Stability analysis of $\theta$ -methods for neutral functional-differential equations

This paper deals with the subject of numerical stability for the neutral functional-differential equation $$ y'(t)=ay(t)+by(qt)+cy'(pt), \qquad t>0. $$ It is proved that numerical solutions generated by $\theta$ -methods are convergent if $|c|<1$ . However, our numerical experiment suggests that they are divergent when $|c|$ is large. In order to obtain convergent numerical solutions when $|c|\geq 1$ , we use $\theta$ -methods 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 consideration has unbounded time lags, it is in general difficult to investigate numerically the long time dynamical behaviour of the exact solution due to limited computer (random access) memory. To avoid this problem we transform the equation under consideration into a neutral equation with constant time lags. Using the later equation as a test model, 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 ${\rm Re} a<0$ and $|a|>|b|$ , if and only if $\theta\geq 1/2$ , 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 ${\rm Re} a<0$ and $|a|=|b|$ .