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
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, \textit{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 \textit{P. J. van der Houwen} and \textit{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].
scientific article; zbMATH DE number 64334
scientific article