Almost optimal convergence of the point vortex method for vortex sheets using numerical filtering

Russel Caflisch, Thomas Y. Hou, John Lowengrub

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)1465-1496
Number of pages32
JournalMathematics of Computation
Volume68
Issue number228
StatePublished - Oct 1999

Fingerprint

Vortex Method
Vortex Sheet
Point Vortex
Rounding error
Vortex flow
Filtering
Unstable
Nonlinear filtering
Singularity
Kelvin-Helmholtz Instability
Cauchy's integral theorem
Hele-Shaw
Nonlinear Filtering
Amplification
Continuous Solution
Numerical methods
Ill-posed Problem
Rayleigh
Analytic function
Eliminate

Keywords

  • Discrete Cauchy-Kowalewski theorem
  • Numerical filtering
  • Point vortices
  • Vortex sheets

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

Cite this

Almost optimal convergence of the point vortex method for vortex sheets using numerical filtering. / Caflisch, Russel; Hou, Thomas Y.; Lowengrub, John.

In: Mathematics of Computation, Vol. 68, No. 228, 10.1999, p. 1465-1496.

Research output: Contribution to journalArticle

Caflisch, Russel ; Hou, Thomas Y. ; Lowengrub, John. / Almost optimal convergence of the point vortex method for vortex sheets using numerical filtering. In: Mathematics of Computation. 1999 ; Vol. 68, No. 228. pp. 1465-1496.
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