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Spectral deferred correction methods for ordinary differential equations. (English) Zbl 0959.65084

From the authors’ abstract: We introduce a new class of methods for the Cauchy problem for Ordinary Differential Equations (ODEs). We begin by converting the original ODE into the corresponding Picard iteration equation and apply a deferred correction procedure in the integral formulation, driven by either the explicit or the implicit Euler marching scheme. The approach results in algorithms of essentially arbitrary order accuracy for both non-stiff and stiff algorithms of essentially arbitrary order accuracy for both non-stiff and stiff problems; their performance is illustrated with several numerical examples.
For non-stiff problems, the stability behavior of the obtained explicit schemes is very satisfactory and algorithms with order between 8 and 20 should be competitive with the best existing ones. In our preliminary experiments with stiff problems, a simple adaptive implementation of the method demonstrates performance comparable to that of a state-of-the art extrapolation code (at least, at moderate to high precision).
Deferred correction methods based on the Picard equation appear to be promising candidates for further investigation.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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