Geodesic
Explicit adaptive Runge–Kutta methods for stiff and oscillation problems
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 8, pp. 1434-1448 Cet article a éte moissonné depuis la source Math-Net.Ru

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Explicit Runge–Kutta methods with the coefficients tuned to the problem of interest are examined. The tuning is based on estimates for the dominant eigenvalues of the Jacobian matrix obtained from the results of the preliminary stages. Test examples demonstrate that methods of this type can be efficient in solving stiff and oscillation problems. 
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L. M. Skvortsov. Explicit adaptive Runge–Kutta methods for stiff and oscillation problems. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 8, pp. 1434-1448. http://geodesic.mathdoc.fr/item/ZVMMF_2011_51_8_a6/

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

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

[3] Fatunla S. O., “Numerical integrators for stiff and highly oscillatory differential equations”, Math. Comput., 34:150 (1980), 373–390 | DOI | MR | Zbl

[4] Bobkov V. V., “Ob odnom sposobe postroeniya metodov chislennogo resheniya differentsialnykh uravnenii”, Differents. ur-niya, 19:7 (1983), 1115–1122 | 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] Saworin A. N., “Über die Effektivitat einiger nichtlinearer Verfahren bei der numerischen Behandlung steifer Differentialgleichungssysteme”, Numer. Math., 40:2 (1982), 169–177 | DOI | MR | Zbl

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

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

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

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

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

[12] Ramos H., “A non-standard explicit integration scheme for initial-value problems”, Appl. Math. Comput., 189:1 (2007), 710–718 | DOI | MR | Zbl

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

[14] Skvortsov L. M., “Adaptivnye metody chislennogo integrirovaniya v zadachakh modelirovaniya dinamicheskikh sistem”, Izv. RAN. Teoriya i sistemy upravleniya, 1999, no. 4, 72–78 | MR | Zbl

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

[16] Skvortsov L. M., “Yavnye metody Runge-Kutty dlya umerenno zhestkikh zadach”, Zh. vychisl. matem. i matem. fiz., 45:11 (2005), 2017–2030 | MR | Zbl

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

[18] Simos T. E., “A modified Runge–Kutta method for the numerical solution of ODE' s with oscillation solutions”, Appl. Math. Letters, 9:6 (1996), 61–66 | DOI | MR | Zbl

[19] Franco J. M., “Runge–Kutta methods adapted to the numerical integration of oscillatory problems”, Appl. Numer. Math., 50:3,4 (2004), 427–443 | DOI | MR | Zbl

[20] Fang Y., Song Y., Wu X., “New embedded pairs of explicit Runge–Kutta methods with FSAL properties adapted to the numerical integration of oscillatory problems”, Phys. Letters A, 372:44 (2008), 6551–6559 | DOI | MR | Zbl

[21] Kosti A. A., Anastassi Z. A., Simos T. E., “An optimized explicit Runge–Kutta method with increased phase-lag order for the numerical solution of the Schrödinger equation and related problems”, J. Math. Chem., 47:1 (2010), 315–330 | DOI | MR | Zbl

[22] Prothero A., Robinson A., “On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations”, Math. Comput., 28:1 (1974), 145–162 | DOI | MR

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

[24] Mazzia F., Magherini C., Test set for initial value problem solvers, Release 2.4, 2008 http://www.pitagora.dm.uniba.it/~testset/ report/testset.pdf

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