Stability of continuous Runge–Kutta-type methods for nonlinear neutral delay-differential equations

Abstract

This paper provides results on the correct simulation, when using continuous Runge–Kutta methods, of certain stability properties of nonlinear neutral delay-differential equations (NDDEs) y ′ ( t ) = f ( t , y ( t ) , y ( t - τ ( t ) ) , y ′ ( t - τ ( t ) ) ) ( t ⩾ t 0 ) . In particular, it is shown that certain continuous Runge–Kutta methods based upon the backward Euler method or the 2-stage Lobatto IIIC method, combined with linear interpolation, are GRN -stable and asymptotically stable for NDDEs.