Integral deferred correction methods constructed with high order Runge–Kutta integrators
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- by Andrew Christlieb, Benjamin Ong and Jing-Mei Qiu PDF
- Math. Comp. 79 (2010), 761-783 Request permission
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
- W. Auzinger, H. Hofstätter, W. Kreuzer, and E. Weinmüller, Modified defect correction algorithms for ODEs. I. General theory, Numer. Algorithms 36 (2004), no. 2, 135–155. MR 2062870, DOI 10.1023/B:NUMA.0000033129.73715.7f
- 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, DOI 10.1016/S0021-9991(03)00251-1
- Andrew Christlieb, Benjamin Ong, and Jing-Mei Qiu, A comment on high order integrators embedded within integral deferred correction methods, in preparation.
- Alok Dutt, Leslie Greengard, and Vladimir Rokhlin, Spectral deferred correction methods for ordinary differential equations, BIT 40 (2000), no. 2, 241–266. MR 1765736, DOI 10.1023/A:1022338906936
- 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. MR 2299441, DOI 10.2140/camcos.2006.1.169
- A. Hansen and J. Strain, Convergence Theory for Spectral Deferred Correction, preprint, University of California at Berkeley, February (2005).
- A. C. Hansen and J. Strain, On the order of deferred correction, http://www.damtp.cam.ac.uk/user/na/people/Anders/Deferred.pdf.
- 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, DOI 10.1016/j.jcp.2005.10.004
- Jingfang Huang, Jun Jia, and Michael Minion, Arbitrary order Krylov deferred correction methods for differential algebraic equations, J. Comput. Phys. 221 (2007), no. 2, 739–760. MR 2293148, DOI 10.1016/j.jcp.2006.06.040
- 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, DOI 10.1016/j.jcp.2003.09.010
- Anita T. Layton and Michael L. Minion, 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, DOI 10.1007/s10543-005-0016-1
- Anita T. Layton, On the choice of correctors for semi-implicit Picard deferred correction methods, Appl. Numer. Math. 58 (2008), no. 6, 845–858. MR 2420621, DOI 10.1016/j.apnum.2007.03.003
- Anita T. Layton and Michael L. Minion, Implications of the choice of predictors for semi-implicit Picard integral deferred correction methods, Commun. Appl. Math. Comput. Sci. 2 (2007), 1–34. MR 2327081, DOI 10.2140/camcos.2007.2.1
- Yuan Liu, Chi-Wang Shu, and Mengping Zhang, Strong stability preserving property of the deferred correction time discretization, J. Comput. Math. 26 (2008), no. 5, 633–656. MR 2444722
- Michael L. Minion, Semi-implicit spectral deferred correction methods for ordinary differential equations, Commun. Math. Sci. 1 (2003), no. 3, 471–500. MR 2069941
- Michael L. Minion, 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, DOI 10.1016/j.apnum.2003.11.005
- 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, DOI 10.1137/0719009
- 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. MR 2328730, DOI 10.3934/dcdsb.2007.8.677
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
- Email: jingqiu@mines.edu
- 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.
- © Copyright 2009 American Mathematical Society
- Journal: Math. Comp. 79 (2010), 761-783
- MSC (2000): Primary 65B05
- DOI: https://doi.org/10.1090/S0025-5718-09-02276-5
- MathSciNet review: 2600542