Stability of solutions of nonlinear neutral differential equations with piecewise constant delay and their discretizations

This paper is devoted to stability analysis of solutions of nonlinear neutral differential equations with piecewise constant delay. These results form the basis for obtaining insight into the analogous properties of numerical solutions generated by BNf- stable Runge-Kutta methods. It is showed, theoretically and numerically, that two classes of Runge-Kutta methods considered in literature are contractive and asymptotically stable. Numerical experiments also demonstrate that a class of Runge-Kutta methods can preserve their original convergence order for ODEs but not the other class of Runge-Kutta methods.