## Communications in Applied Mathematics and Computational Science

### Comments on high-order integrators embedded within integral deferred correction methods

#### Abstract

A class of novel deferred correction methods, integral deferred correction (IDC) methods, is studied. This class of methods is an extension of ideas introduced by Dutt, Greengard and Rokhlin on spectral deferred correction (SDC) methods for solving ordinary differential equations (ODEs). The novel nature of this class of defect correction methods is that the correction of the defect is carried out using an accurate integral form of the residual instead of the more familiar differential form. As a family of methods, these schemes are capable of matching the efficiency of popular high-order RK methods.

The smoothness of the error vector associated with an IDC method is an important indicator of the order of convergence that can be expected from a scheme (Christlieb, Ong, and Qiu; Hansen and Strain; Skeel). It is demonstrated that embedding an $r$-th order integrator in the correction loop of an IDC method does not always result in an $r$-th order increase in accuracy. Examples include IDC methods constructed using non-self-starting multistep integrators, and IDC methods constructed using a nonuniform distribution of quadrature nodes.

Additionally, the integral deferred correction concept is reposed as a framework to generate high-order Runge–Kutta (RK) methods; specifically, we explain how the prediction and correction loops can be incorporated as stages of a high-order RK method. This alternate point of view allows us to utilize standard methods for quantifying the performance (efficiency, accuracy and stability) of integral deferred correction schemes. It is found that IDC schemes constructed using uniformly distributed nodes and high-order integrators are competitive in terms of efficiency with IDC schemes constructed using Gauss–Lobatto nodes and forward Euler integrators. With respect to regions of absolute stability, however, IDC methods constructed with uniformly distributed nodes and high-order integrators are far superior. It is observed that as the order of the embedded integrator increases, the stability region of the IDC method increases.

#### Article information

Source
Commun. Appl. Math. Comput. Sci., Volume 4, Number 1 (2009), 27-56.

Dates
Revised: 20 January 2009
Accepted: 15 March 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.camcos/1513798576

Digital Object Identifier
doi:10.2140/camcos.2009.4.27

Mathematical Reviews number (MathSciNet)
MR2516213

Zentralblatt MATH identifier
1167.65389

#### Citation

Christlieb, Andrew; Ong, Benjamin; Qiu, Jing-Mei. Comments on high-order integrators embedded within integral deferred correction methods. Commun. Appl. Math. Comput. Sci. 4 (2009), no. 1, 27--56. doi:10.2140/camcos.2009.4.27. https://projecteuclid.org/euclid.camcos/1513798576

#### References

• C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods in fluid dynamics, Springer Series in Computational Physics, Springer, New York, 1988.
• A. Christlieb, B. Ong, and J.-M. Qiu, Integral deferred correction methods constructed with high order Runge–Kutta integrators, Math. Comp., accepted.
• A. R. Curtis, An eighth order Runge–Kutta process with eleven function evaluations per step, Numer. Math. 16 (1970), 268–277.
• A. Dutt, L. Greengard, and V. Rokhlin, Spectral deferred correction methods for ordinary differential equations, BIT 40 (2000), 241–266.
• 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, Berlin, 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, Berlin, 1996.
• A. Hansen and J. Strain, On the order of deferred correction, preprint.
• ––––, Convergence theory for spectral deferred correction, preprint, 2005.
• M. E. Hosea, A new recurrence for computing Runge–Kutta truncation error coefficients, SIAM J. Numer. Anal. 32 (1995), no. 6, 1989–2001.
• 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.
• R. Jeltsch and O. Nevanlinna, Largest disk of stability of explicit Runge–Kutta methods, BIT 18 (1978), no. 4, 500–502.
• A. T. Layton, On the choice of correctors for semi-implicit Picard deferred correction methods, Appl. Numer. Math. 58 (2008), no. 6, 845–858.
• A. T. Layton and M. 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.
• ––––, Implications of the choice of predictors for semi-implicit Picard integral deferred correction methods, Commun. Appl. Math. Comput. Sci. 2 (2007), 1–34.
• H. A. Luther, An explicit sixth-order Runge–Kutta formula, Math. Comp. 22 (1968), no. 102, 434–436.
• M. L. Minion, Semi-implicit spectral deferred correction methods for ordinary differential equations, Commun. Math. Sci. 1 (2003), no. 3, 471–500.
• B. Owren and K. Seip, Some stability results for explicit Runge–Kutta methods, BIT 30 (1990), no. 4, 700–706.
• J. Verner, High order Runge–Kutta methods.
• Y. Xia, Y. Xu, and C.-W. Shu, Efficient time discretization for local discontinuous Galerkin methods, Discrete Contin. Dyn. Syst. Ser. B 8 (2007), 677–693.