Abstract
We present an explicit Runge-Kutta scheme devised for the numerical solution ofdelay differential equations (DDEs) where a delayed argument lies in the current Runge-Kutta interval. This can occur when the lag is small relative to the stepsize, and the more obvious extensions of the explicit Runge-Kutta method produce implicit equations. It transpires that the scheme is suitable forparallel implementation for solving both ODEs and more general DDEs. We associate our method with a Runge-Kutta tableau, from which the order of the method can be determined. Stability will affect the usefulness of the scheme and we derive the stability equations of the scheme when applied to the constant-coefficient test DDEu′(t)=λu(t) +μu(t −τ), where the lagτ and the Runge-Kutta stepsizeH n ≡H are both constant. (The caseμ=0 is treated separately.) In the case thatμ ≠ 0, we consider the two distinct possibilities: (i)τ ≥H and (ii)τ<H.
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In memory of Professor Leslie Fox, Balliol College, Oxford
Work performed in part at The University of Auckland, New Zealand.
This paper is presented as an outcome of the LMS Durham Symposium convened by Professor C.T.H. Baker on 4th–14th July 1992 with support from the SERC under Grant reference number GR/H03964.
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Baker, C.T.H., Paul, C.A.H. Parallel continuous Runge-Kutta methods and vanishing lag delay differential equations. Adv Comput Math 1, 367–394 (1993). https://doi.org/10.1007/BF02072017
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DOI: https://doi.org/10.1007/BF02072017
Keywords
- Delay differential equation
- parallel continuous explicit Runge-Kutta methods
- vanishing lag
- iterated continuous extensions