Geodesic
Automatic step size and order control in implicit one-step extrapolation methods
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 9, pp. 1580-1606 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A theory is presented for implicit one-step extrapolation methods for ordinary differential equations. The computational schemes used in such methods are based on the implicit Runge–Kutta methods. An efficient implementation of implicit extrapolation is based on the combined step size and order control. The emphasis is placed on calculating and controlling the global error of the numerical solution. The aim is to achieve the user-prescribed accuracy in an automatic mode (ignoring round-off errors). All the theoretical conclusions of this paper are supported by the numerical results obtained for test problems. 
@article{ZVMMF_2008_48_9_a5,
     author = {G. Yu. Kulikov and E. Y. Khrustaleva},
     title = {Automatic step size and order control in implicit one-step extrapolation methods},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1580--1606},
     year = {2008},
     volume = {48},
     number = {9},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_9_a5/}
}
TY  - JOUR
AU  - G. Yu. Kulikov
AU  - E. Y. Khrustaleva
TI  - Automatic step size and order control in implicit one-step extrapolation methods
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2008
SP  - 1580
EP  - 1606
VL  - 48
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_9_a5/
LA  - ru
ID  - ZVMMF_2008_48_9_a5
ER  - 
%0 Journal Article
%A G. Yu. Kulikov
%A E. Y. Khrustaleva
%T Automatic step size and order control in implicit one-step extrapolation methods
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2008
%P 1580-1606
%V 48
%N 9
%U http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_9_a5/
%G ru
%F ZVMMF_2008_48_9_a5
G. Yu. Kulikov; E. Y. Khrustaleva. Automatic step size and order control in implicit one-step extrapolation methods. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 9, pp. 1580-1606. http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_9_a5/

[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] Shtetter X., Analiz metodov diskretizatsii dlya obyknovennykh differentsialnykh uravnenii, Mir, M., 1978 | MR

[4] Kalitkin H. H., Chislennye metody, Nauka, M., 1978 | MR

[5] Rakitskii Yu. V., Ustinov S. M., Chernorutskii I. G., Chislennye metody resheniya zhestkikh sistem, Nauka, Fizmatlit, M., 1979 | MR

[6] Ortega Dzh., Pul U., Vvedenie v chislennye metody resheniya differentsialnykh uravnenii, Nauka, M., 1986 | MR | Zbl

[7] Bakhvalov N. S., Zhidkov H. P., Kobelkov G. M., Chislennye metody, Nauka, M., 1987 | MR | Zbl

[8] Dekker K., Verver Ya., Ustoichivost metodov Runge–Kutty dlya zhestkikh nelineinykh differentsialnykh uravnenii, Mir, M., 1988 | MR

[9] Marchuk G. I., Metody vychislitelnoi matematiki, Nauka, M., 1989 | MR

[10] Samarskii A. A., Gulin A. B., Chislennye metody, Nauka, M., 1989 | MR

[11] Arushanyan O. B., Zaletkin S. F., Chislennoe reshenie obyknovennykh differentsialnykh uravnenii na Fortrane, Izd-vo MGU, M., 1990 | MR | Zbl

[12] Khairer E., Nersett S., Vanner G., Reshenie obyknovennykh differentsialnykh uravnenii. Nezhestkie zadachi, Mir, M., 1990 | MR

[13] Khairer E., Vanner G., Reshenie obyknovennykh differentsialnykh uravnenii. Zhestkie i differentsialno-algebraicheskie zadachi, Mir, M., 1999

[14] Butcher J. C., Numerical methods for ordinary differential equations, John Wiley and Sons, Chichester, 2003 | MR

[15] Gaiton A., Fiziologiya krovoobrascheniya: Minutnyi ob'em serdtsa i ego regulyatsiya, Meditsina, M., 1969

[16] Grodinz F., Teoriya regulirovaniya i biologicheskie sistemy, Mir, M., 1966

[17] Marchuk G. I., Matematicheskie metody v immunologii. Vychislitelnye metody i eksperimenty, Nauka, M., 1991 | MR

[18] Nerreter V., Raschet elektricheskikh tsepei na personalnoi EVM, Energoatomizdat, M., 1991

[19] Bulirsch R.. Stoer J., “Numerical treatment of ordinary differential equations by extrapolation methods”, Numer. Math., 8 (1966), 1–13 | DOI | MR | Zbl

[20] Deuflhard P., “Order and stepsize control in extrapolation methods”, Numer. Math., 41 (1983), 399–422 | DOI | MR | Zbl

[21] Deuflhard P., “Recent progress in extrapolation methods for ordinary differential equations”, SIAM Rev., 27 (1985), 505–535 | DOI | MR | Zbl

[22] Gragg W. B., Repeated extrapolation to the limit in the numerical solution of ordinary differential equations, Thesis, Univ. of California, 1964

[23] Gragg W. B., “On extrapolation algorithms for ordinary initial value problems”, SIAM J. Numer. Analys. Ser. B, 2 (1965), 384–403 | DOI | MR | Zbl

[24] Kulikov G. Yu., Khrustaleva E. Yu., “Ob avtomaticheskom upravlenii razmerom shaga i poryadkom v yavnykh odnoshagovykh ekstrapolyatsionnykh metodakh”, Zh. vychisl. matem. i matem. fiz., 48:8 (2008), 1392–1405 | MR | Zbl

[25] 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

[26] 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

[27] Kulikov G. Yu., Chislennye metody s kontrolem globalnoi oshibki dlya resheniya differentsialnykh i differentsialno-algebraicheskikh uravnenii indeksa 1, Dis. $\dots$ dokt. fiz.-matem. nauk, Ulyanovskii gos. un-t, Ulyanovsk, 2002

[28] Kulikov G. Yu., “On implicit extrapolation methods for ordinary differential equations”, Rus. J. Numer. Analys. Math. Modelling, 17:1 (2002), 41–69 | MR

[29] Kulikov G. Yu., “O neyavnykh ekstrapolyatsionnykh metodakh dlya sistem differentsialno-algebraicheskikh uravnenii”, Vestn. Mosk. un-ta. Ser. Matem., Mekhan., 2002, no. 5, 3–7 | MR

[30] Kulikov G. Yu., Merkulov A. M., “Ob odnoshagovykh kollokatsionnykh metodakh so starshimi proizvodnymi dlya resheniya obyknovennykh differentsialnykh uravnenii”, Zh. vychisl. matem. i matem. fiz., 44:10 (2004), 1782–1807 | MR | Zbl

[31] Kulikov G. Yu., “One-step methods and implicit extrapolation technique for index 1 differential-algebraic systems”, Rus. J. Numer. Analys. Math. Modelling, 19:6 (2004), 527–553 | DOI | MR | Zbl

[32] Novikov E. A., “Otsenka globalnoi oshibki $A$-ustoichivykh metodov resheniya zhestkikh sistem”, Dokl. RAN, 343:4 (1995), 452–455 | MR | Zbl

[33] Kulikov G. Yu., Shindin S. K., “Ob odnom sposobe upravleniya globalnoi oshibkoi mnogoshagovykh metodov”, Zh. vychisl. matem. i matem. fiz., 40:9 (2000), 1308–1329 | MR | Zbl

[34] Kulikov G. Yu., Shindin S. K., “O mnogoshagovykh ekstrapolyatsionnykh metodakh dlya obyknovennykh differentsialnykh uravnenii”, Dokl. RAN, 372:3 (2000), 301–304 | MR | Zbl

[35] Kulikov G. Yu., Shindin S. K., “O mnogoshagovykh metodakh interpolyatsionnogo tipa s avtomaticheskim kontrolem globalnoi oshibki”, Zh. vychisl. matem. i matem. fiz., 44:8 (2004), 1388–1409 | MR | Zbl

[36] Kulikov G. Yu., Shindin S. K., “Global error estimation and extrapolated multistep methods for index 1 differential-algebraic systems”, BIT, 45 (2005), 517–542 | DOI | MR | Zbl

[37] Kulikov G. Yu., Shindin S. K., “One-leg integration of ordinary differential equations with global error control”, Comput. Methods Appl. Math., 5:1 (2005), 86–96 | MR | Zbl

[38] Kulikov G. Yu., Shindin S. K., “One-leg variable-coefficient formulas for ordinary differential equations and localglobal step size control”, Numer. Algorithms, 43 (2006), 99–121 | DOI | MR | Zbl

[39] Skeel R. D., “Thirteen ways to estimate global error”, Numer. Math., 48 (1986), 1–20 | DOI | MR | Zbl

[40] Aïd R., Levacher L., “Numerical investigations on global error estimation for ordinary differential equations”, J. Comput. Appl. Math., 82 (1997), 21–39 | DOI | MR | Zbl

[41] Kulikov G. Yu., “A theory of symmetric one-step methods for differential-algebraic equations”, Rus. J. Numer. Analys. Math. Modelling, 12:6 (1997), 501–523 | DOI | MR | Zbl

[42] 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

[43] Butcher J. C., Chan R. P. K., “On symmetrizers for Gauss methods”, Numer. Math., 60 (1992), 465–476 | DOI | MR | Zbl

[44] Chan R. P. K., “Generalized symmetric Runge–Kutta methods”, Computing, 50 (1993), 31–49 | DOI | MR | Zbl

[45] Dahlquist G., “A special stability problem for linear multistep methods”, BIT, 3 (1963), 27–43 | DOI | MR | Zbl

[46] Kulikov G. Yu., Shindin S. K., “On a family of cheap symmetric one-step methods of order four”, Comput. Sci. – ICCS 2006. 6th Internat. Conf., Proc, Part I (Reading, UK, May 28–31, 2006), Lect. Notes in Comput. Sci., 3991, 2006, 781–785

[47] Kulikov G. Yu., Shindin S. K., “Numerical tests with Gauss-type nested implicit Runge–Kutta formulas”, Comput. Sci. – ICCS 2007. 7th Internat. Conf., Proc, Part I (Beijing, China, May 27–30, 2007), Lect. Notes in Comput. Sei., 4487, 2007, 136–143

[48] Kulikov G. Yu., Merkulov A. I., Shindin S. K., “Asymptotic error estimate for general Newton-type methods and its application to differential equations”, Rus. J. Numer. Analys. Math. Modelling, 22:6 (2007), 567–590 | DOI | MR | Zbl

[49] Kulikov G. Yu., Shindin S. K., “Adaptive nested implicit Runge–Kutta formulas of Gauss type”, Appl. Numer. Math. (to appear) | MR

[50] Kulikov G. Yu., Khrustaleva E. Yu., Merkulov A. I., “Symmetric Runge–Kutta methods with higher derivatives and quadratic extrapolation”, Comput. Sci. – ICCS 2006. 6th Internat. Conf., Proc, Part I (Reading, UK, May 28–31, 2006), Lect. Notes in Comput. Sci., 3991, 2006, 117–123 | Zbl