Considerations concerning a theory for ode-solvers

Abstract

In today’s general purpose software packages for initial value problems in ODE’s, the course of the computation is normally determined by a tolerance parameter δ: The particular integration procedure and the stepsize to be used in the next step are derived from δ and the local behaviour of the ODE. This control mechanism should imply (for a sufficiently wellbehaved ODE) that the global error ε satisfies ε(t) = v(t)δ + o(δ) where v depends on the problem and the package but not on δ. (Naturally round-off is not considered as δ → 0.) To achieve this "tolerance-convergence", the control procedure has to guarantee that in each step we obtain the exact solution to a δ-perturbed ODE; this might be called "tolerance-consistency". Furthermore, no situation must arise in which the steplengths decrease in a geometric progression. Test computations have established the proportionality requested by tolerance-convergence in a satisfactory manner for the Shampine-Gordon package.