Abstract
The construction of an eighth-order exponentially fitted (EF) two-step hybrid method for the numerical integration of oscillatory second-order initial value problems (IVPs) is considered. The EF two-step hybrid methods integrate exactly differential systems whose solutions can be expressed as linear combinations of exponential or trigonometric functions and have variable coefficients depending on the frequency of each problem. Based on the order conditions and the EF conditions for this class of methods, we construct an explicit EF two-step hybrid method with symmetric nodes and algebraic order eight which only uses seven function evaluations per step. This new method has the highest algebraic order we know for the case of explicit EF two-step hybrid methods. The numerical experiments carried out with several orbital and oscillatory problems show that the new eighth-order EF scheme is more efficient than other standard and EF two-step hybrid codes recently proposed in the scientific literature.
Similar content being viewed by others
References
Landau, L.D., Lifshitz, F.M.: Quantum Mechanics. Pergamon Press, New York (1965)
Liboff, R.L.: Introductory Quantum Mechanics. Addison–Wesley, Reading (1980)
Bettis, D.G.: Runge–kutta algorithms for oscillatory problems. J. Appl. Math. Phys. (ZAMP) 30, 699–704 (1979)
González, A. B., Martín, 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)
Franco, J.M.: Runge-kutta-nyström methods adapted to the numerical integration of perturbed oscillators. Comput. Phys. Comm. 147, 770–787 (2002)
Yang, H., Wu, X., You, X., Fang, Y.: Extended RKN-type methods for numerical integration of perturbed oscillators. Comput. Phys. Comm. 180, 1777–1794 (2009)
Van de Vyver, H.: An explicit Numerov-type method for second-order differential equations with oscillating solutions. Comput. Math. Appl. 53, 1339–1348 (2007)
Papadopoulos, D.F., Anastassi, Z.A., Simos, T.E.: A phase-fitted runge-kutta-nyström method for the numerical solution of initial value problems with oscillating solutions. Comput. Phys. Comm. 180, 1839–1846 (2009)
Coleman, J.P., Duxbury, S.C.: Mixed collocation methods for \(y^{\prime \prime }=f(x, y)\). J. Comput. Appl. Math. 126, 47–75 (2000)
Franco, J.M.: An embedded pair of exponentially fitted explicit Runge–Kutta methods. J. Comput. Appl. Math. 149, 407–414 (2002)
Franco, J.M.: Exponentially fitted explicit Runge–Kutta–Nyström methods. J. Comput. Appl. Math. 167, 1–19 (2004)
Gautschi, W.: Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer. Math. 3, 381–397 (1961)
Ozawa, K.: A functional fitting Runge–Kutta method with variable coefficients. J. Japan Indust. Appl. Math. 18, 107–130 (2001)
Ozawa, K.: A functionally fitted three-stage explicit singly diagonally implicit Runge-Kutta method. J. Japan Indust. Appl. Math. 22, 403–427 (2005)
Paternoster, B.: Runge–Kutta(–Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials. Appl. Numer. Math. 28, 401–412 (1998)
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)
Simos, T.E.: Exponentially–fitted Runge–Kutta–Nyström method for the numerical solution of initial–value problems with oscillating solutions. Appl. Math. Lett. 15, 217–225 (2002)
Vanden Berghe, G., De Meyer, H., Van Daele, M., Van Hecke, T.: Exponentially–fitted explicit Runge–Kutta methods. Comput. Phys. Commun. 123, 7–15 (1999)
Vanden Berghe, G., De Meyer, H., Van Daele, M., Van Hecke, T.: Exponentially fitted Runge–Kutta methods. J. Comput. Appl. Math. 125, 107–115 (2000)
Vanden Berghe, G., Ixaru, Gr L., De Meyer, H.: Frequency determination and step-length control for exponentially fitted Runge–Kutta methods. J. Comput. Appl. Math. 132, 95–105 (2001)
Ixaru, Gr. L., Vanden Berghe, G., De Meyer, H.: Frequency evaluation in exponential fitting algorithms for ODEs. J. Comput. Appl. Math. 140, 423–434 (2002)
Van de Vyver, H.: An embedded exponentially fitted Runge–Kutta–Nyström method for the numerical solution of orbital problems. New Astron. 11, 577–587 (2006)
Fang, Y., Song, Y., Wu, X.: Trigonometrically fitted explicit Numerov-type method for periodic IVPs with two frequencies. Comput. Phys. Commun. 179, 801–811 (2008)
Ramos, H., Vigo-Aguiar, J.: On the frequency choice in trigonometrically fitted methods. Appl. Math. Lett. 23, 1378–1381 (2010)
D’Ambrosio, R., Esposito, E., Paternoster, B.: Exponentially fitted two-step hybrid methods for \(y^{\prime \prime } = f(x, y)\). J. Comput. Appl. Math. 235, 4888–4897 (2011)
Franco, J.M., Rández, L.: Explicit exponentially fitted two-step hybrid methods of high order for second-order oscillatory IVPs. Appl. Math. Comput. 273, 493–505 (2016)
Kalogiratou, Z., Monovasilis, Th., Higinio Ramos, T.E., Simos, A: New approach on the construction of trigonometrically fitted two step hybrid methods. J. Comput. Appl. Math. 303, 146–155 (2016)
Ixaru, L. Gr., Vanden Berghe, G.: Exponential Fitting. Kluwer Academic Publishers (2004)
Coleman, J.P.: Order conditions for a class of two–step methods for \(y^{\prime \prime } = f(x, y)\). IMA J. Numer. Anal. 23, 197–220 (2003)
Franco, J.M.: A class of explicit two–step hybrid methods for second–order IVPs. J. Comput. Appl. Math. 187, 41–57 (2006)
Ixaru, L. Gr.: Operations on oscillatory functions. Comput. Phys. Commun. 105, 1–19 (1997)
Tsitouras, Ch.: Explicit eighth order two-step methods with nine stages for integrating oscillatory problems. Int. J. Mod. Phys. C 17, 861–876 (2006)
Famelis, I. Th.: Explicit eighth order Numerov-type methods with reduced number of stages for oscillatory IVPs. Int. J. Mod. Phys. C 19, 957–970 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Franco, J.M., Rández, L. An eighth-order exponentially fitted two-step hybrid method of explicit type for solving orbital and oscillatory problems. Numer Algor 78, 243–262 (2018). https://doi.org/10.1007/s11075-017-0374-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-017-0374-1