Computing breaking points in implicit delay differential equations
- PDF / 472,469 Bytes
- 19 Pages / 439.37 x 666.142 pts Page_size
- 84 Downloads / 268 Views
Computing breaking points in implicit delay differential equations Nicola Guglielmi · Ernst Hairer
Received: 25 May 2006 / Accepted: 20 November 2006 / Published online: 28 April 2007 © Springer Science + Business Media B.V. 2007
Abstract Systems of implicit delay differential equations, including state-dependent problems, neutral and differential-algebraic equations, singularly perturbed problems, and small or vanishing delays are considered. The numerical integration of such problems is very sensitive to jump discontinuities in the solution or in its derivatives (so-called breaking points). In this article we discuss a new strategy – peculiar to implicit schemes – that allows codes to detect automatically and then to compute very accurately those breaking points which have to be inserted into the mesh to guarantee the required accuracy. In particular for state-dependent delays, where breaking points are not known in advance, this treatment leads to a significant improvement in accuracy. As a theoretical result we obtain a general convergence theorem which was missing in the literature (see Bellen and Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, 2003). Furthermore, as a useful by-product, we design strategies that are able to detect points of non-uniqueness or non-existence of the solution so that the code can terminate when such a situation occurs. A new version of the code RADAR5 together with drivers for some real-life problems is available on the homepages of the authors.
Communicated by A. Iserles. Supported by the Italian M.I.U.R. and G.N.C.S. Supported by the Swiss National Science Foundation, project # 200020-101647. N. Guglielmi (B) Dip. di Matematica Pura e Applicata, Università dell’Aquila, via Vetoio (Coppito), I-67010 L’Aquila, Italy e-mail: [email protected] E. Hairer Department de Mathématiques, Université de Genève, CH-1211 Genève 24, Switzerland e-mail: [email protected]
230
N. Guglielmi, E. Hairer
Keywords Implicit delay differential equations · Runge–Kutta methods · Radau IIA methods · Breaking points · Non-existence and non-uniqueness of solutions · Error control · Numerical well-posedness Mathematics Subject Classifications (2000) AMS(MOS): 65L20 · CR: G1.9
1 Introduction The time evolution of physical systems, whose rate of change also depends on the configuration at previous time instances, is often modelled in terms of delay differential equations. Their solution has a much more complicated dynamics than that for ordinary differential equations (lack of smoothness in the solution, possibility of non-uniqueness or non-existence), and it is rarely possible to find an analytic expression for the solution (see for example [1]). This motivates the need for reliable and efficient numerical integrators for such problems. For a comprehensive introduction to the numerical literature on this subject we refer the reader to the recent books [3, 5] and to the review-paper [2]. In this article we consider the application of stiffly accurate
Data Loading...
