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Mathematics of Computation

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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
Published electronically: May 21, 1999
MathSciNet review: 1651744
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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.

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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

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