Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. W. Auzinger, H. Hofstätter, W. Kreuzer, and E. Weinmüller, Modified defect correction algorithms for ODEs, part I: General theory, Numer. Algorithms 36 (2004), 135-155. MR 2062870 (2005h:65096)
  • 2. Anne Bourlioux, Anita T. Layton, and Michael L. Minion, High-order multi-implicit spectral deferred correction methods for problems of reactive flow, J. Comput. Phys. 189 (2003), no. 2, 651-675. MR 1996061 (2004f:76084)
  • 3. Andrew Christlieb, Benjamin Ong, and Jing-Mei Qiu, A comment on high order integrators embedded within integral deferred correction methods, in preparation.
  • 4. Alok Dutt, Leslie Greengard, and Vladimir Rokhlin, Spectral deferred correction methods for ordinary differential equations, BIT 40 (2000), no. 2, 241-266. MR 1765736 (2001e:65104)
  • 5. Thomas Hagstrom and Ruhai Zhou, On the spectral deferred correction of splitting methods for initial value problems, Commun. Appl. Math. Comput. Sci. 1 (2006), 169-205 (electronic). MR 2299441
  • 6. A. Hansen and J. Strain, Convergence Theory for Spectral Deferred Correction, preprint, University of California at Berkeley, February (2005).
  • 7. A. C. Hansen and J. Strain, On the order of deferred correction,
  • 8. Jingfang Huang, Jun Jia, and Michael Minion, Accelerating the convergence of spectral deferred correction methods, J. Comput. Phys. 214 (2006), no. 2, 633-656. MR 2216607 (2006k:65173)
  • 9. -, Arbitrary order Krylov deferred correction methods for differential algebraic equations, J. Comput. Phys. 221 (2007), no. 2, 739-760. MR 2293148 (2008a:65134)
  • 10. Anita T. Layton and Michael L. Minion, Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics, J. Comput. Phys. 194 (2004), no. 2, 697-715. MR 2034861 (2004k:76089)
  • 11. -, Implications of the choice of quadrature nodes for Picard integral deferred corrections methods for ordinary differential equations, BIT 45 (2005), no. 2, 341-373. MR 2176198 (2006h:65087)
  • 12. A.T. Layton, On the choice of correctors for semi-implicit Picard deferred correction methods, Applied Numerical Mathematics 58 (2008), no. 6, 845-858. MR 2420621
  • 13. A.T. Layton and M.L. Minion, Implications of the choice of predictors for semi-implicit Picard integral deferred corrections methods, Comm. Appl. Math. Comput. Sci. 1 (2007), no. 2, 1-34. MR 2327081 (2008e:65252)
  • 14. Yuan Liu and Chi-Wang Shu, Strong stability preserving property of the deferred correction time discretization, J. Comput. Math. 26 (2008), no. 5, 633-656. MR 2444722
  • 15. Michael L. Minion, Semi-implicit spectral deferred correction methods for ordinary differential equations, Commun. Math. Sci. 1 (2003), no. 3, 471-500. MR 2069941 (2005f:65085)
  • 16. -, Semi-implicit projection methods for incompressible flow based on spectral deferred corrections, Appl. Numer. Math. 48 (2004), no. 3-4, 369-387, Workshop on Innovative Time Integrators for PDEs. MR 2056924
  • 17. Robert D. Skeel, A theoretical framework for proving accuracy results for deferred corrections, SIAM J. Numer. Anal. 19 (1982), no. 1, 171-196. MR 646602 (83d:65184)
  • 18. Yinhua Xia, Yan Xu, and Chi-Wang Shu, Efficient time discretization for local discontinuous Galerkin methods, Discrete Contin. Dyn. Syst. Ser. B 8 (2007), no. 3, 677-693 (electronic). MR 2328730

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65B05

Retrieve articles in all journals with MSC (2000): 65B05

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

American Mathematical Society