A convergent boundary integral method for three-dimensional water waves

Author:
J. Thomas Beale

Journal:
Math. Comp. **70** (2001), 977-1029

MSC (2000):
Primary 65M12, 76B15; Secondary 65D30

Published electronically:
February 17, 2000

MathSciNet review:
1709144

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

We design a boundary integral method for time-dependent, three-dimensional, doubly periodic water waves and prove that it converges with accuracy, without restriction on amplitude. The moving surface is represented by grid points which are transported according to a computed velocity. An integral equation arising from potential theory is solved for the normal velocity. A new method is developed for the integration of singular integrals, in which the Green's function is regularized and an efficient local correction to the trapezoidal rule is computed. The sums replacing the singular integrals are treated as discrete versions of pseudodifferential operators and are shown to have mapping properties like the exact operators. The scheme is designed so that the error is governed by evolution equations which mimic the structure of the original problem, and in this way stability can be assured. The wavelike character of the exact equations of motion depends on the positivity of the operator which assigns to a function on the surface the normal derivative of its harmonic extension; similarly, the stability of the scheme depends on maintaining this property for the discrete operator. With grid points, the scheme can be implemented with essentially operations per time step.

**[1]**Christopher Anderson and Claude Greengard,*On vortex methods*, SIAM J. Numer. Anal.**22**(1985), no. 3, 413–440. MR**787568**, 10.1137/0722025**[2]**V. K. Andreev,*Ustoichivost neustanovivshikhsya dvizhenii zhidkosti so svobodnoi granitsei*, VO “Nauka”, Novosibirsk, 1992 (Russian, with English and Russian summaries). MR**1244798****[3]**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**, 10.1017/S0022112082003164**[4]**Gregory R. Baker, Daniel I. Meiron, and Steven A. Orszag,*Boundary integral methods for axisymmetric and three-dimensional Rayleigh-Taylor instability problems*, Phys. D**12**(1984), no. 1-3, 19–31. MR**762803**, 10.1016/0167-2789(84)90511-6**[5]**J. Thomas Beale, Thomas Y. Hou, and John S. Lowengrub,*Growth rates for the linearized motion of fluid interfaces away from equilibrium*, Comm. Pure Appl. Math.**46**(1993), no. 9, 1269–1301. MR**1231428**, 10.1002/cpa.3160460903**[6]**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**, 10.1137/S0036142993245750**[7]**J. Thomas Beale and Andrew Majda,*Vortex methods. I. Convergence in three dimensions*, Math. Comp.**39**(1982), no. 159, 1–27. MR**658212**, 10.1090/S0025-5718-1982-0658212-5**[8]**J. T. Beale and A. Majda,*High order accurate vortex methods with explicit velocity kernels,*J. Comput. Phys.**58**(1985), 188-208.**[9]**J. Broeze, E. F. G. Van Daalen, and P. J. Zandbergen,*A three-dimensional panel method for nonlinear free surface waves on vector computers,*Comput. Mech.**13**(1993), 12-28.**[10]**David L. Colton and Rainer Kress,*Integral equation methods in scattering theory*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR**700400****[11]**G.-H. Cottet and P.-A. Raviart,*Particle methods for the one-dimensional Vlasov-Poisson equations*, SIAM J. Numer. Anal.**21**(1984), no. 1, 52–76. MR**731212**, 10.1137/0721003**[12]**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**, 10.1002/cpa.3160430305**[13]**I. S. Gradshteyn and I. M. Ryzhik,*Table of integrals, series, and products*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. Corrected and enlarged edition edited by Alan Jeffrey; Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V. Geronimus] and M. Yu. Tseytlin [M. Yu. Tseĭtlin]; Translated from the Russian. MR**582453****[14]**Leslie Greengard and Vladimir Rokhlin,*A new version of the fast multipole method for the Laplace equation in three dimensions*, Acta numerica, 1997, Acta Numer., vol. 6, Cambridge Univ. Press, Cambridge, 1997, pp. 229–269. MR**1489257**, 10.1017/S0962492900002725**[15]**Bertil Gustafsson, Heinz-Otto Kreiss, and Joseph Oliger,*Time dependent problems and difference methods*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication. MR**1377057****[16]**Ole H. Hald,*Convergence of vortex methods for Euler’s equations. II*, SIAM J. Numer. Anal.**16**(1979), no. 5, 726–755. MR**543965**, 10.1137/0716055**[17]**David J. Haroldsen and Daniel I. Meiron,*Numerical calculation of three-dimensional interfacial potential flows using the point vortex method*, SIAM J. Sci. Comput.**20**(1998), no. 2, 648–683 (electronic). MR**1642620**, 10.1137/S1064827596302060**[18]**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**, 10.1080/03605309608821238**[19]**T. Y. Hou and P. Zhang,*Stability of a boundary integral method for 3-D water waves*, submitted to SIAM J. Numer. Anal.**[20]**R. E. Kleinman and G. F. Roach,*Boundary integral equations for the three-dimensional Helmholtz equation*, SIAM Rev.**16**(1974), 214–236. MR**0380087****[21]**M. S. Longuet-Higgins and E. D. Cokelet,*The deformation of steep surface waves on water. I. A numerical method of computation*, Proc. Roy. Soc. London Ser. A**350**(1976), no. 1660, 1–26. MR**0411355****[22]**J. S. Lowengrub, M. J. Shelley, and B. Merriman,*High-order and efficient methods for the vorticity formulation of the Euler equations*, SIAM J. Sci. Comput.**14**(1993), no. 5, 1107–1142. MR**1232178**, 10.1137/0914067**[23]**J. N. Lyness,*An error functional expansion for 𝑁-dimensional quadrature with an integrand function singular at a point*, Math. Comp.**30**(1976), no. 133, 1–23. MR**0408211**, 10.1090/S0025-5718-1976-0408211-0**[24]**J. N. Lyness and Ronald Cools,*A survey of numerical cubature over triangles*, Mathematics of Computation 1943–1993: a half-century of computational mathematics (Vancouver, BC, 1993) Proc. Sympos. Appl. Math., vol. 48, Amer. Math. Soc., Providence, RI, 1994, pp. 127–150. MR**1314845**, 10.1090/psapm/048/1314845**[25]**Andrew Majda, James McDonough, and Stanley Osher,*The Fourier method for nonsmooth initial data*, Math. Comp.**32**(1978), no. 144, 1041–1081. MR**501995**, 10.1090/S0025-5718-1978-0501995-4**[26]**Robert D. Richtmyer and K. W. Morton,*Difference methods for initial-value problems*, Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR**0220455****[27]**J. E. Romate,*The numerical simulation of nonlinear gravity waves*, Engrng. Analysis Bdry. Elts.,**7**(1990), 156-166.**[28]**J. E. Romate and P. J. Zandbergen*Boundary integral equation formulations for free-surface flow problems in two and three dimensions*, Comput. Mech.**4**(1989), 267-282.**[29]**John Strain,*Fast potential theory. II. Layer potentials and discrete sums*, J. Comput. Phys.**99**(1992), no. 2, 251–270. MR**1158209**, 10.1016/0021-9991(92)90206-E**[30]**Michael E. Taylor,*Pseudodifferential operators*, Princeton Mathematical Series, vol. 34, Princeton University Press, Princeton, N.J., 1981. MR**618463****[31]**M. Taylor,*Partial Differential Equations*, Springer, New York, 1996. MR**98b:35002****[32]**Wu-ting Tsai and Dick K. P. Yue,*Computation of nonlinear free-surface flows*, Annual review of fluid mechanics, Vol. 28, Annual Reviews, Palo Alto, CA, 1996, pp. 249–278. MR**1371167****[33]**T. Vinje and P. Brevig,*Numerical simulation of breaking waves*, Adv. Water Resources**4**(1981), 77-82.**[34]**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**, 10.1007/s002220050177**[35]**Sijue Wu,*Well-posedness in Sobolev spaces of the full water wave problem in 3-D*, J. Amer. Math. Soc.**12**(1999), 445-495. CMP**99:08**

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Additional Information

**J. Thomas Beale**

Affiliation:
Department of Mathematics, Duke University, Durham, NC 27708-0320

Email:
beale@math.duke.edu

DOI:
https://doi.org/10.1090/S0025-5718-00-01218-7

Keywords:
Water waves,
boundary integral methods,
integral operators,
quadrature of singular integrals

Received by editor(s):
September 9, 1998

Received by editor(s) in revised form:
June 10, 1999

Published electronically:
February 17, 2000

Additional Notes:
The author was supported in part by NSF Grant #DMS-9870091.

Article copyright:
© Copyright 2000
American Mathematical Society