G., “Runge–Kutta processes with multiple nodes”, Computing, 9 (1972), 9–24 | DOI | MR | Zbl
[20] Kastlunger K. H., Wanner G., “On Turan type implicit Runge–Kutta methods”, Computing, 9 (1972), 317–325 | DOI | MR | Zbl
[21] Nørsett S. P., “One-step methods of Hermite type for numerical integration of stiff systems”, BIT, 14 (1974), 63–77 | DOI | MR
[22] Khovanskii A. N., Prilozheniya tsepnykh drobei i ikh obobschenii k voprosam priblizhennogo analiza, Gostekhizdat, M., 1956
[23] Aulchenko S. M., Latypov A. F., Nikulichev Yu. V., “Metod chislennogo integrirovaniya sistem obyknovennykh differentsialnykh uravnenii s ispolzovaniem interpolyatsionnykh polinomov Ermita”, Zh. vychisl. matem. i matem. fiz., 38:10 (1998), 1665–1670 | MR
[24] Novikov V. A., Novikov E. A., “O povyshenii effektivnosti algoritmov integrirovaniya obyknovennykh differentsialnykh uravnenii za schet kontrolya ustoichivosti”, Zh. vychisl. matem. i matem. fiz., 25:7 (1985), 1023–1030 | MR | Zbl
[25] Hall G., “Equilibrium states of Runge–Kutta schemes”, ACM Trans. Math. Software, 11 (1985), 289–301 | DOI | MR | Zbl
[26] Hall G., “Equilibrium states of Runge–Kutta schemes, II”, ACM Trans. Math. Software, 12 (1986), 183–192 | DOI | MR | Zbl
[27] Hall G., Higham D. J., “Analisys of stepsize selection schemes for Runge–Kutta codes”, IMA J. Numer. Analys., 8 (1988), 305–310 | DOI | MR | Zbl
[28] Enright W. H., “Analysis of error control strategies for continuous Runge–Kutta methods”, SIAM J. Numer. Analys., 26:3 (1989), 588–599 | DOI | MR | Zbl
[29] Higham D. J., “Robust defect control with Runge–Kutta schemes”, SIAM J. Numer. Analys., 26:5 (1989), 1175–1183 | DOI | MR | Zbl
[30] Skeel R. D., “Thirteen ways to estimate global error”, Numer. Math., 48 (1986), 1–20 | DOI | MR | Zbl
[31] Novikov E. A., “Otsenka globalnoi oshibki $A$-ustoichivykh metodov resheniya zhestkikh sistem”, Doklady akademii nauk, 343:4 (1995), 452–455 | MR | Zbl
[32] Skeel R. D., “Global error estimation and the backward differentiation formulas”, Appl. Math. Comput., 31 (1989), 197–208 | DOI | MR | Zbl
[33] Kulikov G. Yu., “Ob odnom sposobe kontrolya oshibki dlya metodov Runge–Kutty”, Zh. vychisl. matem. i matem. fiz., 38:10 (1998), 1651–1653 | MR | Zbl
[34] Kulikov G. Yu., “A local-global version of a stepsize control for Runge–Kutta methods”, Korean J. Comput. Appl. Math., 7:2 (2000), 289–318 | MR | Zbl
[35] Kulikov G. Yu., Chislennye metody s kontrolem globalnoi oshibki dlya resheniya differentsialnykh i differentsialno-algebraicheskikh uravnenii indeksa 1, Dis. ...dokt. fiz.-matem. nauk, Ulyanovskii gos. un-t, Ulyanovsk, 2002
[36] Berezin I. S., Zhidkov N. P., Metody vychislenii, v. 1, Fizmatgiz, M., 1962 | MR
[37] Kulikov G. Yu., “Teoremy skhodimosti dlya iterativnykh metodov Runge–Kutty s postoyannym shagom integrirovaniya”, Zh. vychisl. matem. i matem. fiz., 36:8 (1996), 73–89 | MR | Zbl
[38] Kulikov G. Yu., “Chislennoe reshenie zadachi Koshi dlya sistemy differentsialno-algebraicheskikh uravnenii s pomoschyu neyavnykh metodov Runge–Kutty s netrivialnym prediktorom”, Zh. vychisl. matem. i matem. fiz., 38:1 (1998), 68–84 | MR | Zbl
[39] Dahlquist G., Stability and error bounds in the numerical integration of ordinary differential equations, Trans. Roy. Inst. Technol. Stockholm, 130, 1959 | MR | Zbl
[40] Dahlquist G., “A special stability problem for linear multistep methods”, BIT, 3 (1963), 27–43 | DOI | MR | Zbl
[41] Ehle B. L., On Padé approximathions to the exponential function and $A$-stable methods for the numerical solution of initial value problems, Res. rep. CSRR, Dept. AACS, Univ. of Waterloo, Ontario, Canada, 2010
[42] Butcher J. C., “On $A$-stable implicit Runge–Kutta methods”, BIT, 17 (1977), 375–378 | DOI | MR | Zbl
[43] Wanner G., Hairer E., Nørsett S. P., “Order stars and stability theorems”, BIT, 18 (1977), 475–489 | DOI | MR
[44] Kulikov G. Yu., “Revision of the theory of symmetric one-step methods for ordinary differential equations”, Korean J. Comput. Appl. Math., 5:3 (1998), 579–600 | MR | Zbl
[45] Gragg W. B., Repeated extrapolation to the limit in the numerical solution of ordinary differential equations, Thesis, Univ. of California, 1964
[46] Gragg W. B., “On extrapolation algorithms for ordinary initial value problems”, SIAM J. Numer. Analys. Ser. B, 2 (1965), 384–403 | MR | Zbl