Abstract
A continuous hybrid method using trigonometric basis (CHMTB) with one ‘off-step’ point is developed and used to produce two discrete hybrid methods which are simultaneously applied as numerical integrators by assembling them into a block hybrid method with trigonometric basis (BHMTB) for solving oscillatory initial value problems (IVPs). The stability property of the BHMTB is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.
Similar content being viewed by others
References
Butcher, J.C.: A modified multistep method for the numerical integration of ordinary differential equations. J. Assoc. Comput. Mach. 12, 124–135 (1965)
Dahlquist, G.G.: Numerical integration of ordinary differential equations. Math. Scand. 4, 69–86 (1956)
Fatunla, S.O.: Block methods for second order IVPs. Int. J. Comput. Math. 41, 55–63 (1991)
Gear, C.W.: Hybrid methods for initial value problems in ordinary differential equations. SIAM J. Numer. Anal. 2, 69–86 (1965)
Gragg, W., Stetter, H.J.: Generalized multistep predictor-corrector methods. J. Assoc. Comput. Mach. 11, 188–209 (1964)
Gupta, G.K.: Implementing second-derivative multistep methods using Nordsieck polynomial representation. Math. Comput. 32, 13–18 (1978)
Hairer, E.: A one-step method of order 10 for y′′ = f(x, y). IMA J. Numer. Anal. 2, 83–94 (1982)
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Springer, New York (1996)
Jator, S.N., Swindle, S., French, R.: Trigonometrically fitted block Numerov type method for y” = f(x, y, y’). Numer. Algorithms (2012). doi:10.1007/s11075-012-9562-1
Kohfeld, J.J., Thompson, G.T.: Multistep methods with modified predictors and correctors. J. Assoc. Comput. Mach. 14, 155–166 (1967)
Lambert, J.D.: Computational Methods in Ordinary Differential Equations. Wiley, New York (1973)
Lambert, J.D.: Numerical Methods for Ordinary Differential Systems. Wiley, New York (1991)
Nguyen, H.S., Sidje, R.B., Cong, N.H.: Analysis of trigonometric implicit Runge–Kutta methods. J. Comput. Appl. Math. 198, 187–207 (2007)
Ozawa, K.: A functionally fitted three-stage explicit singly diagonally implicit Runge–Kutta method. Japan J. Indust. Appl. Math. 22, 403–427 (2005)
Shampine, L.F., Watts, H.A.: Block implicit one-step methods. Math. Comput. 23, 731–740 (1969)
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)
Sommeijer, B.P.: Explicit, high-order Runge–Kutta–Nyström methods for parallel computers. Appl. Numer. Math. 13, 221–240 (1993)
Vanden, G., Ixaru, L.G., van Daele, M.: Optimal implicit exponentially-fitted Runge–Kutta. Comput. Phys. Commun. 140, 346–357 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ngwane, F.F., Jator, S.N. Block hybrid method using trigonometric basis for initial value problems with oscillating solutions. Numer Algor 63, 713–725 (2013). https://doi.org/10.1007/s11075-012-9649-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-012-9649-8