Reliable Error Control for ODE Methods

W.H. Enright
Department of Computer Science
University of Toronto
Toronto M5S 1A4

Traditional techniques for error control of ODE methods are based on monitoring some measure of the local error introduced on each step of the integration. In the last decade advances in the analysis and development of numerical methods for ODEs has made it possible to consider a new more natural approach to monitor and control the error. The most relevant such advance is the widespread use of inexpensive method-dependent local interpolants which have been developed to extend a standard 'discrete' method to a 'continuous' method (where the exact solution is approximated by a piecewise polynomial for all values of the independent variable in the range of interest). The 'quality' of the numerical solution can then be measured by determining how well this associated piecewise polynomial satisfies the ODE. This approach is called 'defect' control and its effectiveness depends on three factors: The quality of the underlying discrete method; The cost and accuracy of the local interpolating scheme; and the cost and accuracy of the technique used to estimate the size of the 'defect'. In this talk we will discuss trade-offs associated with these factors and the application of this overall approach to initial value problems, boundary value problems, algebraic differential problems and delay differential problems. For each of these cases we will summarise implementation issues and present numerical results for codes we have implemented.

( slides )