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Convergence of the point vortex method for the 3‐D euler equations
Author(s) -
Hou Thomas Y.,
Lowengrub John
Publication year - 1990
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160430803
Subject(s) - mathematics , euler equations , discretization , vortex , norm (philosophy) , mathematical analysis , vorticity , backward euler method , convergence (economics) , euler method , smoothing , compressibility , euler's formula , physics , political science , law , economics , thermodynamics , economic growth , statistics
We prove consistency, stability, and convergence of a point vortex approximation to the 3‐D incompressible Euler equations with smooth solutions. The 3‐D algorithm we consider here is similar to the corresponding 3‐D vortex blob algorithm introduced by Beale and Majda; see [3]. We first show that the discretization error is second‐order accurate. Then we show that the method is stable in l p norm for the particle trajectories and in w −1. p norm for discrete vorticity. Consequently, the method converges up to any time for which the Euler equations have a smooth solution. One immediate application of our convergence result is that the vortex filament method without smoothing also converges.