Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2016, Article ID 1719846, 16 pages
http://dx.doi.org/10.1155/2016/1719846
Research Article

Optimal 25-Point Finite-Difference Subgridding Techniques for the 2D Helmholtz Equation

1School of Mathematical Sciences, Shandong Normal University, Jinan 250014, China
2Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou 510275, China
3Department of Mathematics, Foshan University, Foshan 528000, China

Received 3 December 2015; Revised 20 February 2016; Accepted 3 March 2016

Academic Editor: Yan-Wu Wang

Copyright © 2016 Tingting Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. C.-H. Jo, C. Shin, and J. H. Suh, “An optimal 9-point, finite-difference, frequency-space, 2-D scalar wave extrapolator,” Geophysics, vol. 61, no. 2, pp. 529–537, 1996. View at Publisher · View at Google Scholar · View at Scopus
  2. C. Shin and H. Sohn, “A frequency-space 2-D scalar wave extrapolator using extended 25-point finite-difference operator,” Geophysics, vol. 63, no. 1, pp. 289–296, 1998. View at Publisher · View at Google Scholar · View at Scopus
  3. A. Deraemaeker, I. Babuška, and P. Bouillard, “Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions,” International Journal for Numerical Methods in Engineering, vol. 46, no. 4, pp. 471–499, 1999. View at Publisher · View at Google Scholar · View at Scopus
  4. R. G. Pratt, C. Shin, and G. J. Hicks, “Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion,” Geophysical Journal International, vol. 133, no. 2, pp. 341–362, 1998. View at Publisher · View at Google Scholar · View at Scopus
  5. R. G. Pratt and M. H. Worthington, “Inverse theory applied to multi-source cross-hole tomography. Part 1: acoustic wave-equation method,” Geophysical Prospecting, vol. 38, no. 3, pp. 287–310, 1990. View at Publisher · View at Google Scholar · View at Scopus
  6. J.-P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” Journal of Computational Physics, vol. 114, no. 2, pp. 185–200, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. S. Tsynkov and E. Turkel, “A Cartesian perfectly matched layer for the Helmholtz equation,” in Absorbing Boundaries and Layers, Domain Decomposition Methods, pp. 279–309, Nova Science Publishers, Huntington, NY, USA, 2001. View at Google Scholar
  8. J. W. Kang and L. F. Kallivokas, “Mixed unsplit-field perfectly matched layers for transient simulations of scalar waves in heterogeneous domains,” Computational Geosciences, vol. 14, no. 4, pp. 623–648, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. J. W. Kang and L. F. Kallivokas, “The inverse medium problem in heterogeneous PML-truncated domains using scalar probing waves,” Computer Methods in Applied Mechanics and Engineering, vol. 200, no. 1–4, pp. 265–283, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. Y. A. Erlangga, C. W. Oosterlee, and C. Vuik, “A novel multigrid based preconditioner for heterogeneous Helmholtz problems,” SIAM Journal on Scientific Computing, vol. 27, no. 4, pp. 1471–1492, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. B. Hustedt, S. Operto, and J. Virieux, “Mixed-grid and staggered-grid finite-difference methods for frequency-domain acoustic wave modelling,” Geophysical Journal International, vol. 157, no. 3, pp. 1269–1296, 2004. View at Publisher · View at Google Scholar · View at Scopus
  12. G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, Clarendon Press, 1985.
  13. J. W. Thomas, Numerical Partial Differential Equations, Finite Difference Methods, Springer, New York, NY, USA, 1995.
  14. Y. Li, L. Sun, and W. Hong, “Helmholtz equation sub-grid method for multi-transition region,” in Proceedings of the National Microwave Meeting, pp. 1150–1152, Qingdao, China, 2011.
  15. P. Moczo, E. Bystrický, J. Kristek, J. M. Carcione, and M. Bouchon, “Hybrid modeling of P-SV seismic motion at inhomogeneous viscoelastic topographic structures,” Bulletin of the Seismological Society of America, vol. 87, no. 5, pp. 1305–1323, 1997. View at Google Scholar · View at Scopus
  16. J. W. Nehrbass and R. Lee, “Optimal finite-difference sub-gridding techniques applied to the Helmholtz equation,” IEEE Transactions on Microwave Theory and Techniques, vol. 48, no. 6, pp. 976–984, 2000. View at Publisher · View at Google Scholar · View at Scopus
  17. S. Wang, F. L. Teixeira, R. Lee, and J.-F. Lee, “Optimization of subgridding schemes for FDTD,” IEEE Microwave and Wireless Components Letters, vol. 12, no. 6, pp. 223–225, 2002. View at Publisher · View at Google Scholar · View at Scopus
  18. B. Donderici and F. L. Teixeira, “Improved FDTD subgridding algorithms via digital filtering and domain overriding,” IEEE Transactions on Antennas and Propagation, vol. 53, no. 9, pp. 2938–2951, 2005. View at Publisher · View at Google Scholar
  19. J. Kristek, P. Moczo, and M. Galis, “Stable discontinuous staggered grid in the finite-difference modelling of seismic motion,” Geophysical Journal International, vol. 183, no. 3, pp. 1401–1407, 2010. View at Publisher · View at Google Scholar · View at Scopus
  20. D. T. Prescott and N. V. Shuley, “A method for incorporating different sized cells into the finite-difference time-domain analysis technique,” IEEE Microwave and Guided Wave Letters, vol. 2, no. 11, pp. 434–436, 1992. View at Publisher · View at Google Scholar · View at Scopus
  21. T. Wu and Z. Chen, “A dispersion minimizing subgridding finite difference scheme for the Helmholtz equation with PML,” Journal of Computational and Applied Mathematics, vol. 267, pp. 82–95, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. Z. Chen, D. Cheng, W. Feng, and T. Wu, “An optimal 9-point finite difference scheme for the Helmholtz equation with PML,” International Journal of Numerical Analysis and Modeling, vol. 10, no. 2, pp. 389–410, 2013. View at Google Scholar · View at MathSciNet · View at Scopus
  23. Z. Chen, D. Cheng, and T. Wu, “A dispersion minimizing finite difference scheme and preconditioned solver for the 3D Helmholtz equation,” Journal of Computational Physics, vol. 231, no. 24, pp. 8152–8175, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  24. Z. Chen, T. Wu, and H. Yang, “An optimal 25-point finite difference scheme for the Helmholtz equation with PML,” Journal of Computational and Applied Mathematics, vol. 236, no. 6, pp. 1240–1258, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. J. Coyle, “Locating the support of objects contained in a two-layered background medium in two dimensions,” Inverse Problems, vol. 16, no. 2, pp. 275–292, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. H. Kaneko and Y. Xu, “Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind,” Mathematics of Computation, vol. 62, no. 206, pp. 739–753, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet