Comments on high-order integrators embedded within integral deferred correction methods

Abstract

A class of novel deferred correction methods, integral deferred correction (IDC) methods, is studied. This class of methods is an extension of ideas introduced by Dutt, Greengard and Rokhlin on spectral deferred correction (SDC) methods for solving ordinary differential equations (ODEs). The novel nature of this class of defect correction methods is that the correction of the defect is carried out using an accurate integral form of the residual instead of the more familiar differential form. As a family of methods, these schemes are capable of matching the efficiency of popular high-order RK methods.

The smoothness of the error vector associated with an IDC method is an important indicator of the order of convergence that can be expected from a scheme (Christlieb, Ong, and Qiu; Hansen and Strain; Skeel). It is demonstrated that embedding an $r$-th order integrator in the correction loop of an IDC method does not always result in an $r$-th order increase in accuracy. Examples include IDC methods constructed using non-self-starting multistep integrators, and IDC methods constructed using a nonuniform distribution of quadrature nodes.

Additionally, the integral deferred correction concept is reposed as a framework to generate high-order Runge–Kutta (RK) methods; specifically, we explain how the prediction and correction loops can be incorporated as stages of a high-order RK method. This alternate point of view allows us to utilize standard methods for quantifying the performance (efficiency, accuracy and stability) of integral deferred correction schemes. It is found that IDC schemes constructed using uniformly distributed nodes and high-order integrators are competitive in terms of efficiency with IDC schemes constructed using Gauss–Lobatto nodes and forward Euler integrators. With respect to regions of absolute stability, however, IDC methods constructed with uniformly distributed nodes and high-order integrators are far superior. It is observed that as the order of the embedded integrator increases, the stability region of the IDC method increases.