### 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 language | English (US) |
---|---|

Pages (from-to) | 1465-1496 |

Number of pages | 32 |

Journal | Mathematics of Computation |

Volume | 68 |

Issue number | 228 |

State | Published - Oct 1999 |

### Fingerprint

### 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

*Mathematics of Computation*,

*68*(228), 1465-1496.

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

Research output: Contribution to journal › Article

*Mathematics of Computation*, vol. 68, no. 228, pp. 1465-1496.

}

TY - JOUR

T1 - Almost optimal convergence of the point vortex method for vortex sheets using numerical filtering

AU - Caflisch, Russel

AU - Hou, Thomas Y.

AU - Lowengrub, John

PY - 1999/10

Y1 - 1999/10

N2 - 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.

AB - 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.

KW - Discrete Cauchy-Kowalewski theorem

KW - Numerical filtering

KW - Point vortices

KW - Vortex sheets

UR - http://www.scopus.com/inward/record.url?scp=0033426460&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033426460&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033426460

VL - 68

SP - 1465

EP - 1496

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 228

ER -