The reliability/cost trade-off for a class of ODE solvers

In the numerical solution of ODEs, it is now possible to develop efficient techniques that will deliver approximate solutions that are piecewise polynomials. The resulting methods can be designed so that the piecewise polynomial will satisfy a perturbed ODE with an associated defect (or residual) that is directly controlled in a consistent fashion. We will investigate the reliability/cost trade off that one faces when implementing and using such methods, when the methods are based on an underlying discrete Runge-Kutta formula. In particular we will identify a new class of continuous Runge-Kutta methods with a very reliable defect estimator and a validity check that reflects the credibility of the estimate. We will introduce different measures of the “reliability” of an approximate solution that are based on the accuracy of the approximate solution; the maximum magnitude of the defect of the approximate solution; and how well the method is able to estimate the maximum magnitude of the defect of the approximate solution. We will also consider how methods can be implemented to detect and cope with special difficulties such as the effect of round-off error (on a single step) or the ability of a method to estimate the magnitude of the defect when the stepsize is large (as might happen when using a high-order method at relaxed accuracy requests). Numerical results on a wide selection of problems will be summarized for methods of orders five, six and eight. It will be shown that a modest increase in the cost per step can lead to a significant improvement in the quality of the approximate solutions and the reliability of the method. For example, the numerical results demonstrate that, if one is willing to increase the cost per step by 50%, then a method can deliver approximate solutions where the reported estimated maximum defect is within 1% of its true value on 95% of the steps.