On the global error control in nested implicit Runge--Kutta methods of Gauss type

G. Yu. Kulikov ; E. B. Kuznetsov ; E. Yu. Khrustaleva

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 14 (2011) no. 3, pp. 245-259.

Résumé

The automatic global error control based on a combined step size and order control presented by Kulikov and Khrustaleva in 2008 is investigated. A special attention is given to the efficiency of computation because the implicit extrapolation based on the multi-stage implicit Runge–Kutta schemes might be expensive. Especially, we discuss the technique of global error estimation and control in order to compute the numerical solution satisfying the user-supplied accuracy conditions (in exact arithmetic) in the automatic mode. The theoretical results of this paper are confirmed by numerical experiments on test problems.

Bibliographie

[1] Kulikov G. Yu., Shindin S. K., “On a family of cheap symmetric one-step methods of order four”, Computational Science ICCS 2006, 6th International Conference (Reading, UK, May 28–31, 2006), Proc. Part I, Lecture Notes in Comp. Sci., 3991, eds. V. N. Alexandrov et al., 2006, 781–785

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

[3] Kulikov G. Yu., Shindin S. K., “Numerical tests with Gauss-type nested implicit Runge–Kutta formulas”, Computational Science ICCS 2007, 7th International Conference (Beijing, China, May 27–30, 2007), Proc. Part I, Lecture Notes in Comp. Sci., 4487, eds. Y Shi et al., 2007, 136–143

[4] Kulikov G. Yu., Shindin S. K., “Adaptive nested implicit Runge–Kutta formulas of Gauss type”, Appl. Numer. Math., 59 (2009), 707–722 | DOI | MR | Zbl

[5] Kulikov G. Yu., “Automatic error control in the Gauss-type nested implicit Runge–Kutta formula of order 6”, Russian J. Numer. Anal. Math. Modelling, 24:2 (2009), 123–144 | DOI | MR | Zbl

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

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

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

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

[10] Kulikov G. Yu., Khrustaleva E. Yu., “Ob avtomaticheskom upravlenii razmerom shaga i poryadkom v neyavnykh odnoshagovykh ekstrapolyatsionnykh metodakh”, Zhurn. vychisl. matem. i mat. fiziki, 48:9 (2008), 1580–1606 | MR

[11] Kulikov G. Yu., Weiner R., “Variable-stepsize interpolating explicit parallel peer methods with inherent global error control”, SIAM J. Sci. Comp., 32:4 (2010), 1695–1723 | DOI | MR | Zbl

[12] Dahlquist G., “Stability and error bounds in the numerical integration of ordinary differential equations”, Trans. Royal Inst. of Technology, 130 (1959), 1–87 | MR

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

[14] Butcher J. C., “On the implementation of implicit Runge–Kutta methods”, BIT, 16 (1976), 237–240 | DOI | MR | Zbl

[15] Bickart T. A., “An efficient solution process for implicit Runge–Kutta methods”, SIAM J. Numer. Anal., 14 (1977), 1022–1027 | DOI | MR | Zbl

[16] Alexander R., “Diagonally implicit Runge–Kutta methods for stiff ODEs”, SIAM J. Numer. Anal., 14 (1977), 1006–1024 | DOI | MR

[17] Alt R., Methodes A-stables pour l'integration de systemes differentiels mal conditionnes, Ph. D. thesis, Universite Paris, Paris, 1971

[18] Crouzeix M., Sur l'approximation des equations differentielles operationnelles lineaires par de methodes de Runge–Kutta, Ph. D. thesis, Universite Paris, Paris, 1975

[19] Kurdi M., Stable high order methods for time discretization of stiff differential quations, Ph. D. thesis, University of California, California, 1974

[20] Kværnø A., “Singly diagonally implicit Runge–Kutta methods with an explicit first stage”, BIT, 44 (2004), 489–502 | DOI | MR | Zbl

[21] Nørsett S. P., Semi-explicit Runge–Kutta methods, Report No 6/74, Dept. of Math., University of Trondheim, Trondheim, 1974

[22] Prothero A., Robinson A., “On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations”, Math. Comp., 24 (1974), 145–162 | DOI | MR

[23] Burrage K., “A special family of Runge–Kutta methods for solving stiff differential equations”, BIT, 18 (1978), 22–41 | DOI | MR | Zbl

[24] Burrage K., Butcher J. C., Chipman F. H., “An implementation of singly implicit Runge–Kutta methods”, BIT, 20 (1980), 326–340 | DOI | MR | Zbl

[25] Nørsett S. P., “Runge–Kutta methods with multiple real eigenvalue only”, BIT, 16 (1976), 388–393 | DOI | MR

[26] Butcher J. C., Cash J. R., “Towards efficient Runge–Kutta methods for stiff systems”, SIAM J. Numer. Anal., 27 (1990), 753–761 | DOI | MR | Zbl

[27] Butcher J. C., Chen D. J. L., “A new type of singly implicit Runge–Kutta methods”, Appl. Numer. Math., 34 (2000), 179–188 | DOI | MR | Zbl

[28] Nørsett S. P., Wolfbrandt A., “Attainable order of rational approximations to the exponential function with only real poles”, BIT, 17 (1977), 200–208 | DOI | MR

[29] Cash J. R., “On a class of implicit Runge–Kutta procedures”, J. Inst. Math. Appl., 19 (1977), 455–470 | DOI | MR | Zbl

[30] Cash J. R., “On a note of the computational aspects of a class of implicit Runge–Kutta procedures”, J. Inst. Math. Appl., 20 (1977), 425–441 | DOI | MR | Zbl

[31] van Bokhoven W. M. G., “Efficient higher order implicit one-step methods for integration of stiff differential equations”, BIT, 20 (1980), 34–43 | DOI | MR | Zbl

[32] Cash J. R., Singhal A., “Mono-implicit Runge–Kutta formulae for the numerical integration of stiff differential systems”, IMA J. Numer. Anal., 2 (1982), 211–227 | DOI | MR | Zbl