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), 14651496
MSC (1991):
Primary 65M25; Secondary 76C05
Published electronically:
May 21, 1999
MathSciNet review:
1651744
Fulltext PDF Free Access
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Abstract: Standard numerical methods for the BirkhoffRott equation for a vortex sheet are unstable due to the amplification of roundoff error by the KelvinHelmholtz instability. A nonlinear filtering method was used by Krasny to eliminate this spurious growth of roundoff error and accurately compute the BirkhoffRott solution essentially up to the time it becomes singular. In this paper convergence is proved for the discretized BirkhoffRott 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 CauchyKowalewski 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 illposed problems such as RayleighTaylor unstable interfaces in incompressible, inviscid, and irrotational fluids, as well as to SaffmanTaylor unstable interfaces in HeleShaw cells.
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Additional Information
Russel E. Caflisch
Affiliation:
Department of Mathematics, UCLA, Box 951555, Los Angeles, California 900951555
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/S0025571899011084
PII:
S 00255718(99)011084
Keywords:
Vortex sheets,
point vortices,
numerical filtering,
discrete CauchyKowalewski 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 #DAAL0391G0162 and #DAAH049510155, the second author’s by ONR Grant N000149610438 and NSF Grant DMS9704976, 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
