New stability results for Runge–Kutta methods adapted to delay differential equations Original Research Article

By using the Kreiss resolvent condition we establish upper bounds for the growth of errors in Runge–Kutta schemes for the numerical solution of delay differential equations. We consider application of such a scheme to the well-known linear test equation Z′(t)=λZ(t)+μZ(t−τ), and prove that at any given point in the so-called stability region of the scheme errors grow at most linearly with the number of timesteps and with the dimension involved. Moreover, we present a sufficient condition on the Runge–Kutta method for which this kind of error growth is valid uniformly within the stability region. For the important case where the Runge–Kutta method is A-stable, we show that this condition is also necessary.