Spectral Deferred Corrections in the framework of Runge-Kutta methods

TL;DR

This paper interprets Spectral Deferred Corrections as Runge-Kutta methods, analyzing their order, stability, and invariant conservation, supported by theoretical proofs and numerical experiments.

Contribution

It provides a unified interpretation of SDC as RKMs, analyzes order jumps, stability, and invariant conservation, introducing new variants with improved properties.

Findings

SDC methods achieve at least order p after p iterations

Order jumps of two are possible with appropriate error discretisations

New SDC variants can conserve quadratic invariants

Abstract

We interpret a wide range of flavors of Spectral Deferred Corrections (SDC) as Runge-Kutta methods (RKM). Using Butcher series, we show that the considered class of SDC methods achieve at least order p after p iterations compared to the underlying RKM, independently of the error discretisation chosen and the choice of nodes. For all collocation RKM, we analyse the phenomenon of order jumps in SDC iterations, where the order is increased by two at each iteration. We prove that it can be obtained by using appropriate inconsistent, implicit, parallelisable error discretisations. We also investigate the stability properties of the new SDC methods which can in general reduce to that of explicit RKM, but it can be improved by suitable combinations of error discretisations. We confirm the convergence analysis with numerical experiments and we apply relaxation RKM to derive SDC variants that conserve quadratic invariants.