Damping and phase analysis for some methods for solving second‐order ordinary differential equations

Abstract

We consider numerical methods for initial value problems for second-order systems of ordinary differential equations, analysing them by applying them to the test equation x + c2x = 0. We discuss conditions which ensure an oscillatory numerical solution and the desirability of such a property. We also use a

Explicit Runge–Kutta (–Nyström) Methods with Reduced Phase Errors for Computing Oscillating Solutions

“…In the case of first-order equations (k = 1), a complete phase analysis has been carried out by Brusa and Nigro [2] for a special third-order implicit one-step method, and for second-order equations (k == 2), a phase-lag analysis may be found in Gladwell and Thomas [8] for linear multistep methods and in Thomas [20] for certain hybrid families related to the multistep Runge-Kutta methods of Cash [3] and Chawla [4]. The papers mentioned above treat both the homogeneous and inhomogeneous components in the phase error.…”

Diagonally Implicit Runge–Kutta–Nyström Methods for Oscillatory Problems

“…A few of them deal with first-order differential equations [l], [9], and [12], but the majority are devoted to the second-order case. In Gladwell and Thomas [7] linear multistep (LM) methods are considered. However, it is well known [14] that the order of such methods is restricted to two, if ?-stability (see § 3.1) is required.…”

“…is frequently used to model forced oscillations (see [7], [ 11 ], and [ 17] ). Application of the RKN method (2.1) yields the inhomogeneous recursion…”

𝑃-stable Obrechkoff methods with minimal phase-lag for periodic initial value problems

“…Gladwell and Thomas [8] have analyzed the conditions which ensure an oscillatory numerical solution. They noted that the symmetric linear multistep methods proposed by Lambert and Watson [11] have no algorithmic damping, bu* do not perform well for problems with high frequency natural nodes.…”

“…Following Gladwell and Thomas [8], Chawla and Rao [5] and Ananthakrishnaiah [1] developed two-step methods with minimal phase-lag errors À6/i6/12096 and a6/j6/42000, respectively. The minimal phase-lag method of Chawla and Rao [5] is of 0(h4) with interval of periodicity (0,2.71) whereas the method of Ananthakrishnaiah [1] is of 0(h2) and is P-stable.…”