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

DOI:
https://doi.org/10.1090/S0025-5718-00-01287-4

Published electronically:
October 18, 2000

MathSciNet review:
1826574

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0025-5718-00-01287-4

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