Stability analysis of two-step Runge-Kutta methods for delay differential equations

We investigate stability properties of two-step Runge-Kutta methods with respect to the linear test equation y′(t) = ay(t) + by(t − τ), t ≥ 0, y(t) = g(t), t ε [−τ,0], where a and b are complex parameters. It is known that the solution y(t) to this equation tends to zero as t → ∞ if b < - Re(a). We will show that under some conditions this property is inherited by any A-stable two-step Runge-Kutta method applied on a constrained mesh to delay differential equations, i.e., that the corresponding method is P-stable.