High Order Compact Mimetic Differences and Discrete Energy Decay in 2D Wave Motions

  • Jose E. CastilloEmail author
  • Guillermo Miranda
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)


Mimetic difference operators Div, Grad and Curl, have been constructed to provide a high order of accuracy in numerical schemes that mimic the properties of their corresponding continuum operators; hence they would be faithful to the physics. However, this faithfulness of the discrete basic operators might not be sufficient if the numerical difference scheme introduces some numerical energy increase, which would obviously result in a potentially unstable performance. We present a high order compact mimetic scheme for 2D wave motions and show that the energy of the system is also conserved in the discrete sense.


  1. 1.
    M. Abouali, J.E. Castillo, High-order compact Castillo-Grone’s operators. Report of Computational Science Research Center at San Diego State University. CSRCR02 1–13 (2012)Google Scholar
  2. 2.
    C. Bazan, M. Abouali, J. Castillo, P. Blomgren, Mimetic finite difference methods in image processing. Comput. Appl. Math. 30(3), 701–720 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    M. Carpenter, D. Gottlieb, S. Abarbanel, Stable and accurate boundary treatments for compact, high-order finite-difference schemes. Appl. Numer. Math. 12, 55–87 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    J.E. Castillo, R. Grone, A matrix analysis to high-order approximations for divergence and gradients satisfying a global conservation law. SIAM Matrix Anal. Appl. 25, 128–142 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    J.E. Castillo, G.F. Miranda, Mimetic Discretization Methods (CRC Press, West Palm Beach, 2013)CrossRefGoogle Scholar
  6. 6.
    J. Castillo, J. Hyman, M. Shashkov, S. Steinberg, Fourth and sixth order conservative finite difference approximations of the divergence and gradient. Appl. Numer. Math. 37(1–2), 171–187 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    L.J. Cordova, O. Rojas, B. Otero, J.E. Castillo, Compact finite difference modeling of 2-D acoustic wave propagation. J. Comput. Appl. Math. (2015). Available online 19 February 2015
  8. 8.
    K. Hoffmann, S. Chiang, Computational Fluid Dynamics, vol. 2, 4th ed. (Engineering Education System Book, Wichita, KS, 2000)Google Scholar
  9. 9.
    S. Lele, Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16–42 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    R. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations (SIAM, Philadelphia, 2007)CrossRefzbMATHGoogle Scholar
  11. 11.
    J. de la Puente, M. Ferrer, M. Hanzich, J.E. Castillo, J.M. Cela, Mimetic seismic wave modeling including topography on deformed staggered grids. Geophysics 79(3), T125-T141 (2014)CrossRefGoogle Scholar
  12. 12.
    O. Rojas, B. Otero, J.E. Castillo, S.M. Day, Low dispersive modeling of Rayleigh waves on partly-staggered grids. Int. J. Comput. Geom. Appl. 18(1), 29–43 (2014)MathSciNetGoogle Scholar
  13. 13.
    R. Aberayatne, Continuum Mechanics, vol. II (Massachusetts Institute of Technology, Cambridge, MA, 2012)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Computational Science Research CenterSan Diego State UniversitySan DiegoUSA

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