Future Generation Computer Systems 19 (2003) 385–393 The global error of Magnus methods based on the Cayley map for some oscillatory problems F. Diele a,∗ , S. Ragni b a Istituto per le Applicazioni del Calcolo M. Picone, CNR, Via Amendola 122, 70126 Bari, Italy b Facoltà di Economia, Università di Bari, Via Camillo Rosalba 53, 70124 Bari, Italy Abstract This paper deals with numerical methods for the discretization of highly oscillatory systems. We approach the problem by writing the solution in terms of the Magnus expansion based on the Cayley map. The global error, obtained when the method is applied to the linear oscillator, is investigated. Moreover, we provide numerical experiments in order to validate our theoretical results. © 2002 Elsevier Science B.V. All rights reserved. Keywords: High oscillatory systems; Magnus methods; Cayley map; Global error 1. Introduction The basic idea underlying this paper is the inves- tigation of Magnus methods when applied to specific problems such as oscillatory ordinary differential equations. The use of these methods arises in the framework of the numerical treatment of linear and non-linear conservative differential equations. This approach is characterized by the assurance that the numerical solution of a Lie-group differential equa- tion evolves on the same group. Although these methods were built in order to repro- duce the qualitative behavior of conservative ODEs, here our interest consists in exploiting them for solv- ing the so-called linear oscillator y ′′ + g(t)y = 0,t ≥ 0, y(0) = y 0 ,y ′ (0) = y ′ 0 , (1) ∗ Corresponding author. Tel.: +39-80-5530719; fax: +39-80-5588235. E-mail addresses: irmafd03@area.ba.cnr.it (F. Diele), irmasr18@area.ba.cnr.it (S. Ragni). where y 2 0 + y ′2 0 > 0, g ∈ C 1 (R + ) is such that g(t) > 0 for t ≫ 1, all the derivatives g (l) (t) are com- paratively small, i.e. 0 ≤|g (l) (t)|/g(t) 1/l ≪ 1 and lim sup t →∞ g(t) = +∞. In matrix form, the previous problem can be written as y ′ (t) =  0 1 -g(t) 0  y(t), t ≥ 0, y(0) = y 0 (2) with y 0 = (y 0 ,y ′ 0 ) T . The solution of Eq. (1) (or equiv- alently (2)) is a highly oscillating function with asymp- totic behavior given by  y(t) y ′ (t)  ≈  g(t) -1/4 0 0 g(t) 1/4  G(t)S 0 ,t ≫ 1 (3) with G(t)=  cos θ(t) sin θ(t) - sin θ(t) cos θ(t)  , θ(t)=  t 0 g(x) 1/2 dx and S 0 a2 × 2 constant matrix (see [4]). More pre- cisely, we are interested in two specific problems 0167-739X/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII:S0167-739X(02)00165-6