Higher Order Time Stepping for Second Order Hyperbolic Problems and Optimal CFL Conditions

  • J. Charles Gilbert
  • Patrick Joly
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 16)


We investigate explicit higher order time discretizations of linear second order hyperbolic problems. We study the even order (2m) schemes obtained by the modified equation method. We show that the corresponding CFL upper bound for the time step remains bounded when the order of the scheme increases. We propose variants of these schemes constructed to optimize the CFL condition. The corresponding optimization problem is analyzed in detail and the analysis results in a specific numerical algorithm. The corresponding results are quite promising and suggest various conjectures.


Chebyshev Polynomial Equation Approach Tangent Point Order Scheme Optimal Polynomial 
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  1. AJT00.
    L. Anné, P. Joly, and Q. H. Tran. Construction and analysis of higher order finite difference schemes for the 1D wave equation. Comput. Geosci., 4(3):207–249, 2000.MATHCrossRefMathSciNetGoogle Scholar
  2. AKM74.
    R. M. Alford, K. R. Kelly, and Boore D. M. Accuracy of finite difference modeling of the acoustic wave equation. Geophysics, 39:834–842, 1974.CrossRefADSGoogle Scholar
  3. BGLS06.
    J. F. Bonnans, J. Ch. Gilbert, C. Lemaréchal, and C. Sagastizábal. Numerical Optimization – Theoretical and Practical Aspects. Universitext. Springer Verlag, Berlin, 2nd edition, 2006.MATHGoogle Scholar
  4. BM05.
    S. Bellavia and B. Morini. An interior global method for nonlinear systems with simple bounds. Optim. Methods Softw., 20(4–5):453–474, 2005.MATHCrossRefMathSciNetGoogle Scholar
  5. CdLBL97.
    R. Carpentier, A. de La Bourdonnaye, and B. Larrouturou. On the derivation of the modified equation for the analysis of linear numerical methods. RAIRO Modél. Math. Anal. Numér., 31(4):459–470, 1997.MATHGoogle Scholar
  6. CF05.
    G. Cohen and S. Fauqueux. Mixed spectral finite elements for the linear elasticity system in unbounded domains. SIAM J. Sci. Comput., 26(3):864–884 (electronic), 2005.MATHCrossRefMathSciNetGoogle Scholar
  7. Che66.
    E. W. Cheney. Introduction to Approximation Theory. McGraw-Hill, 1966.Google Scholar
  8. CJ96.
    G. Cohen and P. Joly. Construction analysis of fourth-order finite difference schemes for the acoustic wave equation in nonhomogeneous media. SIAM J. Numer. Anal., 33(4):1266–1302, 1996.MATHCrossRefMathSciNetGoogle Scholar
  9. CJKMVV99.
    M. J. S. Chin-Joe-Kong, W. A. Mulder, and M. Van Veldhuizen. Higher-order triangular and tetrahedral finite elements with mass lumping for solving the wave equation. J. Engrg. Math., 35(4):405–426, 1999.MATHCrossRefMathSciNetGoogle Scholar
  10. CJRT01.
    G. Cohen, P. Joly, J. E. Roberts, and N. Tordjman. Higher order triangular finite elements with mass lumping for the wave equation. SIAM J. Numer. Anal., 38(6):2047–2078 (electronic), 2001.MATHCrossRefMathSciNetGoogle Scholar
  11. Coh02.
    G. C. Cohen. Higher-order numerical methods for transient wave equations. Scientific Computation. Springer-Verlag, Berlin, 2002.Google Scholar
  12. Dab86.
    M. A. Dablain. The application of high order differencing for the scalar wave equation. Geophysics, 51:54–56, 1986.CrossRefADSGoogle Scholar
  13. Deu04.
    P. Deuflhard. Newton Methods for Nonlinear Problems – Affine Invariance and Adaptative Algorithms. Number 35 in Computational Mathematics. Springer, Berlin, 2004.Google Scholar
  14. DPJ06.
    S. Del Pino and H. Jourdren. Arbitrary high-order schemes for the linear advection and wave equations: application to hydrodynamics and aeroacoustics. C. R. Math. Acad. Sci. Paris, 342(6):441–446, 2006.MATHMathSciNetGoogle Scholar
  15. FLLP05.
    L. Fezoui, S. Lanteri, S. Lohrengel, and S. Piperno. Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes. M2AN Math. Model. Numer. Anal., 39(6):1149–1176, 2005.MATHCrossRefMathSciNetGoogle Scholar
  16. HW96.
    E. Hairer and G. Wanner. Solving ordinary differential equations. II, volume 14 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2nd edition, 1996. Stiff and differential-algebraic problems.Google Scholar
  17. HW02.
    J. S. Hesthaven and T. Warburton. Nodal high-order methods on unstructured grids. I. Time-domain solution of Maxwell’s equations. J. Comput. Phys., 181(1):186–221, 2002.MATHCrossRefADSMathSciNetGoogle Scholar
  18. Jol03.
    P. Joly. Variational methods for time-dependent wave propagation problems. In Topics in computational wave propagation, volume 31 of Lect. Notes Comput. Sci. Eng., pages 201–264. Springer, Berlin, 2003.Google Scholar
  19. Kan01.
    Ch. Kanzow. An active set-type Newton method for constrained nonlinear systems. In M.C. Ferris, O.L. Mangasarian, and J.S. Pang, editors, Complementarity: applications, algorithms and extensions, pages 179–200, Dordrecht, 2001. Kluwer Acad. Publ.Google Scholar
  20. LT86.
    P. Lascaux and R. Théodor. Analyse Numérique Matricielle Appliquée à l’Art de l’Ingénieur. Masson, Paris, 1986.MATHGoogle Scholar
  21. PFC05.
    S. Pernet, X. Ferrieres, and G. Cohen. High spatial order finite element method to solve Maxwell’s equations in time domain. IEEE Trans. Antennas and Propagation, 53(9):2889–2899, 2005.CrossRefADSMathSciNetGoogle Scholar
  22. RM67.
    R. D. Richtmyer and K. W. Morton. Difference methods for initial-value problems, volume 4 of Interscience Tracts in Pure and Applied Mathematics. John Wiley & Sons, Inc., New York, 2nd edition, 1967.Google Scholar
  23. RS78.
    M. Reed and B. Simon. Methods of modern mathematical physics. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978.Google Scholar
  24. SB87.
    G. R. Shubin and J. B. Bell. A modified equation approach to constructing fourth-order methods for acoustic wave propagation. SIAM J. Sci. Statist. Comput., 8(2):135–151, 1987.MATHCrossRefMathSciNetGoogle Scholar
  25. Sch91.
    L. Schwartz. Analyse I – Théorie des Ensembles et Topologie. Hermann, Paris, 1991.MATHGoogle Scholar
  26. TT05.
    E. F. Toro and V. A. Titarev. ADER schemes for scalar non-linear hyperbolic conservation laws with source terms in three-space dimensions. J. Comput. Phys., 202(1):196–215, 2005.MATHCrossRefADSMathSciNetGoogle Scholar
  27. Wei06.
    E. W. Weisstein. Chebyshev polynomial of the first kind. MathWorld. http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html , 2006.

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© Springer Science + Business Media B.V. 2008

Authors and Affiliations

  • J. Charles Gilbert
    • 1
  • Patrick Joly
    • 1
  1. 1.INRIA RocquencourtLe ChesnayFrance

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