Stability Analysis Of Continuous Runge-Kutta Methods For Nonlinear Stiff Delay Differential Equations Of Neutral Type

Stability of theoretical and numerical solutions for nonlinear stiff delay differential equations of neutral type (NDDEs)is discussed in this paper. Because the research of NDDEs is more difficult than that of delay differential equations(DDEs), so far only linear NDDEs(cf.[l-25]) and nonlinear NDDEs(cf.[25-29] with special form have been researched in literature at home and abroad, and in the present paper it is the first time to research NDDEs of the general form (1). Our main results are as follows:(1) Several sufficient conditions for the theoretical solution of problem (1) to be stable or asymptotically stable are obtained, respectively.(2) We introduce the concept of GLW-stability of continuous Runge-Kutta methods for the solutions of the problem (1), and prove that numerical solutions preserve the contractivity properties of the theoretical solution whenever GLW-stable method is used. Then we find that backward Euler method and 2-stage Lobatto IIIC Runge-Kutta method with linear interpolation both are GLW-stable, and therefore the numerical solutions obtained by these methods satisfy a contractivity inequality as well as the theoretical solution, respectively, (note that piecewise constant interpolation can also be used for backward Euler method).(3) It is proved that the numerical solutions obtained by backward Euler method and 2-stage Lobatto IIIC Runge-Kutta method with linear interpolation are also asymptotically stable, as well as the theoretical solution whenever the sufficient condition given in this paper for the asymptotical stability of the theoretical solution is satisfied.(4) Numerical experiments are given for checking the stability propertiesof linear 9 methods and 2-stage Lobatto IIIC Runge-Kutta method, which confirm the theoretical results obtained in this paper.The aforementioned results (2) and (3) can be regarded as extension of the relevant results for NDDEs with special form obtained by Bellen,Guglielmi and Zennaro[26] and Vermiglio and Torelli[29].