Solution of Wave Equation in Rods Using the Wavelet-Galerkin Method for Space Discretization

  • Rodrigo B. BurgosEmail author
  • Marco A. Cetale Santos
  • Raul R. e Silva
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)


The use of multiresolution techniques and wavelets has become increasingly popular in the development of numerical schemes for the solution of partial differential equations (PDEs) in the last three decades. Therefore, the use of wavelets scale functions as a basis in computational analysis holds some promise due to their compact support, orthogonality, localization and multiresolution properties. The present work discusses an alternative to the usual finite difference (FDM) approach to the acoustic wave equation modeling by using a space discretization scheme based on the Galerkin Method. The combination of this method with wavelet analysis using scale functions results in the Wavelet Galerkin Method (WGM) which has been adapted for the direct solution of the wave differential equation in a meshless formulation. This paper presents an extension of previous works which dealt with linear elasticity problems. This work also introduces Deslauriers-Dubuc scaling functions (also known as Interpolets) as interpolating functions in a Galerkin approach considering wave propagation problems. Examples in 1-D were formulated using a central difference (second order) scheme for time differentiation. Encouraging results were obtained when compared with the FDM using the same time steps. The main improvement in the presented formulation was the recognition of a different dispersion pattern when comparing FDM and WGM results using the same space and time grid.



Authors would like to thank CNPq, PETROBRAS and FAPERJ for their financial support.


  1. 1.
    K.R. Kelly, R.W. Ward, S. Treitel, R.M. Alford, Synthetic seismograms: a finite-difference approach. Geophysics 41, 2–27 (1976)CrossRefGoogle Scholar
  2. 2.
    S. Qian, J. Weiss, Wavelets and the numerical solution of partial differential equations. J. Comput. Phys. 106, 155–175 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    X. Chen, S. Yang, J. Ma, Z. He, The construction of wavelet finite element and its application. Finite Elem. Anal. Des. 40, 541–554 (2004)CrossRefGoogle Scholar
  4. 4.
    G. Deslauriers, S. Dubuc, Symmetric iterative interpolation processes. Constr. Approx. 5, 49–68 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    I. Daubechies, Orthonormal bases of compactly supported wavelets. Commun Pure Appl Math 41, 909–996 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    R.B. Burgos, M.A. Cetale Santos, R.R. Silva, Analysis of beams and thin plates using the Wavelet-Galerkin method. Int J Eng Technol 7, 261–266 (2015)CrossRefGoogle Scholar
  7. 7.
    A.J.M. Ferreira, L.M. Castro, S. Bertoluzza, Analysis of plates on Winkler foundation by wavelet collocation. Meccanica 46(4), 865–873 (2011)CrossRefzbMATHGoogle Scholar
  8. 8.
    Z. Shi, D.J. Kouri, G.W. Wei, D.K. Hoffman, Generalized symmetric interpolating wavelets. Comput. Phys. Commun. 119, 194–218 (1999)CrossRefzbMATHGoogle Scholar
  9. 9.
    X. Du, J.C. Bancroft, in Proceedings of the SEG Int’l Exposition and 74th Annual Meeting, 2-D Wave Equation Modeling and Migration By a New Finite Difference Scheme Based on the Galerkin Method, (Denver, USA, 2004)Google Scholar
  10. 10.
    X. Zhou, W. Zhang, The evaluation of connection coefficients on an interval. Commun Nonlinear Sci Numer Simul 3, 252–255 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    R.B. Burgos, M.A. Cetale Santos, R.R. Silva, Deslauriers-Dubuc interpolating wavelet beam finite element. Finite Elem. Anal. Des. 75, 71–77 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    V.P. Nguyen, T. Rabczuk, S. Bordas, M. Duflot, Meshless methods: a review and computer implementation aspects. Math. Comput. Simul. 79, 763–813 (2008)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Rodrigo B. Burgos
    • 1
    Email author
  • Marco A. Cetale Santos
    • 2
  • Raul R. e Silva
    • 3
  1. 1.Department of Structures and FoundationsUERJRio de JaneiroBrazil
  2. 2.Department of Geology and GeophysicsUFFNiteróiBrazil
  3. 3.Department of Civil EngineeringPUC-RioRio de JaneiroBrazil

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