Skip to main content
Springer Nature Link
Log in

Deferred Correction Methods for Ordinary Differential Equations

  • Review
  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Deferred correction is a well-established method for incrementally increasing the order of accuracy of a numerical solution to a set of ordinary differential equations. Because implementations of deferred corrections can be pipelined, multi-core computing has increased the importance of deferred correction methods in practice, especially in the context of solving initial-value problems. In this paper, we review the theoretical underpinnings of deferred correction methods in a unified manner, specifically the classical algorithm of Zadunaisky/Stetter, the method of Dutt, Greengard and Rokhlin, spectral deferred correction, and integral deferred correction. We highlight some nuances of their implementations, including the choice of quadrature nodes, interpolants, and combinations of discretization methods, in a unified notation. We analyze how time-integration methods based on deferred correction can be effective solvers on modern computer architectures and demonstrate their performance. Lightweight and flexible Matlab software is provided for exploration with modern variants of deferred correction methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. Aggul, M., Labovsky, A.: A high accuracy minimally invasive regularization technique for Navier–Stokes equations at high Reynolds number. Numer. Methods Partial Differ. Equ. 33(3), 814–839 (2017). https://doi.org/10.1002/num.22124

    Article  MathSciNet  MATH  Google Scholar 

  2. Auzinger, W., Hofstätter, H., Kreuzer, W., Weinmüller, E.: Modified defect correction algorithms for ODEs. I. General theory. Numer. Algorithms 36(2), 135–155 (2004). https://doi.org/10.1023/B:NUMA.0000033129.73715.7f

    Article  MathSciNet  MATH  Google Scholar 

  3. Auzinger, W., Hofstätter, H., Kreuzer, W., Weinmüller, E.: Modified defect correction algorithms for ODEs. II. Stiff initial value problems. Numer. Algorithms 40(3), 285–303 (2005). https://doi.org/10.1007/s11075-005-5327-4

    Article  MathSciNet  MATH  Google Scholar 

  4. Bolten, M., Moser, D., Speck, R.: A multigrid perspective on the parallel full approximation scheme in space and time. Numer. Linear Algebra Appl. 24(6), e2110 (2017). https://doi.org/10.1002/nla.2110

    Article  MathSciNet  MATH  Google Scholar 

  5. Bolten, M., Moser, D., Speck, R.: Asymptotic convergence of the parallel full approximation scheme in space and time for linear problems. Numer. Linear Algebra Appl. 25(6), e2208 (2018). https://doi.org/10.1002/nla.2208

    Article  MathSciNet  MATH  Google Scholar 

  6. Boscarino, S., Qiu, J.M.: Error estimates of the integral deferred correction method for stiff problems. ESAIM Math. Model. Numer. Anal. 50(4), 1137–1166 (2016). https://doi.org/10.1051/m2an/2015072

    Article  MathSciNet  MATH  Google Scholar 

  7. Boscarino, S., Qiu, J.M., Russo, G.: Implicit–explicit integral deferred correction methods for stiff problems. SIAM J. Sci. Comput. 40(2), A787–A816 (2018). https://doi.org/10.1137/16M1105232

    Article  MathSciNet  MATH  Google Scholar 

  8. Bottasso, C.L., Ragazzi, A.: Deferred-correction optimal control with applications to inverse problems in flight mechanics. J. Guidance Control Dyn. 24(1), 101–108 (2001). https://doi.org/10.2514/2.4681

    Article  Google Scholar 

  9. Bourlioux, A., Layton, A.T., Minion, M.L.: High-order multi-implicit spectral deferred correction methods for problems of reactive flow. J. Comput. Phys. 189(2), 651–675 (2003). https://doi.org/10.1016/S0021-9991(03)00251-1

    Article  MathSciNet  MATH  Google Scholar 

  10. Briggs, W.L., Henson, V.E., McCormick, S.E.: A Multigrid Tutorial, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000). https://doi.org/10.1137/1.9780898719505

    Book  MATH  Google Scholar 

  11. Bu, S., Huang, J., Minion, M.L.: Semi-implicit Krylov deferred correction methods for differential algebraic equations. Math. Comput. 81(280), 2127–2157 (2012). https://doi.org/10.1090/S0025-5718-2012-02564-6

    Article  MathSciNet  MATH  Google Scholar 

  12. Causley, M.F., Seal, D.C.: On the convergence of spectral deferred correction methods. Commun. Appl. Math. Comput. Sci. 14(1), 33–64 (2019). https://doi.org/10.2140/camcos.2019.14.33

    Article  MathSciNet  MATH  Google Scholar 

  13. Christlieb, A., Ong, B., Qiu, J.M.: Comments on high-order integrators embedded within integral deferred correction methods. Commun. Appl. Math. Comput. Sci. 4, 27–56 (2009). https://doi.org/10.2140/camcos.2009.4.27

    Article  MathSciNet  MATH  Google Scholar 

  14. Christlieb, A., Ong, B., Qiu, J.M.: Integral deferred correction methods constructed with high order Runge–Kutta integrators. Math. Comput. 79(270), 761–783 (2010). https://doi.org/10.1090/S0025-5718-09-02276-5

    Article  MathSciNet  MATH  Google Scholar 

  15. Christlieb, A.J., Guo, W., Morton, M., Qiu, J.M.: A high order time splitting method based on integral deferred correction for semi-Lagrangian Vlasov simulations. J. Comput. Phys. 267, 7–27 (2014). https://doi.org/10.1016/j.jcp.2014.02.012

    Article  MathSciNet  MATH  Google Scholar 

  16. Christlieb, A.J., Macdonald, C.B., Ong, B.W.: Parallel high-order integrators. SIAM J. Sci. Comput. 32(2), 818–835 (2010). https://doi.org/10.1137/09075740X

    Article  MathSciNet  MATH  Google Scholar 

  17. Christlieb, A.J., Macdonald, C.B., Ong, B.W., Spiteri, R.J.: Revisionist integral deferred correction with adaptive step-size control. Commun. Appl. Math. Comput. Sci. 10(1), 1–25 (2015). https://doi.org/10.2140/camcos.2015.10.1

    Article  MathSciNet  MATH  Google Scholar 

  18. Christlieb, A.J., Morton, M., Ong, B., Qiu, J.M.: Semi-implicit integral deferred correction constructed with additive Runge–Kutta methods. Commun. Math. Sci. 9(3), 879–902 (2011). https://doi.org/10.4310/CMS.2011.v9.n3.a10

    Article  MathSciNet  MATH  Google Scholar 

  19. Chu, K.W., Spence, A.: Deferred correction for the integral equation eigenvalue problem. J. Aust. Math. Soc. Ser. B 22(4), 474–487 (1980/81). https://doi.org/10.1017/S0334270000002812

  20. Dutt, A., Greengard, L., Rokhlin, V.: Spectral deferred correction methods for ordinary differential equations. BIT 40(2), 241–266 (2000). https://doi.org/10.1023/A:1022338906936

    Article  MathSciNet  MATH  Google Scholar 

  21. Eijkhout, V.: Introduction to high performance scientific computing. Lulu.com (2012)

  22. Emmett, M., Minion, M.: Toward an efficient parallel in time method for partial differential equations. Commun. Appl. Math. Comput. Sci. 7(1), 105–132 (2012). https://doi.org/10.2140/camcos.2012.7.105

    Article  MathSciNet  MATH  Google Scholar 

  23. Fox, L.: Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations. Proc. R. Soc. Lond. Ser. A 190, 31–59 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  24. Fox, L., Goodwin, E.T.: Some new methods for the numerical integration of ordinary differential equations. Proc. Camb. Philos. Soc. 45, 373–388 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  25. Frank, R., Ueberhuber, C.W.: Iterated defect correction for the efficient solution of stiff systems of ordinary differential equations. BIT Numer. Math. 17(2), 146–159 (1977). https://doi.org/10.1007/BF01932286

    Article  MathSciNet  MATH  Google Scholar 

  26. Götschel, S., Minion, M.L.: Parallel-in-time for parabolic optimal control problems using PFASST. In: Domain Decomposition Methods in Science and Engineering XXIV. Lecture Notes on Computer Science Engineering, vol. 125, pp. 363–371. Springer, Cham (2018)

  27. Grout, R., Kolla, H., Minion, M., Bell, J.: Achieving algorithmic resilience for temporal integration through spectral deferred corrections. Commun. Appl. Math. Comput. Sci. 12(1), 25–50 (2017). https://doi.org/10.2140/camcos.2017.12.25

    Article  MathSciNet  Google Scholar 

  28. Güttel, S., Klein, G.: Efficient high-order rational integration and deferred correction with equispaced data. Electron. Trans. Numer. Anal. 41, 443–464 (2014)

    MathSciNet  MATH  Google Scholar 

  29. Hackbusch, W.: Multigrid Methods and Applications, Springer Series in Computational Mathematics, vol. 4. Springer, Berlin (1985). https://doi.org/10.1007/978-3-662-02427-0

    Book  Google Scholar 

  30. Hagstrom, T., Zhou, R.: On the spectral deferred correction of splitting methods for initial value problems. Commun. Appl. Math. Comput. Sci. 1, 169–205 (2006). https://doi.org/10.2140/camcos.2006.1.169

    Article  MathSciNet  MATH  Google Scholar 

  31. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations. I, Springer Series in Computational Mathematics, vol. 8, 2nd edn. Springer, Berlin (1993). (Nonstiff problems)

    MATH  Google Scholar 

  32. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. II, Springer Series in Computational Mathematics, vol. 14, 2nd edn. Springer, Berlin (1996). https://doi.org/10.1007/978-3-642-05221-7. (Stiff and differential-algebraic problems)

    Book  MATH  Google Scholar 

  33. Hamon, F.P., Schreiber, M., Minion, M.L.: Multi-level spectral deferred corrections scheme for the shallow water equations on the rotating sphere. J. Comput. Phys. 376, 435–454 (2019). https://doi.org/10.1016/j.jcp.2018.09.042

    Article  MathSciNet  MATH  Google Scholar 

  34. Hansen, A.C., Strain, J.: On the order of deferred correction. Appl. Numer. Math. 61(8), 961–973 (2011). https://doi.org/10.1016/j.apnum.2011.04.001

    Article  MathSciNet  MATH  Google Scholar 

  35. Hofstätter, H., Koch, O.: Analysis of a defect correction method for geometric integrators. Numer. Algorithms 41(2), 103–126 (2006). https://doi.org/10.1007/s11075-005-9001-7

    Article  MathSciNet  MATH  Google Scholar 

  36. Huang, J., Jia, J., Minion, M.: Accelerating the convergence of spectral deferred correction methods. J. Comput. Phys. 214(2), 633–656 (2006). https://doi.org/10.1016/j.jcp.2005.10.004

    Article  MathSciNet  MATH  Google Scholar 

  37. Kress, W., Gustafsson, B.: Deferred correction methods for initial boundary value problems. In: Proceedings of the 5th International Conference on Spectral and High Order Methods (ICOSAHOM-01) (Uppsala), vol. 17, pp. 241–251 (2002). https://doi.org/10.1023/A:1015113017248

  38. Layton, A.T., Minion, M.L.: Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics. J. Comput. Phys. 194(2), 697–715 (2004). https://doi.org/10.1016/j.jcp.2003.09.010

    Article  MathSciNet  MATH  Google Scholar 

  39. Layton, A.T., Minion, M.L.: Implications of the choice of quadrature nodes for Picard integral deferred corrections methods for ordinary differential equations. BIT 45(2), 341–373 (2005). https://doi.org/10.1007/s10543-005-0016-1

    Article  MathSciNet  MATH  Google Scholar 

  40. Layton, A.T., Minion, M.L.: Implications of the choice of predictors for semi-implicit Picard integral deferred correction methods. Commun. Appl. Math. Comput. Sci. 2, 1–34 (2007). https://doi.org/10.2140/camcos.2007.2.1

    Article  MathSciNet  MATH  Google Scholar 

  41. Lindberg, B.: Error estimation and iterative improvement for the numerical solution of operator equations. Technical Report, Illinois University at Urbana-Champaign Department of Computer Science (1976)

  42. Lions, J., Maday, Y., Turinici, G.: A “parareal” in time discretization of PDEs. C. R. l’Acad. Sci. Ser. I Math. 332(7), 661–668 (2001)

    MATH  Google Scholar 

  43. Lions, J.L., Maday, Y., Turinici, G.: Résolution d’EDP par un schéma en temps “pararéel”. C. R. Acad. Sci. Paris Sér. I Math. 332(7), 661–668 (2001). https://doi.org/10.1016/S0764-4442(00)01793-6

    Article  MathSciNet  MATH  Google Scholar 

  44. Liu, F., Shen, J.: Stabilized semi-implicit spectral deferred correction methods for Allen–Cahn and Cahn–Hilliard equations. Math. Methods Appl. Sci. 38(18), 4564–4575 (2015). https://doi.org/10.1002/mma.2869

    Article  MathSciNet  MATH  Google Scholar 

  45. Liu, Y., Shu, C.W., Zhang, M.: Strong stability preserving property of the deferred correction time discretization. J. Comput. Math. 26(5), 633–656 (2008)

    MathSciNet  MATH  Google Scholar 

  46. Minion, M.L.: Semi-implicit spectral deferred correction methods for ordinary differential equations. Commun. Math. Sci. 1(3), 471–500 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  47. Minion, M.L.: Semi-implicit projection methods for incompressible flow based on spectral deferred corrections. Appl. Numer. Math. 48(3–4), 369–387 (2004). https://doi.org/10.1016/j.apnum.2003.11.005. (Workshop on innovative time integrators for PDEs)

    Article  MathSciNet  MATH  Google Scholar 

  48. Minion, M.L.: A hybrid parareal spectral deferred corrections method. Commun. Appl. Math. Comput. Sci. 5(2), 265–301 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  49. Minion, M.L., Speck, R., Bolten, M., Emmett, M., Ruprecht, D.: Interweaving PFASST and parallel multigrid. SIAM J. Sci. Comput. 37(5), S244–S263 (2015). https://doi.org/10.1137/14097536X

    Article  MathSciNet  MATH  Google Scholar 

  50. Minion, M.L., Williams, S.A.: Parareal and spectral deferred corrections. AIP Conference Proceedings 1048(1), 388–391 (2008). https://doi.org/10.1063/1.2990941

    Article  Google Scholar 

  51. Ong, B.W., Haynes, R.D., Ladd, K.: Algorithm 965: RIDC methods: a family of parallel time integrators. ACM Trans. Math. Softw. 43(1), 8:1–8:13 (2016). https://doi.org/10.1145/2964377

    Article  MathSciNet  MATH  Google Scholar 

  52. Pazner, W.E., Nonaka, A., Bell, J.B., Day, M.S., Minion, M.L.: A high-order spectral deferred correction strategy for low mach number flow with complex chemistry. Combust. Theory Model. 20(3), 521–547 (2016). https://doi.org/10.1080/13647830.2016.1150519

    Article  MathSciNet  Google Scholar 

  53. Pereyra, V.: On improving an approximate solution of a functional equation by deferred corrections. Numer. Math. 8, 376–391 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  54. Pereyra, V.: Iterated deferred corrections for nonlinear boundary value problems. Numer. Math. 11, 111–125 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  55. Qu, W., Brandon, N., Chen, D., Huang, J., Kress, T.: A numerical framework for integrating deferred correction methods to solve high order collocation formulations of ODEs. J. Sci. Comput. 68(2), 484–520 (2016). https://doi.org/10.1007/s10915-015-0146-9

    Article  MathSciNet  MATH  Google Scholar 

  56. Rangan, A.V.: Adaptive solvers for partial differential and differential-algebraic equations. Ph.D. thesis, University of California, Berkeley (2003)

  57. Ruprecht, D., Speck, R.: Spectral deferred corrections with fast-wave slow-wave splitting. SIAM J. Sci. Comput. 38(4), A2535–A2557 (2016). https://doi.org/10.1137/16M1060078

    Article  MathSciNet  MATH  Google Scholar 

  58. Schild, K.H.: Gaussian collocation via defect correction. Numer. Math. 58(4), 369–386 (1990). https://doi.org/10.1007/BF01385631

    Article  MathSciNet  MATH  Google Scholar 

  59. Skeel, R.D.: A theoretical framework for proving accuracy results for deferred corrections. SIAM J. Numer. Anal. 19(1), 171–196 (1982). https://doi.org/10.1137/0719009

    Article  MathSciNet  MATH  Google Scholar 

  60. Speck, R.: Parallelizing spectral deferred corrections across the method (2017). ArXiv e-prints

  61. Speck, R.: Algorithm 997: pySDC—prototyping spectral deferred corrections. ACM Trans. Math. Software 45(3), Art. 35, 23 (2019). https://doi.org/10.1145/3310410

  62. Speck, R., Ruprecht, D.: Toward fault-tolerant parallel-in-time integration with PFASST. Parallel Comput. 62, 20–37 (2017). https://doi.org/10.1016/j.parco.2016.12.001

    Article  MathSciNet  Google Scholar 

  63. Speck, R., Ruprecht, D., Emmett, M., Minion, M., Bolten, M., Krause, R.: A multi-level spectral deferred correction method. BIT 55(3), 843–867 (2015). https://doi.org/10.1007/s10543-014-0517-x

    Article  MathSciNet  MATH  Google Scholar 

  64. Speck, R., Ruprecht, D., Krause, R., Emmett, M., Minion, M., Winkel, M., Gibbon, P.: Integrating an \(N\)-body problem with SDC and PFASST. In: Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes on Computer Science Engineering, vol. 98, pp. 637–645. Springer, Cham (2014)

  65. Speck, R., Ruprecht, D., Minion, M., Emmett, M., Krause, R.: Inexact spectral deferred corrections. In: Dickopf, T., Gander, M.J., Halpern, L., Krause, R., Pavarino, L.F. (eds.) Domain Decomposition Methods in Science and Engineering XXII, pp. 389–396. Springer International Publishing, Cham (2016)

    Chapter  Google Scholar 

  66. Stetter, H.J.: Economical Global Error Estimation, pp. 245–258. Springer, Boston (1974). https://doi.org/10.1007/978-1-4684-2100-2_19

    Book  Google Scholar 

  67. Sun, G.: The deferred correction procedure for linear multistep formulas. J. Comput. Math. 3(1), 41–49 (1985)

    MathSciNet  MATH  Google Scholar 

  68. Tang, T., Zhao, W., Zhou, T.: Deferred correction methods for forward backward stochastic differential equations. Numer. Math. Theory Methods Appl. 10(2), 222–242 (2017). https://doi.org/10.4208/nmtma.2017.s02

    Article  MathSciNet  MATH  Google Scholar 

  69. Trefethen, L.N.: Spectral Methods in MATLAB, Software, Environments, and Tools, vol. 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000). https://doi.org/10.1137/1.9780898719598

    Book  Google Scholar 

  70. Verwer, J.G., Sommeijer, B.P.: An implicit–explicit Runge–Kutta–Chebyshev scheme for diffusion–reaction equations. SIAM J. Sci. Comput. 25(5), 1824–1835 (2004). https://doi.org/10.1137/S1064827503429168

    Article  MathSciNet  MATH  Google Scholar 

  71. Weiser, M.: Faster SDC convergence on non-equidistant grids by DIRK sweeps. BIT Numer. Math. 55(4), 1219–1241 (2015). https://doi.org/10.1007/s10543-014-0540-y

    Article  MathSciNet  MATH  Google Scholar 

  72. Weiser, M., Ghosh, S.: Theoretically optimal inexact spectral deferred correction methods. Commun. Appl. Math. Comput. Sci. 13(1), 53–86 (2018). https://doi.org/10.2140/camcos.2018.13.53

    Article  MathSciNet  MATH  Google Scholar 

  73. Winkel, M., Speck, R., Ruprecht, D.: A high-order Boris integrator. J. Comput. Phys. 295, 456–474 (2015). https://doi.org/10.1016/j.jcp.2015.04.022

    Article  MathSciNet  MATH  Google Scholar 

  74. Xin, J., Huang, J., Zhao, W., Zhu, J.: A spectral deferred correction method for fractional differential equations. Abstr. Appl. Anal., Art. ID 139530, 6 (2013)

  75. Zadunaisky, P.E.: A method for the estimation of errors propagated in the numerical solution of a system of ordinary differential equations. In: G.I. Kontopoulos (ed.) The Theory of Orbits in the Solar System and in Stellar Systems, IAU Symposium, vol. 25, pp. 281–287 (1966)

  76. Zadunaisky, P.E.: On the estimation of errors propagated in the numerical integration of ordinary differential equations. Numer. Math. 27(1), 21–39 (1976/77)

Download references

Funding

Funding was provided by Natural Sciences and Engineering Research Council of Canada (Grant No. 228090-2013).

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931, USA

    Benjamin W. Ong

  2. Department of Computer Science, University of Saskatchewan, 176 Thorvaldson Building, 110 Science Place, Saskatoon, SK, S7N 5C9, Canada

    Raymond J. Spiteri

Authors
  1. Benjamin W. Ong
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Raymond J. Spiteri
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Raymond J. Spiteri.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ong, B.W., Spiteri, R.J. Deferred Correction Methods for Ordinary Differential Equations. J Sci Comput 83, 60 (2020). https://doi.org/10.1007/s10915-020-01235-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01235-8

Keywords

Mathematics Subject Classification

Profiles

  1. Raymond J. Spiteri View author profile