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

Full-text PDF Free Access

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.

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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

DOI:
http://dx.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