Stability of Runge-Kutta methods for delay integro-differential equations (Q1612419)

The author studies the stability of Runge-Kutta (RK) methods for delay integro-differential equations with a constant delay \(\tau>0\) on the basis of the linear test equation \[ \frac{du}{dt}=Lu(t)+Mu(t-\tau)+K\int_{t-\tau}^tu(\theta) d\theta \] where \(L\), \(M\), \(K\) are constant complex matrices. It is shown that every \(A\)-stable RK method preserves the delay-independent stability of the exact solution whenever a step-size of the form \(h = \tau/m\) is used, where \(m\) is a positive integer.