STABILITY ANALYSIS OF THETA-METHODS FOR NEUTRAL FUNCTIONAL-DIFFERENTIAL EQUATIONS

This paper deals with the subject of numerical stability for the neutr al functional-differential equation Y'(t)=ay(t)+by(qt)+cy'(pt), t>0. I t is proved that numerical solutions generated by theta-methods are co nvergent if \c\ < 1 However, our numerical experiment suggests that th ey are divergent when \c\ is large. In order to obtain convergent nume rical solutions when \c\ greater than or equal to 1, we use theta-meth ods to obtain approximants to some high order derivative of the exact solution, then we use the Taylor expansion with integral remainder to obtain approximants to the exact solution. Since the equation under co nsideration has unbounded time lags, it is in general difficult to inv estigate numerically the long time dynamical behaviour of the exact so lution due to limited computer (random access) memory. To avoid this p roblem we transform the equation under consideration into a neutral eq uation with constant time lags. Using the later equation as a test mod el, we prove that the linear theta-method is Lambda-stable, i.e., the numerical solution tends to zero for any constant stepsize as long as Re a < 0 and \a\ > \b\, if and only if theta greater than or equal to 1/2, and that the one-leg theta-method is lambda-stable if theta=1. We also find out that inappropriate stepsize causes spurious solution in the marginal case where Rea < 0 and \a\=\b\.