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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