Construction of high-order Runge–Kutta methods which preserve delay-dependent stability of DDEs

Abstract

This paper is concerned with the construction of some high-order Runge–Kutta methods, which preserve delay-dependent stability of delay differential equations. The methods of the first kind are developed by extending the ideas of Brugnano et al., while the methods of the second kind are developed according to the structure of the stability matrix. We show that the derived methods are τ(0)-stable for delay differential equations. Meanwhile, the Runge–Kutta methods can own the same order of the accuracy as the Radau methods or Gauss methods if the parameters are adequately defined. These results not only improve the order of accuracy of the methods investigated by Huang, but also open an interesting route of finding new τ(0)-stable Runge–Kutta methods. Finally, numerical experiments are proposed to illustrate the theoretical results.