Skip to main content
SpringerLink
Log in

A Finite Difference Method with Optimized Dispersion Correction for the Helmholtz Equation

  • Conference paper
  • First Online:
Domain Decomposition Methods in Science and Engineering XXIV (DD 2017)

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 125))

Included in the following conference series:

Abstract

We propose a new finite difference method (FDM) with optimized dispersion correction for the Helmholtz equation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    We could also use different norms leading to different optimized dispersion corrections.

References

  1. I.M. Babuska, S.A. Sauter, Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM J. Numer. Anal. 34(6), 2392–2423 (1997)

    Article  MathSciNet  Google Scholar 

  2. I. Babuška, F. Ihlenburg, E.T. Paik, S.A. Sauter, A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution. Comput. Methods Appl. Mech. Eng. 128(3–4), 325–359 (1995)

    Article  MathSciNet  Google Scholar 

  3. Z. Chen, D. Cheng, T. Wu, A dispersion minimizing finite difference scheme and preconditioned solver for the 3d Helmholtz equation. J. Comput. Phys. 231(24), 8152–8175 (2012)

    Article  MathSciNet  Google Scholar 

  4. D. Cheng, X. Tan, T. Zeng, A dispersion minimizing finite difference scheme for the Helmholtz equation based on point-weighting. Comput. Math. Appl. 314, 618–646 (2017)

    MathSciNet  MATH  Google Scholar 

  5. O.G. Ernst, M.J. Gander, Multigrid methods for Helmholtz problems: a convergent scheme in 1D using standard components, in Direct and Inverse Problems in Wave Propagation and Applications, ed. by I. Graham, U. Langer, M. Melenk, M. Sini (DeGruyter, 2013), pp. 135–186.

    Google Scholar 

  6. F. Ihlenburg, I. Babuška, Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation. Int. J. Numer. Methods Eng. 38(22), 3745–3774 (1995)

    Article  MathSciNet  Google Scholar 

  7. F. Ihlenburg, I. Babuška, Finite element solution of the Helmholtz equation with high wave number part i: the h-version of the FEM. Comput. Math. Appl. 30(9), 9–37 (1995)

    Article  MathSciNet  Google Scholar 

  8. C.-H. Jo, C. Shin, J.H. Suh, An optimal 9-point, finite-difference, frequency-space, 2-d scalar wave extrapolator. Geophysics 61(2), 529–537 (1996)

    Article  Google Scholar 

  9. C. Shin, H. Sohn, A frequency-space 2-d scalar wave extrapolator using extended 25-point finite-difference operator. Geophysics 63(1), 289–296 (1998)

    Article  Google Scholar 

  10. C.C. Stolk, A dispersion minimizing scheme for the 3-d Helmholtz equation based on ray theory. J. Comput. Phys. 314, 618–646 (2016)

    Article  MathSciNet  Google Scholar 

  11. C.C. Stolk, M. Ahmed, S.K. Bhowmik, A multigrid method for the Helmholtz equation with optimized coarse grid corrections. SIAM J. Sci. Comput. 36(6), A2819–A2841 (2014)

    Article  MathSciNet  Google Scholar 

  12. E. Turkel, D. Gordon, R. Gordon, S. Tsynkov, Compact 2d and 3d sixth order schemes for the Helmholtz equation with variable wave number. J. Comput. Phys. 232(1), 272–287 (2013)

    Article  MathSciNet  Google Scholar 

  13. T. Wu, A dispersion minimizing compact finite difference scheme for the 2d Helmholtz equation. J. Comput. Appl. Math. 311, 497–512 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université de la Réunion, PIMENT, Sainte-Clotilde, Reunion, France

    Pierre-Henri Cocquet

  2. Université de Genève, Genève, Switzerland

    Martin J. Gander

  3. Qian Xuesen Laboratory of Space Technology, Beijing, China

    Xueshuang Xiang

Authors
  1. Pierre-Henri Cocquet
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin J. Gander
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xueshuang Xiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Pierre-Henri Cocquet , Martin J. Gander or Xueshuang Xiang .

Editor information

Editors and Affiliations

  1. Department of Informatics, University of Bergen, Bergen, Norway

    Petter E. Bjørstad

  2. Department of Mathematics, Louisana State University, Baton Rouge, LA, USA

    Susanne C. Brenner

  3. Laboratoire Analyse, Géométrie et Applications, Université Paris 13, Villetaneuse, France

    Lawrence Halpern

  4. Department of Applied Mathematics, Kyung Hee University, Yongin, Korea (Republic of)

    Hyea Hyun Kim

  5. FB Mathematik und Informatik, Freie Universität Berlin, Berlin, Germany

    Ralf Kornhuber

  6. Western Norway University of Applied Sciences, Bergen, Norway

    Talal Rahman

  7. Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

    Olof B. Widlund

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Cocquet, PH., Gander, M.J., Xiang, X. (2018). A Finite Difference Method with Optimized Dispersion Correction for the Helmholtz Equation. In: Bjørstad, P., et al. Domain Decomposition Methods in Science and Engineering XXIV . DD 2017. Lecture Notes in Computational Science and Engineering, vol 125. Springer, Cham. https://doi.org/10.1007/978-3-319-93873-8_18

Download citation

Publish with us

Policies and ethics

Search

Navigation