×

An explicit almost \(P\)-stable two-step method with phase-lag of order infinity for the numerical integration of second-order periodic initial- value problems. (English) Zbl 0765.65083

Special second order problems, \(y''=f(x,y)\), \(y(x_ 0)=y_ 0\), \(y'(x_ 0)=y_ 0'\), are of particular interest when either free or forced periodic solutions exist. For such problems, a variety of both explicit and implicit methods are known. When a method is applied to the test problem \(y''=-c^ 2y\), the phase lag measures the difference in phase between actual and approximate solutions, and the interval of periodicity identifies the steplengths for which the numerical solution is bounded and not decaying. A method is \(P\)-stable if its interval of periodicity is semi-infinite.
For a problem with a solution of known period, methods exist for which the phase lag is zero. The author develops an explicit method for which the arbitrary parameter may be selected to satisfy this property. On application to two problems selected for illustration, the new method is shown to be more accurate than two other known methods.
The comparison of the new method with other known methods is incomplete. For one of the selected problems, R. M. Thomas [BIT 24, 225-238 (1984; Zbl 0569.65052)] has shown more accurate results are obtained for several implicit methods. In spite of its lower accuracy, some advantage may accrue to the new method since it is explicit. In this respect, corresponding results for the same problems using the explicit methods developed by P. J. van der Houwen and B. P. Sommeijer [SIAM J. Numer. Anal. 24, 595-617 (1987; Zbl 0624.65058)] would have been illuminating. A distinction between the implementation of explicit and implicit methods for such problems would have added perspective to the paper. Finally, the almost \(P\)-stability property of the new method is different than the property identified by Thomas [loc. cit.].
[Reviewer’s remark: Because the proposed method depends substantially on an a priori evaluation of the period, it will be of limited interest. In the paper, a clearer distinction between \(\bar y_{n+1}\) and \(\bar y_{n-1}\) (and also their derivatives), and a correction to the L.T.E. in (2.5) (which is missing a term) are needed].

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brusa, L.; Nigro, L., A one-step method for direct integration of structural dynamic equations, Int. J. Numer. Methods Engrg., 15, 685-699 (1980) · Zbl 0426.65034
[2] Chawla, M. M.; Rao, P. S., A Noumerov-type method with minimal phase-lag for the numerical integration of second order periodic initial-value problem, J. Comput. Appl. Math., 11, 277-281 (1984) · Zbl 0565.65041
[3] Chawla, M. M.; Rao, P. S., Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problem, J. Comput. Appl. Math., 15, 329-337 (1986), Part II: Explicit method · Zbl 0598.65054
[4] Chawla, M. M.; Rao, P. S.; Neta, B., Two-step fourth-order \(P\)-stable methods with phase-lag of order six for \(y\)″\(^n=f(t,y)\), J. Comput. Appl. Math., 16, 233-236 (1986) · Zbl 0596.65047
[5] Chawla, M. M.; Rao, P. S., An explicit sixth-order method with phase-lag of order eight for \(y\)″\(^n=f(t,y)\), J. Comput. Appl. Math., 17, 365-368 (1987) · Zbl 0614.65084
[6] Coleman, J. P., Numerical methods for \(y^n=f(x,y)\) via rational approximations for the cosine, IMA J. Num. Anal., 9, 145-165 (1989) · Zbl 0675.65072
[7] P.J. van der Houwen and B.P. Sommeijer, Predictor-corrector methods for periodic second-order initial-value problems, IMA J. Num. Anal. 7:407-422.; P.J. van der Houwen and B.P. Sommeijer, Predictor-corrector methods for periodic second-order initial-value problems, IMA J. Num. Anal. 7:407-422. · Zbl 0631.65074
[8] Houwen, P. J.van der; Sommeijer, B. P., Explicity Runge-Kutta (-Nystrom) methods with reduced phase errors for computing oscillating solutions, SIAM J. Numer. Anal., 24, 595-617 (1987) · Zbl 0624.65058
[9] Raptis, A. D.; Simos, T. E., A four-step phase-fitted method for the numerical integration of second order initial-value problem, BIT, 31, 160-168 (1991) · Zbl 0726.65089
[10] Stiefel, E.; Bettis, D. G., Stabilization of Cowell’s method, Num. Math., 13, 154-175 (1969) · Zbl 0219.65062
[11] R.M. Thomas, Phase properties of high order almost \(P\); R.M. Thomas, Phase properties of high order almost \(P\) · Zbl 0569.65052
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.