Almost optimal convergence

of the point vortex method

for vortex sheets using numerical filtering

Authors:
Russel E. Caflisch, Thomas Y. Hou and John Lowengrub

Journal:
Math. Comp. **68** (1999), 1465-1496

MSC (1991):
Primary 65M25; Secondary 76C05

DOI:
https://doi.org/10.1090/S0025-5718-99-01108-4

Published electronically:
May 21, 1999

MathSciNet review:
1651744

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Standard numerical methods for the Birkhoff-Rott equation for a vortex sheet are unstable due to the amplification of roundoff error by the Kelvin-Helmholtz instability. A nonlinear filtering method was used by Krasny to eliminate this spurious growth of round-off error and accurately compute the Birkhoff-Rott solution essentially up to the time it becomes singular. In this paper convergence is proved for the discretized Birkhoff-Rott equation with Krasny filtering and simulated roundoff error. The convergence is proved for a time almost up to the singularity time of the continuous solution. The proof is in an analytic function class and uses a discrete form of the abstract Cauchy-Kowalewski theorem. In order for the proof to work almost up to the singularity time, the linear and nonlinear parts of the equation, as well as the effects of Krasny filtering, are precisely estimated. The technique of proof applies directly to other ill-posed problems such as Rayleigh-Taylor unstable interfaces in incompressible, inviscid, and irrotational fluids, as well as to Saffman-Taylor unstable interfaces in Hele-Shaw cells.

**1.**G. R. Baker, Russel E. Caflisch, and Michael Siegel,*Singularity formation during Rayleigh-Taylor instability*, Journal of Fluid Mechanics**252**(1993), 51-78. MR**94e:76034****2.**J. T. Beale, T. Y. Hou, and J. Lowengrub,*Growth rates for the linearized motion of fluid interfaces away from equilibrium*, Comm. Pure Appl. Math.**46**(1993), 1269-1301. MR**95c:76016****3.**-,*On the well-posedness of two fluid interfacial flows with surface tension*, in*Singularities in Fluids, Plasmas and Optics*(R. C. Caflisch and G. Papanicolau, editors), NATO ASI Series, Kluwer, 1993, pp. 11-38.**4.**-,*Convergence of boundary integral methods for water waves*, SIAM J. Num. Anal.**33**(1996), 1797-1843. MR**89b:76009****5.**R. Caflisch and O. Orellana,*Long time existence for a slightly perturbed vortex sheet*, Comm. Pure Appl. Math.**39**(1986), 807-838. MR**87m:76018****6.**-,*Singular solutions and ill-posedness for the evolution of vortex sheets*, SIAM J. Math. Anal.**20**(1989), 293-307. MR**90d:76020****7.**R. E. Caflisch,*A simplified version of the abstract Cauchy-Kowalewski Theorem with weak singularities*, Bull. Amer. Math. Soc.**23**(1990), 495-500. MR**91f:35008****8.**Russel E. Caflisch and John S. Lowengrub,*Convergence of the vortex method for vortex sheets*, SIAM J. Numer. Anal.**26**(1989), 1060-1080. MR**91g:76073****9.**S. J. Cowley, G. R. Baker, and S. Tanveer,*On the formation of Moore curvature singularities in vortex sheets*, J. Fluid Mech., submitted.**10.**W. S. Dai and M. J. Shelley,*A numerical study of the effect of surface tension and noise on an expanding Hele-Shaw bubble*, Phys. Fluids A**5**(1993), pp. 1465-1496.**11.**A. J. DeGregoria and L. W. Schwartz,*A boundary integral method for two phase displacement in Hele-Shaw cells*, J. Fluid Mech.**164**(1986), 383-400. MR**87h:76073****12.**J. Duchon and R. Robert,*Global vortex sheet solutions of Euler equations in the plane*, J. Diff. Eqn.**73**(1988), 215-224. MR**89h:35262****13.**H. Dym and H. McKean,*Fourier Series and Integrals*, Academic Press, New York, 1972. MR**56:945****14.**T. Y. Hou, J. Lowengrub, and R. Krasny,*Convergence of a point vortex method for vortex sheets*, SIAM J. Numer. Anal.**28**(1991), 308-320. MR**92d:65153****15.**R. Krasny,*A study of singularity formation in a vortex sheet by the point-vortex approximation*, J. Fluid Mechanics**167**(1986), 65-93. MR**87g:76028****16.**D. I. Meiron, G. R. Baker, and S. A. Orszag,*Analytic structure of vortex sheet dynamics*,*part*1,*Kelvin-Helmholtz Instability*, J. Fluid Mech.**114**(1982), 283-298. MR**83c:76017****17.**D. W. Moore,*The spontaneous appearance of a singularity in the shape of an evolving vortex sheet*, Proc. Roy. Soc. London A**365**(1979), 105-119. MR**80b:76006****18.**L. Nirenberg,*An abstract form of the nonlinear Cauchy-Kowalewski theorem*, J. Diff. Geom.**6**(1972), 561-576. MR**48:683****19.**T. Nishida,*A note on a theorem of Nirenberg*, J. Diff. Geom.**12**(1977), 629-633. MR**80a:58013****20.**P. G. Saffman and G. I. Taylor,*The penetration of a fluid into a porous medium of a Hele-Shaw cell containing a more viscous fluid*, Proc. Royal Soc. London Ser. A**245**(1958), 312-329. MR**20:3697****21.**M. V. Safonov,*The abstract Cauchy-Kovalevskaya theorem in a weighted Banach space*, Comm. Pure Appl. Math.**48**(1995), 629-637. MR**96g:35002****22.**M. J. Shelley,*A study of singularity formation in vortex sheet motion by a spectrally accurate vortex method*, J. Fluid Mech.**244**(1992), 493-526. MR**93g:73065****23.**C. Sulem, P. L. Sulem, C. Bardos, and U. Frisch,*Finite time analyticity for the two and three dimensional Kelvin-Helmholtz instability*, Comm. Math. Phys.**80**(1981), 485-516. MR**83d:76012****24.**G. Tryggvason,*Numerical simulations of the Rayleigh-Taylor problem*, J. Computational Phys.**75**(1988), 253.**25.**G. Tryggvason and H. Aref,*Numerical experiments on Hele-Shaw flows with a sharp interface*, J. Fluid Mech.**136**(1983), 1-30.

Retrieve articles in *Mathematics of Computation of the American Mathematical Society*
with MSC (1991):
65M25,
76C05

Retrieve articles in all journals with MSC (1991): 65M25, 76C05

Additional Information

**Russel E. Caflisch**

Affiliation:
Department of Mathematics, UCLA, Box 951555, Los Angeles, California 90095-1555

**Thomas Y. Hou**

Affiliation:
Department of Applied Mathematics, California Institute of Technology, Pasadena, California 91125

**John Lowengrub**

Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Address at time of publication:
Department of Mathematics, University of North Carolina, Phillips Hall, Chapel Hill, North Carolina 27599

DOI:
https://doi.org/10.1090/S0025-5718-99-01108-4

Keywords:
Vortex sheets,
point vortices,
numerical filtering,
discrete Cauchy-Kowalewski theorem

Received by editor(s):
December 16, 1997

Published electronically:
May 21, 1999

Additional Notes:
The first author’s research was supported in part by the Army Research Office under grants #DAAL03-91-G-0162 and #DAAH04-95-1-0155, the second author’s by ONR Grant N00014-96-1-0438 and NSF Grant DMS-9704976, and the third author’s by the McKnight Foundation, the National Science Foundation, the Sloan Foundation, the Department of Energy, and the University of Minnesota Supercomputer Institute

Article copyright:
© Copyright 1999
American Mathematical Society