article

Generalized integrating factor methods for stiff PDEs

Author:

S. KrogstadAuthors Info & Claims

Journal of Computational Physics, Volume 203, Issue 1

Pages 72 - 88

https://doi.org/10.1016/j.jcp.2004.08.006

Published: 10 February 2005 Publication History

Abstract

The integrating factor (IF) method for numerical integration of stiff nonlinear PDEs has the disadvantage of producing large error coefficients when the linear term has large norm. We propose a generalization of the IF method, and in particular construct multistep-type methods with several orders of magnitude improved accuracy. We also consider exponential time differencing (ETD) methods, and point out connections with a particular application of the commutator-free Lie group methods. We present a new fourth order ETDRK method with improved accuracy. The methods considered are compared in several numerical examples.

References

[1]

{1} G. Beyklin, J.M. Keiser, L. Vozovoi, A new class of time discretization schemes for the solution of nonlinear PDEs, J. Comput. Phys. 147 (1998) 362-387.]]

Digital Library

[2]

{2} J.P. Boyd, Chebyshev and Fourier Spectral Methods, Dover, New York, 2001, Available from: 〈http://www-personal.engin.umich.edu/~jpboyd〉.]]

[3]

{3} E. Celledoni, A. Martinsen, B. Owren, Commutator-free Lie group methods, FGCS 19 (3) (2003) 341-352.]]

Digital Library

[4]

{4} S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys. 176 (2002) 430-455.]]

Digital Library

[5]

{5} P.E. Crouch, R. Grossman, Numerical integration of ordinary differential equations on manifolds, J. Nonlinear Sci. 3 (1993) 1-33.]]

[6]

{6} P. Davies, N. Higham, A Schur-Parlett algorithm for computing matrix functions, SIAM J. Matrix Anal. Appl. 25 (2) (2003) 464-485.]]

Digital Library

[7]

{7} B. Fornberg, T.A. Driscoll, A fast spectral algorithm for nonlinear wave equations with linear dispersion, J. Comput. Phys. 155 (1999) 456-467.]]

Digital Library

[8]

{8} E. Hairer, S.P. Nørsett, G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, second ed.Springer Series in Computational Mathematics, vol. 8, Springer, Berlin, 1993.]]

[9]

{9} E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic ProblemsSpringer Series in Computational Mathematics, vol. 14, Springer, Berlin, 1996.]]

[10]

{10} M. Hochbruck, C.h. Lubich, On Krylov subspace approximations to the matrix exponential operator, SIAM J. Sci. Comput. 34 (5) (1997) 1911-1925.]]

Digital Library

[11]

{11} M. Hochbruck, C. Lubich, H. Selhofer, Exponential integrators for large systems of differential equations, SIAM J. Numer. Anal. 19 (5) (1998) 1552-1574.]]

Digital Library

[12]

{12} M. Hochbruck, A. Ostermann, Exponential Runge-Kutta methods for parabolic problems, Appl. Numer. Math. (to appear).]]

[13]

{13} A. Iserles, H.Z. Munthe-Kaas, S.P. Nørsett, A. Zanna, Lie-group methods, Acta Numer. (2000) 215-236.]]

[14]

{14} A.K. Kassam, L.N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM J. Sci. Comput. (to appear).]]

[15]

{15} S. Krogstad, RKMK-related methods for stiff nonlinear PDEs, Report, University of Bergen, 2003.]]

[16]

{16} E. Lodden, Geometric integration of the heat equation, Masters Thesis, University of Bergen, 2000.]]

[17]

{17} Y. Maday, A.T. Patera, E.M. Rønquist, An operator-integration-factor splitting method for time-dependent problems: application to incompressible fluid flow, J. Sci. Comp. 5 (4) (1990) 263-292.]]

Digital Library

[18]

{18} H. Munthe-Kaas, High order Runge-Kutta methods on manifolds, J. Appl. Numer. Math. 29 (1999) 115-127.]]

Digital Library

[19]

{19} B. Owren, A. Marthinsen, Runge-Kutta methods adapted to manifolds and based on rigid frames, BIT 39 (1) (1999) 116-142.]]

[20]

{20} S. Nørsett, An A-stable modification of the Adams-Bashforth methodsLecture Notes in Mathematics, vol. 109, Springer, Berlin, 1969, pp. 214-219.]]

[21]

{21} A. Suslowicz, Application of numerical Lie group integrators to parabolic PDE's, Technical Report No. 013, University of Bergen, 2001.]]

[22]

{22} L.N. Trefethen, Spectral methods in MATLAB, SIAM, Philadelphia, 2000.]]

Digital Library

Cited By

Shen MWang H(2024)An efficient spectral method for the fractional Schrödinger equation on the real lineJournal of Computational and Applied Mathematics10.1016/j.cam.2024.115774444:COnline publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1016/j.cam.2024.115774
Zhang HZhang GLiu ZQian XSong S(2024)On the maximum principle and high-order, delay-free integrators for the viscous Cahn–Hilliard equationAdvances in Computational Mathematics10.1007/s10444-024-10143-650:3Online publication date: 3-May-2024
https://dl.acm.org/doi/10.1007/s10444-024-10143-6
Zhang HYan JQian XSong S(2023)Temporal high-order, unconditionally maximum-principle-preserving integrating factor multi-step methods for Allen-Cahn-type parabolic equationsApplied Numerical Mathematics10.1016/j.apnum.2022.12.020186:C(19-40)Online publication date: 1-Apr-2023
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Index Terms

Generalized integrating factor methods for stiff PDEs
1. Mathematics of computing
  1. Mathematical analysis
    1. Nonlinear equations
    2. Numerical analysis
      1. Numerical differentiation
  2. Probability and statistics
    1. Probabilistic reasoning algorithms
      1. Markov-chain Monte Carlo methods
        Markov-chain Monte Carlo convergence measures
2. Theory of computation
  1. Design and analysis of algorithms
  2. Randomness, geometry and discrete structures
    1. Error-correcting codes

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cover image Journal of Computational Physics

Journal of Computational Physics Volume 203, Issue 1

10 February 2005

373 pages

ISSN:0021-9991

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Copyright © Elsevier Inc. © 2004.

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Academic Press Professional, Inc.

United States

Publication History

Published: 10 February 2005

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Cited By

Shen MWang H(2024)An efficient spectral method for the fractional Schrödinger equation on the real lineJournal of Computational and Applied Mathematics10.1016/j.cam.2024.115774444:COnline publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1016/j.cam.2024.115774
Zhang HZhang GLiu ZQian XSong S(2024)On the maximum principle and high-order, delay-free integrators for the viscous Cahn–Hilliard equationAdvances in Computational Mathematics10.1007/s10444-024-10143-650:3Online publication date: 3-May-2024
https://dl.acm.org/doi/10.1007/s10444-024-10143-6
Zhang HYan JQian XSong S(2023)Temporal high-order, unconditionally maximum-principle-preserving integrating factor multi-step methods for Allen-Cahn-type parabolic equationsApplied Numerical Mathematics10.1016/j.apnum.2022.12.020186:C(19-40)Online publication date: 1-Apr-2023
https://dl.acm.org/doi/10.1016/j.apnum.2022.12.020
Cheng LXu K(2023)Solving Time-Dependent PDEs with the Ultraspherical Spectral MethodJournal of Scientific Computing10.1007/s10915-023-02287-296:3Online publication date: 20-Jul-2023
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Buvoli TMinion M(2022)On the stability of exponential integrators for non-diffusive equationsJournal of Computational and Applied Mathematics10.1016/j.cam.2022.114126409:COnline publication date: 1-Aug-2022
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Luan VMichels D(2022)Efficient exponential time integration for simulating nonlinear coupled oscillatorsJournal of Computational and Applied Mathematics10.1016/j.cam.2021.113429391:COnline publication date: 23-Apr-2022
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Souleymane KAmmari K(2022)Boundary value problems initial condition identification by a wavelet-based Galerkin methodNumerical Algorithms10.1007/s11075-022-01421-993:1(397-414)Online publication date: 10-Oct-2022
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Phan DOstermann A(2022)Exponential Integrators for Second-Order in Time Partial Differential EquationsJournal of Scientific Computing10.1007/s10915-022-02018-z93:2Online publication date: 1-Nov-2022
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Owolabi KKaraagac BBaleanu D(2021)Dynamics of pattern formation process in fractional-order super-diffusive processes: a computational approachSoft Computing - A Fusion of Foundations, Methodologies and Applications10.1007/s00500-021-05885-025:16(11191-11208)Online publication date: 1-Aug-2021
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Bhatt HKhaliq AFurati K(2020)Efficient high-order compact exponential time differencing method for space-fractional reaction-diffusion systems with nonhomogeneous boundary conditionsNumerical Algorithms10.1007/s11075-019-00729-383:4(1373-1397)Online publication date: 1-Apr-2020
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