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Time Approximation

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Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations

Part of the book series: Scientific Computation ((SCIENTCOMP))

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Abstract

This chapter deals with time approximation for the methods described in this book, including local time-stepping. The first part is devoted to the schemes with a constant time-step adapted to wave phenomena. After describing their construction, stability analysis is studied by using both plane wave analysis and energy techniques. Moreover, a link with the approximation of unbounded domains is studied. The second part introduces three efficient local time-stepping schemes which can be used in order to speed up the methods.

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Notes

  1. 1.

    Actually, for a second order operator, for any complex \(\omega _{h}\) involved in (7.23), \(\overline{\omega _h}\) is also involved. This implies that a complex value of \(\omega _h\) always produces exponentially growing solutions.

  2. 2.

    Abbreviation of Courant–Friedrichs–Lewy condition.

  3. 3.

    It is a structure associated to a manifold (whose the cotangent bundle is the phase space of the Hamiltonian system) which is defined by a skew-symmetric closed non-degenerate differential 2-form.

References

  1. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations. Series in Computational Mathematics. Springer, Heidelberg (1991)

    Book  MATH  Google Scholar 

  2. Dablain, M.A.: The application of high order differencing for the scalar wave equation. Geophysics 51(1), 54–66 (1986)

    Article  Google Scholar 

  3. Cohen, G., Joly, P.: Construction and analysis of fourth-order finite difference schemes for the acoustic wave equation in inhomogeneous media. SIAM J. Numer. Anal. 33(4), 1266–1302 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Shubin, G.R., Bell, J.B.: A modified equation approach to constructing fourth order methods for acoustic wave propagation. SIAM J. Sci. Comput. 8(2), 135–151 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Tal-Ezer, H.: Spectral methods in time for hyperbolic equations. SIAM J. Numer. Anal. 23(1), 11–26 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cohen, G.: High Order Numerical Methods for Transient Wave Equations. Scientific Computation. Springer, Heidelberg (2001)

    Google Scholar 

  7. Montseny, E., Pernet, S., Ferrires, X., Cohen, G.: Dissipative terms and local time-stepping improvements in a spatial high order Discontinuous Galerkin scheme for the time-domain Maxwell’s equations. J. Comput. Phys. 227(14), 6795–6820 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chevalier, M., Luebbers, R., Cable, V.: FDTD local grid with material traverse. IEEE Trans. Antennas Propag. 45(3), 411–421 (1997)

    Article  Google Scholar 

  9. Kunz, K.S., Simpson, L.: A technique for increasing the resolution of finite difference solutions of the Maxwell equation. IEEE Trans. Electromagn. Compat. 23(4), 419–422 (1981)

    Article  Google Scholar 

  10. Kim, I., Hoefer, W.J.R.: A local mesh refinement algorithm for the time domain finite difference method using Maxwell’s curl equations. IEEE Trans. Microw. Theory Tech. 38(6), 812–815 (1990). Jun

    Google Scholar 

  11. Prescott, D., Shuley, N.: A method for incorporating different sized cells into the finite-difference time-domain analysis technique. IEEE Microw. Guided Wave Lett. 2(11), 434–436 (1992). Nov

    Google Scholar 

  12. Dumbser, M., Käser, M., Toro, E.F.: An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - V. Local time stepping and p-adaptivity. Geophys. J. Int. 171(2), 695–717 (2007)

    Article  Google Scholar 

  13. Piperno, S.: Symplectic local time-stepping in non-dissipative DGTD methods applied to wave propagation problems. ESAIM Math. Model. Numer. Anal. 40(5), 815–841 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Collino, F., Fouquet, T., Joly, P.: Conservative space-time mesh refinement methods for the FDTD solution of Maxwell’s equations. J. Comput. Phys. 211(1), 9–35 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bécache, E., Joly, P., Rodriguez, J.: Space-time mesh refinement for elastodynamics. Numerical results. Comput. Methods Appl. Mech. Eng. 194, 355–366 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Rodriguez, J.: A spurious-free space-time mesh refinement for elastodynamics. Int. J. Multiscale Comput. Eng. 6(3), 263–279 (2008)

    Article  Google Scholar 

  17. Ezziani, A., Joly, P.: Space-time mesh refinement for discontunuous Galerkin methods for symmetric hyperbolic systems. J. Comput. Appl. Math. 234(6), 1886–1895 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rodriguez, J.: Raffinement de Maillage Spatio-Temporel pour les Equations de l’Elastodynamique, thèse de doctorat, U. Paris-Dauphine (2004)

    Google Scholar 

  19. Joly, P., Rodriguez, J.: An error analysis of conservative space-time mesh refinement methods for the 1D wave equation. SIAM J. Numer. Anal. 43(2), 825–859 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Diaz, J., Grote, M.J.: Energy conserving explicit local time stepping for second-order wave equations. SIAM J. Sci. Comput. 31(3), 1985–2014 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins, Baltimore (1996)

    MATH  Google Scholar 

  22. Dudouit, Y.: Spatio-temporal refinement using a discontinuous Galerkin approach for elastodynamic in a high performance computing framework, thèse de doctorat, U. de Bordeaux (2014)

    Google Scholar 

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Authors and Affiliations

  1. Inria, Saclay, France

    Gary Cohen

  2. The French Aerospace Lab, ONERA, Toulouse, France

    Sébastien Pernet

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  1. Gary Cohen
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  2. Sébastien Pernet
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Correspondence to Gary Cohen .

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Cohen, G., Pernet, S. (2017). Time Approximation. In: Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations. Scientific Computation. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-7761-2_7

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  • DOI: https://doi.org/10.1007/978-94-017-7761-2_7

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  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-017-7759-9

  • Online ISBN: 978-94-017-7761-2

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