# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A 5(3) pair of explicit ARKN methods for the numerical integration of perturbed oscillators. (English) Zbl 1037.65073
Summary: A new embedded pair of explicit adapted Runge-Kutta-Nyström (ARKN) methods specially adapted to the numerical integration of perturbed oscillators is obtained. This pair depends on a parameter $\nu = \omega h > 0$ ($h$ is the integration step and $\omega$ is the dominant frequency), and it has four stages and algebraic orders five and three. The numerical experiments carried out show that the new pair is very competitive when it is compared with high-quality codes proposed in the scientific literature.

##### MSC:
 65L06 Multistep, Runge-Kutta, and extrapolation methods 65L05 Initial value problems for ODE (numerical methods) 34A34 Nonlinear ODE and systems, general
Full Text:
##### References:
 [1] Avdelas, G.; Simos, T. E.; Vigo-Aguiar, J.: An embedded exponentially-fitted Runge--Kutta method for the numerical solution of the Schrödinger equation and related periodic initial-value problems. Comput. phys. Commun. 131, 52-67 (2000) · Zbl 0982.65079 [2] Bettis, D. G.: Numerical integration of products of Fourier and ordinary polynomials. Numer. math. 14, 421-434 (1970) · Zbl 0198.49601 [3] Bettis, D. G.: Runge--Kutta algorithms for oscillatory problems. J. appl. Math. phys. (ZAMP) 30, 699-704 (1979) · Zbl 0412.65038 [4] Dormand, J. R.; El-Mikkawy, M. E. A.; Prince, P. J.: Families of Runge--Kutta--Nyström formulae. IMA J. Numer. anal. 7, 235-250 (1987) · Zbl 0624.65059 [5] Franco, J. M.: Runge--Kutta--Nyström methods adapted to the numerical integration of perturbed oscillators. Comput. phys. Commun. 147, 770-787 (2002) · Zbl 1019.65050 [6] J.M. Franco, Embedded pairs of explicit ARKN methods for the numerical integration of perturbed oscillators, in: Proceedings of the 2002 Conference on Computational and Mathematical Methods on Science and Engineering, Vol. I, Alicante, 2002, pp. 92--101. [7] Franco, J. M.: An embedded pair of exponentially fitted explicit Runge--Kutta methods. J. comput. Appl. math. 149, 407-414 (2002) · Zbl 1014.65061 [8] Franco, J. M.; Palacián, J. F.: High order adaptive methods of Nyström--cowell type. J. comput. Appl. math. 81, 115-134 (1997) · Zbl 0879.65050 [9] Garcı\acute{}a, A.; Martı\acute{}n, P.; González, A. B.: New methods for oscillatory problems based on classical codes. Appl. numer. Math. 42, 141-157 (2002) · Zbl 0998.65069 [10] González, A. B.; Martı\acute{}n, P.; Farto, J. M.: A new family of Runge--Kutta type methods for the numerical integration of perturbed oscillators. Numer. math. 82, 635-646 (1999) · Zbl 0935.65075 [11] Hairer, E.; Nørsett, S. P.; Wanner, S. P.: Solving ordinary differential equations I, nonstiff problems. (1993) · Zbl 0789.65048 [12] Jain, M. K.: A modification of the Stiefel--bettis method for nonlinear damped oscillators. Bit 28, 302-307 (1998) · Zbl 0646.65063 [13] Paternoster, B.: Runge--$Kutta(--Nystr\"om)$ methods for odes with periodic solutions based on trigonometric polynomials. Appl. numer. Math. 28, 401-412 (1998) · Zbl 0927.65097 [14] Simos, T. E.: An exponentially-fitted Runge--Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions. Comput. phys. Commun. 115, 1-8 (1998) · Zbl 1001.65080 [15] Stiefel, E.; Bettis, D. G.: Stabilization of cowell’s method. Numer. math. 13, 154-175 (1969) · Zbl 0219.65062 [16] Berghe, G. Vanden; De Meyer, H.; Van Daele, M.; Van Hecke, T.: Exponentially-fitted explicit Runge--Kutta methods. Comput. phys. Commun. 123, 7-15 (1999) · Zbl 0948.65066 [17] Berghe, G. Vanden; De Meyer, H.; Van Daele, M.; Van Hecke, T.: Exponentially fitted Runge--Kutta methods. J. comput. Appl. math. 125, 107-115 (2000) · Zbl 0999.65065