Journal ArticleDOI
Convergence of the point vortex method for the 2-D euler equations
TLDR
This work proves consistency, stability and convergence of the point vortex approximation to the 2-D incompressible Euler equations with smooth solutions to be stable in l p norm for all time.Abstract:
We prove consistency, stability and convergence of the point vortex approximation to the 2-D incompressible Euler equations with smooth solutions. We first show that the discretization error is second-order accurate. Then we show that the method is stable in l p norm. Consequently the method converge in l p norm for all time. The convergence is also illustrated by a numerical experimentread more
Citations
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Journal ArticleDOI
Quadrature by expansion
TL;DR: This paper presents a systematic, high-order approach that works for any singularity (including hypersingular kernels), based only on the assumption that the field induced by the integral operator is locally smooth when restricted to either the interior or the exterior.
Book ChapterDOI
On the Dynamics of Large Particle Systems in the Mean Field Limit
TL;DR: In this paper, the authors explain how the usual mean field evolution partial differential equations (PDEs) in Statistical Physics can be rigorously derived from the fundamental microscopic equations that govern the evolution of large, interacting particle systems.
Book ChapterDOI
The derivation of Swarming models: Mean-Field Limit and Wasserstein distances
TL;DR: A summary of the kinetic models derived as continuum versions of second order models for swarming, focusing on the question of passing from the discrete to the continuum model in the Dobrushin framework, gives qualitative bounds on the approximation of initial data by particles to obtain the mean-field limit of large ensembles of interacting particles.
Book ChapterDOI
The derivation of swarming models: Mean-field limit and Wasserstein distances
TL;DR: In this paper, a summary on the mean field limit of large ensembles of interacting particles with applications in swarming models is given, along with qualitative bounds on the approximation of initial data by particles to obtain the mean-field limit for radial singular potentials up to the Newtonian singularity.
Journal ArticleDOI
A review of the mean field limits for Vlasov equations
TL;DR: In this paper, the authors review some classical and more recent results on the mean field limit and propagation of chaos for systems of many particles, leading to Vlasov or macroscopic equations.
References
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Book
Introduction to partial differential equations
TL;DR: The Introduction to Partial Differential Equations (IDEQE) as discussed by the authors is the most widely used partial differential equation (PDE) formalism for algebraic partial differential equations.
Journal ArticleDOI
Numerical study of slightly viscous flow
TL;DR: In this paper, a numerical method for solving the time-dependent Navier-Stokes equations in two space dimensions at high Reynolds number is presented, where the crux of the method lies in the numerical simulation of the process of vorticity generation and dispersal, using computer-generated pseudo-random numbers.
Journal ArticleDOI
Vortex methods for flow simulation
TL;DR: Recent progress in the development of vortex methods and their applications to the numerical simulation of incompressible fluid flows are reviewed in this article, with a focus on recent results concerning the accuracy of these methods, improvements in computational efficiency, and development of three-dimensional vortex methods.
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