Title
Parallel Continuous Runge-Kutta Methods and Vanishing Lag Delay Differential Equations
Authors
Christopher T.H. Baker & Christopher A.H. Paul
Journal
Advances in Computational Mathematics
Volume
1
Year
1993
Pages
367-394
File size
1049Kb
Abstract
We present an explicit Runge-Kutta scheme devised for the numerical solution of delay 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 for parallel 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 DDE

y'(t) = l y(t) + u y(t-T),

where the lag T and the Runge-Kutta stepsize Hn = H are both constant. (The case u = 0 is treated separately.) In the case that u <> 0, we consider the two distinct possibilities: (i) T >= H and (ii) T < H.
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