Comments on High Order Integrators Embedded within Integral Deferred Correction Methods

Spectral deferred correction (SDC) methods for solving ordinary differential equations (ODEs) were introduced by Dutt, Greengard and Rokhlin, [3]. In this paper, we study the properties of these integral deferred correction methods, constructed using high order integrators in the prediction and correction loops, and various distributions of quadrature nodes. The smoothness of the error vector associated with a method, is a key indicator of the order of convergence that can be expected from a scheme, [1, 7, 19]. We will demonstrate that using an rth order method in the correction loop, doesn’t always result in an r-order increase in accuracy. Examples include integral deferred correction methods constructed using non self-starting multi-step integrators, and methods constructed using a non-uniform 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. In brief, known RK methods are more efficient than integral correction methods for low order (< 6th order ) schemes. For high order schemes (≥ 8th order), comparable efficiency is observed numerically. However, it should be noted that integral deferred correction methods offer a much larger region of absolute stability, which may be beneficial in many problems. Additionally, as the order of the embedded integrator is increased, the stability region of these integral deferred correction method increases.