A new stabilizing technique for boundary integral methods for water waves
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- by Thomas Y. Hou and Pingwen Zhang PDF
- Math. Comp. 70 (2001), 951-976 Request permission
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.References
- Gregory R. Baker, Daniel I. Meiron, and Steven A. Orszag, Generalized vortex methods for free-surface flow problems, J. Fluid Mech. 123 (1982), 477–501. MR 687014, DOI 10.1017/S0022112082003164
- Gregory Baker and André Nachbin, Stable methods for vortex sheet motion in the presence of surface tension, SIAM J. Sci. Comput. 19 (1998), no. 5, 1737–1766. MR 1617821, DOI 10.1137/S1064827595296562
- J. Thomas Beale, Thomas Y. Hou, and John Lowengrub, Convergence of a boundary integral method for water waves, SIAM J. Numer. Anal. 33 (1996), no. 5, 1797–1843. MR 1411850, DOI 10.1137/S0036142993245750
- Russel E. Caflisch and John S. Lowengrub, Convergence of the vortex method for vortex sheets, SIAM J. Numer. Anal. 26 (1989), no. 5, 1060–1080. MR 1014874, DOI 10.1137/0726059
- J. W. Dold, An efficient surface-integral algorithm applied to unsteady gravity waves, J. Comput. Phys. 103 (1992), no. 1, 90–115. MR 1188090, DOI 10.1016/0021-9991(92)90327-U
- Jonathan Goodman, Thomas Y. Hou, and John Lowengrub, Convergence of the point vortex method for the $2$-D Euler equations, Comm. Pure Appl. Math. 43 (1990), no. 3, 415–430. MR 1040146, DOI 10.1002/cpa.3160430305
- Thomas Y. Hou, Numerical solutions to free boundary problems, Acta numerica, 1995, Acta Numer., Cambridge Univ. Press, Cambridge, 1995, pp. 335–415. MR 1352474, DOI 10.1017/S0962492900002567
- Thomas Y. Hou, Zhen-huan Teng, and Pingwen Zhang, Well-posedness of linearized motion for $3$-D water waves far from equilibrium, Comm. Partial Differential Equations 21 (1996), no. 9-10, 1551–1585. MR 1410841, DOI 10.1080/03605309608821238
- A. I. Markushevich, Theory of functions of a complex variable. Vol. I, II, III, Second English edition, Chelsea Publishing Co., New York, 1977. Translated and edited by Richard A. Silverman. MR 0444912
- A.J. Roberts, A stable and accurate numerical method to calculate the motion of a sharp interface between fluids, I.M.A. J. Appl. Math 31, 13-35 (1983).
- L. Rosenhead, The point vortex approximation of a vortex sheet, Proc. Roy. Soc. London Ser. A, 134, 170-192 (1932).
- Gilbert Strang, Accurate partial difference methods. II. Non-linear problems, Numer. Math. 6 (1964), 37–46. MR 166942, DOI 10.1007/BF01386051
- G. B. Whitham, Linear and nonlinear waves, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0483954
- Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in $2$-D, Invent. Math. 130 (1997), no. 1, 39–72. MR 1471885, DOI 10.1007/s002220050177
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
- 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. - © Copyright 2000 American Mathematical Society
- Journal: Math. Comp. 70 (2001), 951-976
- MSC (2000): Primary 65M12, 76B15
- DOI: https://doi.org/10.1090/S0025-5718-00-01287-4
- MathSciNet review: 1826574