Nonlinear stability of Runge–Kutta methods for neutral delay differential equations

This paper deals with stability properties of Runge–Kutta methods for the initial value problem in nonlinear neutral delay differential equationsy′(t)=f(t,y(t),y(t-τ),y′(t-τ)),t⩾0,y(t)=φ(t),t∈[-τ,0].The new concepts of GS(l)GS(l)-stability, GAS(l)GAS(l)-stability and Weak GAS(l)GAS(l)-stability are introduced, and it is shown that (k,l)(k,l)-algebraically stable Runge–Kutta methods with piecewise linear interpolation are GS(l)GS(l)-, GAS(l)GAS(l)- and Weakly GAS(l)GAS(l)-stable. Two numerical examples are given in the end of this paper which confirm our results.