Vortex methods. II. Higher order accuracy in two and three dimensions

Beale, J. Thomas; Majda, Andrew J.

doi:10.1090/s0025-5718-1982-0658213-7

Cited by 106 publications

(132 citation statements)

References 11 publications

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“…Some of the notable results for the vortex blob methods are due to Hald [13], Beale and Majda [2], and Cottet [8], and the results for the PVM are due to Goodman, Hou, and Lowengrub [12]. For the discontinuous flows, we refer to Beale [1], Brenier and Cottet [4], Caflisch and Lowengrub [5], Liu and Xin [16] and Schochet [22].…”

Section: Introductionmentioning

confidence: 99%

Convergence of the point vortex method for 2-D vortex sheet

Liu

Xin

2000

Math. Comp.

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Abstract. We give an elementary proof of the convergence of the point vortex method (PVM) to a classical weak solution for the two-dimensional incompressible Euler equations with initial vorticity being a finite Radon measure of distinguished sign and the initial velocity of locally bounded energy. This includes the important example of vortex sheets, which exhibits the classical Kelvin-Helmholtz instability. A surprise fact is that although the velocity fields generated by the point vortex method do not have bounded local kinetic energy, the limiting velocity field is shown to have a bounded local kinetic energy.

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Section: Introductionmentioning

confidence: 99%

Convergence of the point vortex method for 2-D vortex sheet

Liu

Xin

2000

Math. Comp.

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“…In fact our techniques also yield stable, convergent vortex methods of arbitrary accuracy in 2-D and answer a variety of questions raised by Hald [10] and Leonard [16]. We postpone a detailed discussion of the 2-D algorithms until Part II of this work [1].…”

mentioning

confidence: 56%

Untitled

1982

Math. Comp.

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Abstract. Recently several different approaches have been developed for the simulation of three-dimensional incompressible fluid flows using vortex methods. Some versions use detailed tracking of vortex filament structures and often local curvatures of these filaments, while other methods require only crude information, such as the vortex blobs of the two-dimensional case. Can such "crude" algorithms accurately account for vortex stretching and converge? We answer this question affirmatively by constructing a new class of "crude" three-dimensional vortex methods and then proving that these methods are stable and convergent, and can even have arbitrarily high order accuracy without being more expensive than other "crude" versions of the vortex algorithm.1. Introduction. Vortex methods have provided an attractive and successful approach for the numerical simulation of incompressible fluid flow at high Reynolds number in two dimensions. They have several distinctive features: (1) the interactions of the computational vortices mimic the physical mechanisms in actual fluid flow; (2) vortex methods are automatically adaptive, since the vortex "blobs" concentrate in the regions of physical interest; and (3) there are no inherent errors which behave like the numerical viscosity of conventional Eulerian difference methods. Such diffusive errors can swamp the effects of physical viscosity in high Reynolds number flow similation.The earliest attempts to represent flows by a vortex method used point vortices, and thus permitted arbitrarily large local velocities (Rosenhead [20]). Chorin [3] and Kuwahara and Takami [24] introduced the idea of using a finite vortex core, rather than a point vortex, to stabilize the method. Subsequently there have been a large

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“…One can formulate the existence in law of a diffusion process solution of a stochastic differential equation like (3), in terms of a martingale problem. We introduce the measurable space (C([0, +∞); R), B(C([0, +∞); R))) and now we denote X the canonical process on C([0, +∞); R).…”

Section: The Associated Martingale Problemmentioning

confidence: 99%

“…We introduce the measurable space (C([0, +∞); R), B(C([0, +∞); R))) and now we denote X the canonical process on C([0, +∞); R). Finding a solution in law to Equation (3) consists in finding a probability measure P on (C([0, +∞); R), B(C([0, +∞); R))) such that the canonical process (X t , t ≥ 0) is a solution of (3). In this formulation, the Brownian motion and the probability space (Ω,F,P ) on which it lives, must be specified in terms of (C([0, +∞); R), B(C([0, +∞); R)), P ) (see e.g.…”

Section: The Associated Martingale Problemmentioning

confidence: 99%

Some stochastic particle methods for nonlinear parabolic PDEs

Bossy

2005

ESAIM: Proc.

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Abstract. We introduce, on some examples, the main tools to analyze the convergence rate of stochastic particle methods for the numerical approximation of some nonlinear parabolic PDEs. We treat the case of Lipschitz coefficients as well as the case of viscous scalar conservation laws. We also discuss some particular aspects like the case of a bounded spatial domain or the use of a Romberg extrapolation as a speed up procedure.Résumé. On introduit, sur certains exemples, les outils principaux pour l'analyse du taux de convergence des méthodes particulaires stochastiques pour l'approximation numérique de quelques EDPs paraboliques non-linéaires. On traite le cas de coefficients lipschitziens, ainsi que le cas des lois de conservation scalaires. On discute quelques aspects particuliers comme le cas d'un domaine spatial borné ou l'utilisation de l'extrapolation de Romberg comme procédé d'accélération.

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Vortex methods. II. Higher order accuracy in two and three dimensions

Cited by 106 publications

References 11 publications

Convergence of the point vortex method for 2-D vortex sheet

Convergence of the point vortex method for 2-D vortex sheet

Untitled

Some stochastic particle methods for nonlinear parabolic PDEs

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