×

Dissipative trigonometrically-fitted methods for linear second-order IVPs with oscillating solution. (English) Zbl 1062.65075

Summary: A dissipative trigonometrically-fitted two-step explicit hybrid method is constructed. This method is based on a dissipative explicit two-step method developed recently by C. Tsitouras [Comput. Math. Appl. 43, No. 8–9, 943–949 (2002; Zbl 1050.65071)]. Numerical examples show that the procedure of trigonometrical fitting is an efficient way for one to produce numerical methods for the solution of second-order linear initial value problems (IVPs) with oscillating solutions.

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems

Citations:

Zbl 1050.65071
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Tsitouras, Ch, Explicit two-step methods for second-order linear IVPs, Computers Math. Applic., 43, 8/9, 943-949 (2002) · Zbl 1050.65071
[2] (Prigogine, I.; Rice, S., Advances in Chemical Physics, Volume 93: New Methods in Computational Quantum Mechanics (1997), John Wiley & Sons)
[3] Simos, T. E., Atomic structure computations, (Hinchliffe, A., Chemical Modelling: Applications and Theory, Volume 1 (2000), The Royal Society of Chemistry), 38-142, UMIST
[4] Simos, T. E., Numerical methods for the solution of 1D, 2D and 3D differential equations arising in chemical problems, (Hinchliffe, A., Chemical Modelling: Applications and Theory, Volume 2 (2002), The Royal Society of Chemistry), 170-270, UMIST
[5] Simos, T. E.; Williams, P. S., On finite difference methods for the solution of the Schrödinger equation, Computers & Chemistry, 23, 513-554 (1999) · Zbl 0940.65082
[6] Simos, T. E., Numerical solution of ordinary differential equations with periodical solution, (Doctoral Dissertation (1990), National Technical University of Athens: National Technical University of Athens Greece), (in Greek) · Zbl 0915.65084
[7] Ixaru, L. Gr, Numerical Methods for Differential Equations and Applications (1984), Reidel: Reidel Dordrecht · Zbl 0301.34010
[8] Lyche, T., Chebyshevian multistep methods for ordinary differential equations, Numerische Mathematik, 10, 65-75 (1972) · Zbl 0221.65123
[9] Raptis, A. D.; Allison, A. C., Exponential-fitting methods for the numerical solution of the Schrödinger equation, Computer Physics Communications, 14, 1-5 (1978)
[10] Franco, J. M.; Palacios, M., High-order \(P\)-stable multistep methods, J. Comput. Appl. Math., 30, 1-10 (1990) · Zbl 0726.65091
[11] Stiefel, E.; Bettis, D. G., Stabilization of Cowell’s method, Numerische Mathematik, 13, 154-175 (1969) · Zbl 0219.65062
[12] Chawla, M. M., Numerov made explicit has better stability, BIT, 24, 117-118 (1984) · Zbl 0568.65042
[13] Chawla, M. M.; Rao, P. S., An explicit sixth-order method with phase-lag of order eight for \(y\)″ = \(f(t,y)\), Journal of Computational and Applied Mathematics, 17, 365-368 (1987) · Zbl 0614.65084
[14] Dormand, J. R.; El-Mikkawy, M. E.A; Prince, P. J., High-order embedded Runge-Kutta-Nyström formulae, IMA Journal of Numerical Analysis, 7, 423-430 (1987) · Zbl 0627.65085
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.