Integral deferred correction methods constructed with high order Runge–Kutta integrators

Abstract

Abstract. Spectral deferred correction (SDC) methods for solving ordinary differential equations (ODEs) were introduced by Dutt, Greengard and Rokhlin (2000). It was shown in that paper that SDC methods can achieve arbitrary high order accuracy and possess nice stability properties. Their SDC methods are constructed with low order integrators, such as forward Euler or backward Euler, and are able to handle stiff and non-stiff terms in the ODEs. In this paper, we use high order Runge-Kutta (RK) integrators to construct a family of related methods, which we refer to as integral deferred correction (IDC) methods. The distribution of quadrature nodes is assumed to be uniform, and the corresponding local error analysis is given. The smoothness of the error vector associated with an IDC method, measured by the discrete Sobolev norm, is a crucial tool in our analysis. The expected order of accuracy is demonstrated through several numerical examples. Superior numerical stability and accuracy regions are observed when high order RK integrators are used to construct IDC methods.

Semi-implicit integral deferred correction constructed with additive Runge–Kutta methods

“…A corollary of the theorem provides the truncation error results for IDC methods constructed using implicit RK integrators in the prediction and correction loops. The analysis of the truncation error closely follows the analysis in [9], and, since only the essential differences are presented in this work, it is recommended that the interested reader first reads [9] before reading Section 3. Numerical results in Section 5 support the conclusion of the theorem.…”

“…Krylov deferred correction, which uses SDC with an Euler integrator as a preconditioner for a Newton-Krylov method, was developed to handle differential algebraic equations [17,18]. In [9], a variant of SDC, integral deferred correction (IDC), constructed using uniform nodes and high order explicit RK integrators in both the prediction and corrections, referred to as IDC-RK, was introduced. In [9], it was established that using an explicit RK method of order r in the correction results in r more degrees of accuracy with each successive correction.…”

“…In [9], a variant of SDC, integral deferred correction (IDC), constructed using uniform nodes and high order explicit RK integrators in both the prediction and corrections, referred to as IDC-RK, was introduced. In [9], it was established that using an explicit RK method of order r in the correction results in r more degrees of accuracy with each successive correction. In [8] it was demonstrated that IDC-RK methods are more efficient than SDC methods (constructed with Euler integrators) of equivalent order.…”

“…In [8] it was demonstrated that IDC-RK methods are more efficient than SDC methods (constructed with Euler integrators) of equivalent order. IDC-RK methods were also found to possess better stability properties than SDC [9] and can be written as RK methods in Butcher table form [8]. Semi-implicit IDC with backward and forward Euler was developed as an IMEX method to handle IVPs of the form (1.1) [25], while in [22], semi-implicit IDC methods using second order base schemes (BDF and RK) in both the prediction and correction loops were implemented.…”

“…An IMEX form of Runge-Kutta methods known as ARK is explained in Section 2.1. Our formulation of semi-implicit IDC-ARK schemes, based on [25] and [9], can be found in Section 2.2. Section 3 contains the analysis of the truncation error.…”