Communications in Applied Mathematics and Computational Science

On the convergence of spectral deferred correction methods

Mathew F. Causley and David C. Seal

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In this work we analyze the convergence properties of the spectral deferred correction (SDC) method originally proposed by Dutt et al. (BIT 40 (2000), no. 2, 241–266). The framework for this high-order ordinary differential equation (ODE) solver is typically described as a low-order approximation (such as forward or backward Euler) lifted to higher-order accuracy by applying the same low-order method to an error equation and then adding in the resulting defect to correct the solution. Our focus is not on solving the error equation to increase the order of accuracy, but on rewriting the solver as an iterative Picard integral equation solver. In doing so, our chief finding is that it is not the low-order solver that picks up the order of accuracy with each correction, but it is the underlying quadrature rule of the right-hand-side function that is solely responsible for picking up additional orders of accuracy. Our proofs point to a total of three sources of errors that SDC methods carry: the error at the current time point, the error from the previous iterate, and the numerical integration error that comes from the total number of quadrature nodes used for integration. The second of these two sources of errors is what separates SDC methods from Picard integral equation methods; our findings indicate that as long as the difference between the current and previous iterates always gets multiplied by at least a constant multiple of the time step size, then high-order accuracy can be found even if the underlying ODE “solver” is inconsistent. From this vantage, we solidify the prospects of extending spectral deferred correction methods to a larger class of solvers, of which we present some examples.

Article information

Commun. Appl. Math. Comput. Sci., Volume 14, Number 1 (2019), 33-64.

Received: 19 June 2017
Revised: 7 November 2018
Accepted: 2 December 2018
First available in Project Euclid: 25 July 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65L05: Initial value problems 65L20: Stability and convergence of numerical methods

initial-value problems spectral deferred correction Picard integral semi-implicit methods


Causley, Mathew F.; Seal, David C. On the convergence of spectral deferred correction methods. Commun. Appl. Math. Comput. Sci. 14 (2019), no. 1, 33--64. doi:10.2140/camcos.2019.14.33.

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  • J. C. Butcher, Implicit Runge–Kutta processes, Math. Comp. 18 (1964), 50–64.
  • T. Buvoli, A class of exponential integrators based on spectral deferred correction, preprint, 2015.
  • A. Christlieb, W. Guo, M. Morton, and J.-M. Qiu, A high order time splitting method based on integral deferred correction for semi-Lagrangian Vlasov simulations, J. Comput. Phys. 267 (2014), 7–27.
  • A. Christlieb, M. Morton, B. Ong, and J.-M. Qiu, Semi-implicit integral deferred correction constructed with additive Runge–Kutta methods, Commun. Math. Sci. 9 (2011), no. 3, 879–902.
  • A. Christlieb and B. Ong, Implicit parallel time integrators, J. Sci. Comput. 49 (2011), no. 2, 167–179.
  • A. Christlieb, B. Ong, and J.-M. Qiu, Comments on high-order integrators embedded within integral deferred correction methods, Commun. Appl. Math. Comput. Sci. 4 (2009), 27–56.
  • ––––, Integral deferred correction methods constructed with high order Runge–Kutta integrators, Math. Comp. 79 (2010), no. 270, 761–783.
  • A. J. Christlieb, Y. Liu, and Z. Xu, High order operator splitting methods based on an integral deferred correction framework, J. Comput. Phys. 294 (2015), 224–242.
  • A. J. Christlieb, C. B. Macdonald, and B. W. Ong, Parallel high-order integrators, SIAM J. Sci. Comput. 32 (2010), no. 2, 818–835.
  • A. J. Christlieb, C. B. Macdonald, B. W. Ong, and R. J. Spiteri, Revisionist integral deferred correction with adaptive step-size control, Commun. Appl. Math. Comput. Sci. 10 (2015), no. 1, 1–25.
  • M. Duarte and M. Emmett, High order schemes based on operator splitting and deferred corrections for stiff time dependent PDEs, preprint, 2014.
  • A. Dutt, L. Greengard, and V. Rokhlin, Spectral deferred correction methods for ordinary differential equations, BIT 40 (2000), no. 2, 241–266.
  • M. Emmett and M. L. Minion, Toward an efficient parallel in time method for partial differential equations, Commun. Appl. Math. Comput. Sci. 7 (2012), no. 1, 105–132.
  • I. Faragó, Note on the convergence of the implicit Euler method, Numerical analysis and its applications (I. Dimov, I. Faragó, and L. Vulkov, eds.), Lecture Notes in Comput. Sci., no. 8236, Springer, 2013, pp. 1–11.
  • T. Hagstrom and R. Zhou, On the spectral deferred correction of splitting methods for initial value problems, Commun. Appl. Math. Comput. Sci. 1 (2006), 169–205.
  • E. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential equations, I: Nonstiff problems, 2nd ed., Springer Series in Computational Mathematics, no. 8, Springer, 1993.
  • E. Hairer and G. Wanner, Solving ordinary differential equations, II: Stiff and differential-algebraic problems, 2nd ed., Springer Series in Computational Mathematics, no. 14, Springer, 1996.
  • E. Hairer, C. Lubich, and G. Wanner, Geometric numerical integration: structure-preserving algorithms for ordinary differential equations, 2nd ed., Springer Series in Computational Mathematics, no. 31, Springer, 2006.
  • A. C. Hansen and J. Strain, On the order of deferred correction, Appl. Numer. Math. 61 (2011), no. 8, 961–973.
  • J. Huang, J. Jia, and M. Minion, Accelerating the convergence of spectral deferred correction methods, J. Comput. Phys. 214 (2006), no. 2, 633–656.
  • ––––, Arbitrary order Krylov deferred correction methods for differential algebraic equations, J. Comput. Phys. 221 (2007), no. 2, 739–760.
  • J. Jia, J. C. Hill, K. J. Evans, G. I. Fann, and M. A. Taylor, A spectral deferred correction method applied to the shallow water equations on a sphere, Mon. Weather Rev. 141 (2013), 3435–3449.
  • S. Y. Kadioglu and V. Colak, An essentially non-oscillatory spectral deferred correction method for conservation laws, Int. J. Comput. Methods 13 (2016), no. 5, art. id. 1650027.
  • J. Kuntzmann, Neuere Entwicklungen der Methode von Runge und Kutta, Z. Angew. Math. Mech. 41 (1961), no. S1, T28–T31.
  • A. T. Layton, On the choice of correctors for semi-implicit Picard deferred correction methods, Appl. Numer. Math. 58 (2008), no. 6, 845–858.
  • ––––, On the efficiency of spectral deferred correction methods for time-dependent partial differential equations, Appl. Numer. Math. 59 (2009), no. 7, 1629–1643.
  • A. T. Layton and M. L. Minion, Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics, J. Comput. Phys. 194 (2004), no. 2, 697–715.
  • ––––, Implications of the choice of quadrature nodes for Picard integral deferred corrections methods for ordinary differential equations, BIT 45 (2005), no. 2, 341–373.
  • ––––, Implications of the choice of predictors for semi-implicit Picard integral deferred correction methods, Commun. Appl. Math. Comput. Sci. 2 (2007), 1–34.
  • M. L. Minion, Semi-implicit spectral deferred correction methods for ordinary differential equations, Commun. Math. Sci. 1 (2003), no. 3, 471–500.
  • ––––, A hybrid parareal spectral deferred corrections method, Commun. Appl. Math. Comput. Sci. 5 (2010), no. 2, 265–301.
  • M. M. Morton, Integral Deferred Correction methods for scientific computing, Ph.D. thesis, Michigan State University, 2010.
  • W. Pazner and P.-O. Persson, Stage-parallel fully implicit Runge–Kutta solvers for discontinuous Galerkin fluid simulations, J. Comput. Phys. 335 (2017), 700–717.
  • W. Qu, N. Brandon, D. Chen, J. Huang, and T. Kress, A numerical framework for integrating deferred correction methods to solve high order collocation formulations of ODEs, J. Sci. Comput. 68 (2016), no. 2, 484–520.
  • D. Ruprecht and R. Speck, Spectral deferred corrections with fast-wave slow-wave splitting, SIAM J. Sci. Comput. 38 (2016), no. 4, A2535–A2557.
  • R. Speck, D. Ruprecht, M. Emmett, M. Minion, M. Bolten, and R. Krause, A multi-level spectral deferred correction method, BIT 55 (2015), no. 3, 843–867.
  • T. Tang, H. Xie, and X. Yin, High-order convergence of spectral deferred correction methods on general quadrature nodes, J. Sci. Comput. 56 (2013), no. 1, 1–13.
  • M. Weiser, Faster SDC convergence on non-equidistant grids by DIRK sweeps, BIT 55 (2015), no. 4, 1219–1241.
  • Y. Xia, Y. Xu, and C.-W. Shu, Efficient time discretization for local discontinuous Galerkin methods, Discrete Contin. Dyn. Syst. Ser. B 8 (2007), no. 3, 677–693.
  • P. E. Zadunaisky, On the estimation of errors propagated in the numerical integration of ordinary differential equations, Numer. Math. 27 (1976), no. 1, 21–39.