An implicit finite-difference operator for the Helmholtz equation

Chunlei Chu, Paul L. Stoffa

Research output: Contribution to journalArticlepeer-review

28 Scopus citations


We have developed an implicit finite-difference operator for the Laplacian and applied it to solving the Helmholtz equation for computing the seismic responses in the frequency domain. This implicit operator can greatly improve the accuracy of the simulation results without adding significant extra computational cost, compared with the corresponding conventional explicit finite-difference scheme. We achieved this by taking advantage of the inherently implicit nature of the Helmholtz equation and merging together the two linear systems: one from the implicit finite-difference discretization of the Laplacian and the other from the discretization of the Helmholtz equation itself. The end result of this simple yet important merging manipulation is a single linear system, similar to the one resulting from the conventional explicit finite-difference discretizations, without involving any differentiation matrix inversions. We analyzed grid dispersions of the discrete Helmholtz equation to show the accuracy of this implicit finite-difference operator and used two numerical examples to demonstrate its efficiency. Our method can be extended to solve other frequency domain wave simulation problems straightforwardly. © 2012 Society of Exploration Geophysicists.
Original languageEnglish (US)
Pages (from-to)T97-T107
Number of pages1
Issue number4
StatePublished - Jul 2012
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: Paul Stoffa would like to acknowledge the King Abdullah University of Science and Technology (KAUST) for their support of his research. We are grateful for the constructive comments from the reviewers and associate editor Jeffrey Shragge, which helped improve the original manuscript significantly. We thank ConocoPhillips for permission to publish this work.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.


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