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 equations y ′ ( 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 ) -stability, GAS ( l ) -stability and Weak GAS ( l ) -stability are introduced, and it is shown that ( k , l ) -algebraically stable Runge–Kutta methods with piecewise linear interpolation are GS ( l ) -, GAS ( l ) - and Weakly GAS ( l ) -stable. Two numerical examples are given in the end of this paper which confirm our results.