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

Authors: Andrew Christlieb, Benjamin Ong and Jing-Mei Qiu
Journal: Math. Comp. 79 (2010), 761-783
MSC (2000): Primary 65B05
Published electronically: September 21, 2009
MathSciNet review: 2600542
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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.

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Additional Information

Andrew Christlieb
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

Benjamin Ong
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

Jing-Mei Qiu
Affiliation: Department of Mathematics and Computer Sciences, Colorado School of Mines, Golden, Colorado 80401

Keywords: Integral deferred correction methods, Runge-Kutta methods, discrete Sobolev norm, local error, accuracy, stability
Received by editor(s): April 3, 2008
Received by editor(s) in revised form: October 12, 2008, and March 21, 2008
Published electronically: September 21, 2009
Additional Notes: Research supported by Air Force Office of Scientific Research and Air Force Research Labs (Edwards and Kirtland). Grant Numbers FA9550-07-1-0092 and FA9550-07-1-0144.
Article copyright: © Copyright 2009 American Mathematical Society

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