Error control for initial value problems with discontinuities and delays

When using software for ordinary differential equation (ODE) initial value problems, it is not unreasonable to expect the global error to decrease linearly with the user-supplied error tolerance. For standard ODEs, conditions on an algorithm that guarantee such 'tolerance proportionality' asymptotically (as the error tolerance tends to zero) were derived by Stetter. Here we extend the analysis to cover a certain class of ODEs with low-order derivative discontinuities, and the class of ODEs with constant delays. We show that standard error control techniques will be successful if discontinuities are handled correctly and delay terms are calculated with sufficiently accurate interpolants. It is perhaps surprising that several delay ODE algorithms that have been proposed do not use sufficiently accurate interpolants to guarantee asymptotic proportionality. Our theoretical results are illustrated numerically.