Dissipativity of multistep runge-kutta methods for dynamical systems with delays

This paper is concerned with the numerical solution of dissipative initial value problems with delays by multistep Runge-Kutta methods. We investigate the dissipativity properties of (k, l)-algebraically stable multistep Runge-Kutta methods with constrained grid and linear interpolation procedure. In particular, it is proved that an algebraically stable, irreducible multistep Runge-Kutta method is dissipative for finite-dimensional dynamical systems with delays, which extends and unifies some extant results. In addition, we obtain dissipativity results of A-stable linear multistep methods by using the relationship between one-leg methods and linear multistep methods.