Deferred Correction Methods for Ordinary Differential Equations

Deferred correction is a well-established method for incrementally increasing the order of accuracy of a numerical solution to a set of ordinary differential equations. Because implementations of deferred corrections can be pipelined, multi-core computing has increased the importance of deferred correction methods in practice, especially in the context of solving initial-value problems. In this paper, we review the theoretical underpinnings of deferred correction methods in a unified manner, specifically the classical algorithm of Zadunaisky/Stetter, the method of Dutt, Greengard and Rokhlin, spectral deferred correction, and integral deferred correction. We highlight some nuances of their implementations, including the choice of quadrature nodes, interpolants, and combinations of discretization methods, in a unified notation. We analyze how time-integration methods based on deferred correction can be effective solvers on modern computer architectures and demonstrate their performance. Lightweight and flexible Matlab software is provided for exploration with modern variants of deferred correction methods.

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