Uniform ultimate boundedness of numerical solutions to nonlinear neutral delay differential equations

This paper is concerned with the long-time behaviour of the numerical solutions generated by Runge–Kutta (RK) methods for nonlinear neutral delay differential equations (NDDEs). It is proved that the numerical solutions produced by (k,lk,l)-algebraically stable RK methods are uniformly ultimately bounded. Some examples reveal that some RK methods completely preserve the long-time behaviour of the exact solutions to NDDEs for sufficiently small time stepsize hh. As a comparison with the previous results, a numerical example which further illustrates our theoretical results is provided.