Stability of Runge–Kutta methods for delay integro-differential equations

We study stability of Runge–Kutta (RK) methods for delay integro-differential equations with a constant delay on the basis of the linear equationdudt=Lu(t)+Mu(t−τ)+K∫t−τt u(θ) dθ,where L,M,K are constant complex matrices. In particular, we show that the same result as in the case K=0 (Koto, BIT 34 (1994) 262–267) holds for this test equation, i.e., every A-stable RK method preserves the delay-independent stability of the exact solution whenever a step-size of the form h=τ/m is used, where m is a positive integer.