Abstract
A Class of Continuous Extended Backward Differentiation Formula using Trigonometric Basis functions (CEBDTB) with one superfuture point is constructed and used to generate a family of Trigonometrically Fitted Extended Backward Differentiation Formula (TFEBDF) and other discrete methods as by-products. This family of discrete schemes together with the TFEBDF are simultaneously applied as numerical integrators by assembling them into a Block Trigonometrically Fitted Extended Backward Differentiation Method (BTFEBDM). The error analysis, stability properties, and implementation of the BTFEBDM are discussed, and the class of methods is shown to be suitable for oscillatory and/or stiff Initial Value Problems (IVPs). The performance of the method is demonstrated on some numerical examples to show its accuracy and computational efficiency.
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References
Cash, J.R.: On the integration of stiff systems of O.D.E.s using extended backward differentiation formulae. Numer. Math. 34, 235–246 (1980)
Jator, S.N., Swindle, S., French, R.: Trigonometrically fitted block Numerov type method for \(y^{\prime \prime }=f(x, y, y^{\prime })\). Numerical Algorithms 62, 13–26 (2013)
Nguyen, H.S., Sidje, R.B., Cong, N.H.: Analysis of trigonometric implicit Runge-Kutta methods. J. Comput. Appl. Math. 198, 187–207 (2007)
Dahlquist, G.G.: A special stability problem for linear multistep methods. BIT 3, 27–43 (1963)
Enright, W.H.: Second derivative multistep methods for stiff ordinary differential equations. SIAM J. Numer. Anal. 11, 321–331 (1974)
Fang, Y., Song, Y., Wu, X.: A robust trigonometrically fitted embedded pair for perturbed oscillators. J. Comput. Appl. Math. 225, 347–355 (2009)
Fatunla, S.O.: Block methods for second order IVPs. Intern. J. Comput. Math. 41, 55–63 (1991)
Franco, J.M.: Runge-kutta-nyström methods adapted to the numerical intergration of perturbed oscillators. Comput. Phys. Comm. 147, 770–787 (2002)
Gear, C.W.: Algorithm 407, Difsub for solution of ordinary differential equations. Comm. ACM 14, 185–190 (1971)
Vanden Berghe, G., Ixaru, L.G., van Daele, M.: Optimal implicit exponentially-fitted Runge-Kutta method. Comput. Phys. Commun. 140, 346–357 (2001)
Milne, W.E.: Numerical Solution of Differential Equations. Wiley, New York (1953)
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Springer, New York (1996)
Henrici, P.: Discrete Variable Methods in ODEs. Wiley, New York (1962)
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)
Ixaru, L., Berghe, G.V.: Exponential Fitting. Kluwer, Dordrecht, Netherlands (2004)
Ngwane, F.F., Jator, S.N.: Block hybrid method using trigonometric basis for initial value problems with oscillating solutions. Numerical Algorithms 63, 713–725 (2013)
Lambert, J.D.: Numerical Methods for Ordinary Differential Systems. Wiley, New York (1991)
Lambert, J.D.: Computational Methods in Ordinary Differential Equations. Wiley, New York (1973)
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. Comp. 23, 731–740 (1969)
Rosser, J.B.: A Runge-Kutta for all seasons. SIAM Rev. 9, 417–452 (1967)
Coleman, J., Ixaru, L.G.: P-stability and exponential-fitting methods for \(y^{\prime \prime } = f(x,y)\). IMA J. Numer. Anal. 16, 179–199 (1996)
Sommeijer, B.P.: Explicit high-order Runge-Kutta-Nyström methods for parallel computers. Appl. Numer. Math. 13, 221–240 (1993)
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Ndukum, P.L., Biala, T.A., Jator, S.N. et al. On a family of trigonometrically fitted extended backward differentiation formulas for stiff and oscillatory initial value problems. Numer Algor 74, 267–287 (2017). https://doi.org/10.1007/s11075-016-0148-1
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DOI: https://doi.org/10.1007/s11075-016-0148-1