\(\mathcal H\)-stability of Runge-Kutta methods with general variable stepsize for pantograph equation. (Q1421287)

The paper deals with the \({\mathcal H}\)-stability of the Runge-Kutta methods with variable stepsize applied to the pantograph equation \[ y^\prime (t) = \lambda y(t) + \mu y(qt), \qquad t > 0, \] where \(\lambda\), \(\mu\) are complex constants, \(q \in (0,1)\). The authors show that the Runge-Kutta methods with a regular matrix \(A\) are \({\mathcal H}\)-stable if and only if the modulus of the stability function at infinity is less than 1. It is shown also that the same conclusion is valid if the Runge-Kutta methods are stiffly accurate.