Explicit Nystrom Methods Research Articles

Continuous extensions to Nyström methods for second order initial value problems

In this work we consider interpolants for Nystrom methods, i.e., methods for solving second order initial value problems. We give a short introduction to the theory behind the discrete methods, and extend some of the work to continuous, explicit Nystrom methods. Interpolants for continuous, explicit Runge-Kutta methods have been intensively studied by several authors, but there has not been much effort devoted to continuous Nystrom methods. We therefore extend some of the work by Owren.

Phase-lag analysis of explicit Nyström methods fory″=f(x,y)

We examine phase-lag (frequency distortion) of the two-parameter familyM4(α1, α3) of fourth order explicit Nystrom methods of [1] by applying these to the test equation:y″+λ2y=0, λ>0. While the methodM4(1/6, 5/6) possessing the largest interval of periodicity of size 3.46 has a phase-lag of (1/4320)H (H4=λh, h is the step-size), we show that there exist two fourth order methods ofM4(α1, α3) for which the phase-lag is minimal and of size (1/40320)H6; interestingly, both methods also possess a sizable interval of periodicity of length 2.75 each.

On the Order and Attainable Intervals of Periodicity of Explicit Nyström Methods for $y'' = f(t,y)$

We consider the class of m-stage explicit Nystrom methods for $y'' = f(t,y)$ which are at least order one. We first show that the length of the interval of periodicity (see our definition in § 2) scaled by dividing by m for any such method cannot exceed two. While there exist methods of order one and two possessing this optimal length of interval of periodicity; we show that if a method of order higher than two possessses an interval of periodicity, then the length of its scaled interval of periodicity is less than two.

Intervals of periodicity and absolute stability of explicit nyström methods fory″=f(x,y)

We examine intervals of periodicity and absolute stability of explicit Nystrom methods fory″=f(x,y) by applying these methods to the test equationy″=−λ2y,λ>0. We consider in detail general families of fourth-order explicit Nystrom methods; necessary and sufficient conditions are given to characterize methods which possess non-vanishing intervals of periodicity and absolute stability. We establish closed-form expressions giving intervals of periodicity and/or absolute stability, in case these exist, for any fourth-order method. We then show that the methodM4 (1/6, 5/6) has the largest interval of periodicity out of all fourth-order methods; we also obtain the fourth-order method with the largest interval of absolute stability. The corresponding results for second and third-order explicit Nystrom methods are also included.