Analyse d’une équation de vitesse de diffusion

Lacombe, Gilles

doi:10.1016/s0764-4442(00)88610-3

Cited by 11 publications

(7 citation statements)

References 4 publications

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“…The proof follows the arguments of Lacombe and MasGallic [25] for the diffusion-velocity transport equation (see also [24]). Here, however, we improve the result of [25] by observing that the resulting solution has the same regularity as the initial data.…”

Section: The Dispersion-velocity Method: Linear Problemsmentioning

confidence: 72%

“…Particles carrying fixed masses will be then convected with speed a(u). The convergence properties of the diffusion-velocity method were investigated, e.g., in [24,25], where short time existence and uniqueness of solutions to the resulting diffusion-velocity transport equation were proved. The diffusion-velocity method serves as the basic tool for the derivation of our particle methods in the dispersive world.…”

Section: Introductionmentioning

confidence: 99%

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Particle Methods for Dispersive Equations

Chertock¹,

Levy²

2001

Journal of Computational Physics

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We introduce a new dispersion-velocity particle method for approximating solutions of linear and nonlinear dispersive equations. This is the first time in which particle methods are being used for solving such equations. Our method is based on an extension of the diffusion-velocity method of Degond and Mustieles (SIAM J. Sci. Stat. Comput. 11(2), 293 (1990)) to the dispersive framework. The main analytical result we provide is the short time existence and uniqueness of a solution to the resulting dispersion-velocity transport equation. We numerically test our new method for a variety of linear and nonlinear problems. In particular we are interested in nonlinear equations which generate structures that have nonsmooth fronts. Our simulations show that this particle method is capable of capturing the nonlinear regime of a compacton-compacton type interaction.

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Section: The Dispersion-velocity Method: Linear Problemsmentioning

confidence: 72%

Section: Introductionmentioning

confidence: 99%

Particle Methods for Dispersive Equations

Chertock¹,

Levy²

2001

Journal of Computational Physics

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“…Indeed, Equation can be written as where it is readily seen that the i ‐th component of v ( X , t ) defined as provides a form similar to Equation . The diffusion velocity v has a term divided by ψ , but there is no hidden difficulties. Indeed, if ψ represents a density of probability, that is, a quantity of probability divided by a given volume, a particle cannot have a density equal to zero because it cannot have a mass equal to zero (as shown in Section 3.3); nor can it occupy an infinite volume.…”

Section: Sph Formulation Of Fpk Equationmentioning

confidence: 99%

Transient Fokker–Planck–Kolmogorov equation solved with smoothed particle hydrodynamics method

Canor

Denoël

2013

Numerical Meth Engineering

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Probabilistic theories aim at describing the properties of systems subjected to random excitations by means of statistical characteristics such as the probability density function ψ (pdf). The time evolution of the pdf of the response of a randomly excited deterministic system is commonly described with the transient Fokker-Planck-Kolmogorov equation (FPK). The FPK equation is a conservation equation of a hypothetical or abstract fluid, which models the transport of probability. This paper presents a generalized formalism for the resolution of the transient FPK equation using the well-known mesh-free Lagrangian method, Smoothed Particle Hydrodynamics (SPH).Numerical implementation shows notable advantages of this method in an unbounded state space: (i) the conservation of total probability in the state space is explicitly written, (ii) no artifact is required to manage far-field boundary conditions , (iii) the positivity of the pdf is ensured and (iv) the extension to higher dimensions is straightforward.Furthermore, thanks to the moving particles, this method is adapted for a large kind of initial conditions, even slightly dispersed distributions. The FPK equation is solved without any a priori knowledge of the stationary distribution; just a precise representation of the initial distribution is required.

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“…It was first introduced by Fronteau & Combis [1] in 1984 and popularized by Degond & Mustieles [2,3], Ogami & Akamatsu [4] and Kempka & Strickland [5] in the early 1990s. This method was then largely analysed [6][7][8][9][10], adapted to dispersion equations [10][11][12], coupled with turbulence models [13,14] and extended to the diffusion of a vector field and to axisymmetric flows [15][16][17].…”

Section: Introductionmentioning

confidence: 99%

A self-regularising DVM–PSE method for the modelling of diffusion in particle methods

Mycek

Pinon

Germain

et al. 2013

Comptes Rendus. Mécanique

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The modelling of diffusive terms in particle methods is a delicate matter and several models have been proposed in the literature. The Diffusion Velocity Method (DVM) consists in rewriting these terms in an advective way, thus defining a so-called diffusion velocity. In addition to the actual velocity, it is used to compute the particles displacement. On the other hand, the well-known and commonly used Particle Strength Exchange method (PSE) uses an approximation of the Laplacian operator in order to model diffusion. This approximation is based on an exchange of particles strength. Although DVM is particularly well suited to particle methods since it preserves their Lagrangian aspect, its major drawback stems in the fact that it suffers from severe singular behaviours. This paper intends to give insights and ideas for coping with these issues, based on an exact decomposition of the diffusion coefficient allowing a hybrid DVM-PSE treatment of diffusive terms.

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Analyse d’une équation de vitesse de diffusion

Cited by 11 publications

References 4 publications

Particle Methods for Dispersive Equations

Particle Methods for Dispersive Equations

Transient Fokker–Planck–Kolmogorov equation solved with smoothed particle hydrodynamics method

A self-regularising DVM–PSE method for the modelling of diffusion in particle methods

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