SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Solving delay differential equations with diagonally implicit Runge-Kutta methods

A. Eremin and A.R. Humphries

Abstract
In case of so-called overlapping, when in delay differential equations a delay becomes smaller than the step-size, the standard implementation of continuous Runge-Kutta methods (CRKs) makes the methods fully implicit. Pioneering works by L. Tavernini and recent comprehensive papers by a group of Italian authors show how explicit methods can be applied explicitly even with overlapping. In search of better stability properties we adapt the same technique for Diagonally Implicit CRKs preserving their diagonal structure in the case of overlapping. Methods of different orders are constructed. Problems of practical implementation are also studied: error estimation, automatic step-size control, interpolant overshoot, combining methods for better efficiency etc.

Bibliography
[1] A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Great Clarendon Street, Oxford 0X2 6DP, 2003.
[2] S. Maset, L. Torelli and R. Vermiglio, Runge-Kutta Methods for Retarded Functional Differential Equations, M3AS, vol. 15, no. 8, 2005, pp. 1203-1251.
[3] A. Bellen, S. Maset, M. Zennaro and N. Guglielmi, Recent trends in the numerical solution of retarded functional differential equations, Acta Numer., 2009, pp. 1-110.

Organized by         Universidad de Valladolid     IMUVA