Explicit adaptive Runge--Kutta methods
Matematičeskoe modelirovanie, Tome 23 (2011) no. 7, pp. 73-87.

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Adjusting the parameters of the integration formula to the problem under solution allows us to increase considerably efficiency of explicit Runge–Kutta methods when solving stiff problems. Construction of such adaptive methods is considered. Results of the solution of test problems in comparison with known methods are presented.
Keywords: explicit Runge–Kutta methods, stiff problems, nonstiff problems.
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L. M. Skvortsov. Explicit adaptive Runge--Kutta methods. Matematičeskoe modelirovanie, Tome 23 (2011) no. 7, pp. 73-87. http://geodesic.mathdoc.fr/item/MM_2011_23_7_a5/

[1] Fowler M. E., Warten R. M., “A numerical integration technique for ordinary differential equations with widely separated eigenvalues”, IBM J. Res. and Development, 11:5 (1967), 537–543 | DOI | MR | Zbl

[2] Lambert J. D., “Nonlinear methods for stiff systems of ordinary differential equations”, Lect. Notes in Math., 363, 1974, 75–88 | DOI | MR | Zbl

[3] Wambecq A., “Rational Runge-Kutta methods for solving systems of ordinary differential equations”, Computing, 20:4 (1978), 333–342 | DOI | MR | Zbl

[4] Bobkov V. V., “Novye yavnye A-ustoichivye metody chislennogo resheniya differentsialnykh uravnenii”, Differents. ur-niya, 14:12 (1978), 2249–2251 | MR | Zbl

[5] Zavorin A. N., “Primenenie nelineinykh metodov dlya rascheta perekhodnykh protsessov v elektricheskikh tsepyakh”, Izv. vuzov. Radioelektronika, 26:3 (1983), 35–41

[6] Ashour S. S., Hanna O. T., “Explicit exponential method for the integration of stiff ordinary differential equations”, J. of Guidance, Control and Dynamics, 14:6 (1991), 1234–1239 | DOI | MR | Zbl

[7] Wu X. Y., Xia J. L., “Two low accuracy methods for stiff systems”, Appl. Math. Comput., 123:2 (2001), 141–153 | DOI | MR | Zbl

[8] Skvortsov L. M., “Adaptivnye metody tsifrovogo modelirovaniya dinamicheskikh sistem”, Izv. RAN. Teoriya i sistemy upravleniya, 1995, no. 4, 180–190 | MR

[9] Skvortsov L. M., “Yavnye adaptivnye metody chislennogo resheniya zhestkikh sistem”, Matem. modelirovanie, 12:12 (2000), 97–107 | MR | Zbl

[10] Skvortsov L. M., “Yavnyi mnogoshagovyi metod chislennogo resheniya zhestkikh differentsialnykh uravnenii”, Zh. vychisl. matem. i matem. fiz., 47:6 (2007), 959–967 | MR

[11] Skvortsov L. M., “Prostye yavnye metody chislennogo resheniya zhestkikh obyknovennykh differentsialnykh uravnenii”, Vychisl. metody i programmirovanie, 9 (2008), 154–162 ; http://www.num-meth.srcc.msu.ru/

[12] Kozlov O. S., Skvortsov L. M., Khodakovskii V. V., Reshenie differentsialnykh i differentsialno-algebraicheskikh uravnenii v programmnom komplekse MVTU http://www.model.exponenta.ru/mvtu/20051121.html

[13] Khairer E., Vanner G., Reshenie obyknovennykh differentsialnykh uravnenii. Zhestkie i differentsialno-algebraicheskie zadachi, Mir, M., 1999, 685 pp.

[14] Ralston A., “Runge-Kutta methods with minimum error bounds”, Math. Comput., 16:80 (1962), 431–437 | DOI | MR | Zbl

[15] Alshina E. A., Zaks E. M., Kalitkin N. N., “Optimalnye parametry yavnykh skhem Runge-Kutty nevysokikh poryadkov”, Matem. modelirovanie, 18:2 (2006), 3–15 | MR | Zbl

[16] Skvortsov L. M., “Modelnye uravneniya dlya issledovaniya tochnosti metodov Runge-Kutty”, Matem. modelirovanie, 22:5 (2010), 146–160 | MR | Zbl

[17] Khairer E., Nërsett S., Vanner G., Reshenie obyknovennykh differentsialnykh uravnenii. Nezhestkie zadachi, Mir, M., 1990, 512 pp. | MR

[18] http://www.web.math.unifi.it/users/brugnano/ BiM/BiMD/index_BiMD.htm/

[19] Lebedev V. I., “Kak reshat yavnymi metodami zhestkie sistemy differentsialnykh uravnenii”, Vychislitelnye protsessy i sistemy, vyp. 8, Nauka, M., 1991, 237–291 | MR

[20] Verwer J. G., “Explicit Runge-Kutta methods for parabolic partial differential equations”, Appl. Numer. Math., 22:1–3 (1996), 359–379 | DOI | MR | Zbl