Theoretically optimal inexact spectral deferred correction methods

Abstract

In several initial value problems with particularly expensive right-hand side evaluation or implicit step computation, there is a tradeoff between accuracy and computational effort. We consider inexact spectral deferred correction (SDC) methods for solving such initial value problems. SDC methods are interpreted as fixed-point iterations and, due to their corrective iterative nature, allow one to exploit the accuracy-work tradeoff for a reduction of the total computational effort. First we derive error models bounding the total error in terms of the evaluation errors. Then we define work models describing the computational effort in terms of the evaluation accuracy. Combining both, a theoretically optimal local tolerance selection is worked out by minimizing the total work subject to achieving the requested tolerance. The properties of optimal local tolerances and the predicted efficiency gain compared to simpler heuristics, and reasonable practical performance, are illustrated with simple numerical examples.