On the Newton iteration in the application of collocation methods to implicit delay equations

Authors:
Nicola Guglielmi
Affiliations:
Dipartimento di Matematica Pura e Applicata, Università dell'Aquila, L'Aquila, Italy
Venue:
Applied Numerical Mathematics - Tenth seminar on and differential-algebraic equations (NUMDIFF-10)
Year:
2005

Citing 4
Cited 0

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Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Stiff differential equations solved by Radau methods

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Implementing Radau IIA methods for stiff delay differential equations

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Differential algebraic equations with after-effect

Journal of Computational and Applied Mathematics - Special issue: Proceedings of the 9th International Congress on computational and applied mathematics

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Abstract

We consider stiffly accurate collocation methods based on Radau nodes for the integration of initial value problems for implicit delay differential equations of the form My'(t) = f(t, y(t), y(α1(t, y(t))),...,y(αp(t, y(t)))), where M is a constant matrix and αi(t, y(t)) (i=1,....,p) denote the deviating arguments, which are assumed to satisfy the inequalities αi (t, y(t))≤t for all i. In a recent paper [Computing 67 (2001) 1-12] we have described how collocation methods based on Radau nodes can be applied to solve problems of this type.The aim of this paper is that of explaining the difficulties arising when solving the Runge-Kutta equations using stepsizes larger than delays and to design techniques able to efficiently overcome them.