Stability And Dissipativity Of Runge-kutta Methods For A Class Of Neutral Integro-differential Equations

In this paper,we consider the numerical stability of the Runge-Kutta method initial value problems(IVPs)of neutral integral-differential equations(NIDES).(?)where ?1,?2>0,?=max{?1,?2},f:[t0,T]× Cd × Cd?Cd,g:D × Cd?Cd,?:[t0-?,t0]?Cd are continuous mappings,and D={(t,?):t?[t0,T],??[t-?,t]}.Under two different conditions,the numerical stability of the Runge-Kutta method is given.The first kind condition Re<(u1-u2)-(w1-w2),f(t,u1,v1)-f(t,u2,v2)>???u1-u2?2+??v1-v2?2+??w1-w2?2,(?)t ?[t0,T],u1,u2,v1,v2,w1,w2 ? Cd,?g(t,?,u1)-g(t,?,u2)????u1-u2?,(?)t ?[t0,T],(t,?)?D,u1,u2?Cd.Sufficent conditions for stability and asymptotic stability of algebraically stable Runge-Kutta method are given.The second condition Re<u-w,f(t,u,v)>??+??u?2+??v?2+??w?2,(?)t ?[t0,+?),u,v,w ? Cd.?g(t,?,u)????u?,(?)t?[t0,+?),(t,?)?D,u?Cd.Sufficient conditions for dissipativity of the IVPs itself and the algebraically stable RungeKutta method are given.Finally,numerical experiments are carried out for the initial value problems under two kinds of conditions,and the theoretical results are verified.