Spectral Deferred Corrections in the framework of Runge-Kutta methods
This work provides incremental theoretical insights into numerical methods for solving differential equations, primarily benefiting researchers in computational mathematics.
The paper tackles the interpretation of Spectral Deferred Corrections (SDC) methods as Runge-Kutta methods, proving that these SDC methods achieve at least order p after p iterations and can exhibit order jumps of two per iteration under specific conditions, with numerical experiments confirming the analysis.
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.