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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

A new stabilizing technique for boundary integral methods for water waves


Authors: Thomas Y. Hou and Pingwen Zhang
Journal: Math. Comp. 70 (2001), 951-976
MSC (2000): Primary 65M12, 76B15
Published electronically: October 18, 2000
MathSciNet review: 1826574
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Abstract:

Boundary integral methods to compute interfacial flows are very sensitive to numerical instabilities. A previous stability analysis by Beale, Hou and Lowengrub reveals that a very delicate balance among terms with singular integrals and derivatives must be preserved at the discrete level in order to maintain numerical stability. Such balance can be preserved by applying suitable numerical filtering at certain places of the discretization. While this filtering technique is effective for two-dimensional (2-D) periodic fluid interfaces, it does not apply to nonperiodic fluid interfaces. Moreover, using the filtering technique alone does not seem to be sufficient to stabilize 3-D fluid interfaces.

Here we introduce a new stabilizing technique for boundary integral methods for water waves which applies to nonperiodic and 3-D interfaces. A stabilizing term is added to the boundary integral method which exactly cancels the destabilizing term produced by the point vortex method approximation to the leading order. This modified boundary integral method still has the same order of accuracy as the point vortex method. A detailed stability analysis is presented for the point vortex method for 2-D water waves. The effect of various stabilizing terms is illustrated through careful numerical experiments.


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Additional Information

Thomas Y. Hou
Affiliation: Applied Mathematics, California Institute of Technology, Pasadena, California 91125
Email: hou@ama.caltech.edu

Pingwen Zhang
Affiliation: School of Mathematical Science, Peking University, Beijing 100871, China
Email: pzhang@sxx0.math.pku.edu.cn

DOI: http://dx.doi.org/10.1090/S0025-5718-00-01287-4
PII: S 0025-5718(00)01287-4
Keywords: Boundary integral method, stability, water waves
Received by editor(s): August 14, 1998
Received by editor(s) in revised form: August 5, 1999
Published electronically: October 18, 2000
Additional Notes: The first author was partially supported by the National Science Foundation under grant DMS-9704976, by the Office of Naval Research under grant N00014-96-1-0438, and by the Army Research Office under grant DAAD19-99-1-0141.
The second author was partially supported by the National Science Foundation under grant DMS-9704976 and by the Office of Naval Research under grant N00014-96-1-0438. He was also supported by the National Natural Science Foundation of China and the China State Major Key Project for Basic Research.
Article copyright: © Copyright 2000 American Mathematical Society



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