A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. (English) Zbl 0565.65041

Authors’ summary: We consider a one-parameter family of Noumerov-type methods for the integration of second order periodic initial-value problems: \(y''=f(t,y)\), \(y(t_ 0)=y_ 0\), \(y'(t_ 0)=y'\!_ 0\). By applying these methods to the test equation: \(y''+\lambda^ 2y=0\), we determine the parameter of the family so that the phase-lag (frequency distortion) for the methods is minimal. The resulting method has a very small phase-lag of size \((1/12096)\lambda^ 6h^ 6\) (h is the step- size); interestingly, this method also possesses an interval of periodicity of size 2.71. Noumerov’s method has a phase-lag of size \((1/480)\lambda^ 4h^ 4\) and an interval of periodicity of size 2.449. The superiority of our method over Noumerov’s method is illustrated by two examples.
Reviewer: Z.Jackiewicz


65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34C25 Periodic solutions to ordinary differential equations
Full Text: DOI


[1] Brusa, L.; Nigro, L., A one-step method for direct integration of structural dynamic equations, Internat. J. Number. Methods Engrg., 15, 685-699 (1980) · Zbl 0426.65034
[2] Chawla, M. M.; Sharma, S. R., Intervals of periodicity and absolute stability of explicit Nyström methods, BIT, 21, 455-464 (1981) · Zbl 0469.65048
[3] Chawla, M. M., Unconditionally stable Noumerov-type methods for second order differential equations, BIT, 23, 541-542 (1983) · Zbl 0523.65055
[4] Fried, I., Numerical Solution of Differential Equations (1978), Academic Press: Academic Press New York
[5] Gladwell, I.; Thomas, R. M., Damping and phase analysis for some methods for solving second-order ordinary differential equations, Internat. J. Numer. Methods Engrg., 19, 493-503 (1983) · Zbl 0513.65053
[6] Hairer, E., Unconditionally stable methods for second order differential equations, Numer. Math., 32, 373-379 (1979) · Zbl 0393.65035
[7] Lambert, J. D.; Watson, I. A., Symmetric multistep methods for periodic initial-value problems, J. Inst. Math. Appl., 18, 189-202 (1976) · Zbl 0359.65060
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.