Sticky Particles and Scalar Conservation Laws

Brenier, Yann; Grenier, Emmanuel

doi:10.1137/s0036142997317353

Cited by 305 publications

(289 citation statements)

References 10 publications

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“…Solution of (10) [30]. Equation (10) must be considered in a weak sense: we look for t → U t , probability measure on…”

Section: Particle Methods For the Mckean-valsov Modelmentioning

confidence: 99%

“…where P s = P • X −1 s and where we impose Moreover, there exists a unique solution pathwise (given X 0 ∈ L 2 (Ω) and W ) and in law, to the stochastic differential equation, Taking the expectation in (12), it is not difficult to conclude that t → P t is a weak solution of the McKeanVlasov equation (10), where P t are the time marginals of the unique solution P of (MP).…”

Section: Particle Methods For the Mckean-valsov Modelmentioning

confidence: 99%

“…The propagation of chaos suggests that 1/N N i=1 δ X i t approximates the solution U t of Equation (10). To get a simulation procedure for a trajectory of each (X i t ), we discretize in time: ∆t > 0 and K ∈ N are chosen such that T = K∆t; the discrete times are denoted by t k := k∆t, with 1 ≤ k ≤ K. The Euler scheme leads to the following discrete-time system:…”

Section: Numerical Algorithmmentioning

confidence: 99%

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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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“…Solution of (10) [30]. Equation (10) must be considered in a weak sense: we look for t → U t , probability measure on…”

Section: Particle Methods For the Mckean-valsov Modelmentioning

confidence: 99%

Section: Particle Methods For the Mckean-valsov Modelmentioning

confidence: 99%

Section: Numerical Algorithmmentioning

confidence: 99%

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Some stochastic particle methods for nonlinear parabolic PDEs

Bossy

2005

ESAIM: Proc.

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“…Let us remark that the system (7) for the spray contains the pressureless gas dynamics system which has been studied by several authors [2,4]. Indeed, this system can be written:…”

Section: Pressureless Gas Dynamicsmentioning

confidence: 99%

A high order moment method simulating evaporation and advection of a polydisperse liquid spray

Kah

Laurent

Massot

et al. 2012

Journal of Computational Physics

109

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In this paper, we tackle the modeling and numerical simulation of sprays and aerosols, that is dilute gasdroplet flows for which polydispersity description is of paramount importance. Starting from a kinetic description for point particles experiencing transport either at the carrier phase velocity for aerosols or at their own velocity for sprays as well as evaporation, we focus on an Eulerian high order moment method in size and consider a system of partial differential equations (PDEs) on a vector of successive integer size moments of order 0 to N , N > 2, over a compact size interval. There exists a stumbling block for the usual approaches using high order moment methods resolved with high order finite volume methods: the transport algorithm does not preserve the moment space. Indeed, reconstruction of moments by polynomials inside computational cells coupled to the evolution algorithm can create N -dimensional vectors which fail to be moment vectors: it is impossible to find a size distribution for which there are the moments. We thus propose a new approach as well as an algorithm which is second order in space and time with very limited numerical diffusion and allows to accurately describe the advection process and naturally preserves the moment space. The algorithm also leads to a natural coupling with a recently designed algorithm for evaporation which also preserves the moment space; thus polydispersity is accounted for in the evaporation and advection process, very accurately and at a very reasonable computational cost. These modeling and algorithmic tools are referred to as the EMSM (Eulerian Multi Size Moment) model. We show that such an approach is very competitive compared to multi-fluid approaches, where the size phase space is discretized into several sections and low order moment methods are used in each section, as well as with other existing high order moment methods. An accuracy study assesses the order of the method as well as the low level of numerical diffusion on structured meshes. Whereas the extension to unstructured meshes is provided, we focus in this paper on cartesian meshes and two 2D test-cases are presented: Taylor-Green vortices and turbulent free jets, where the accuracy and efficiency of the approach are assessed.

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“…They also can be obtained from Boltzmann equations [2] and the flux-splitting scheme of the full compressible Euler equations [3,4]. System (1.1) is used to model the motion of free particles which stick under collision [5] and the formation of large-scale structures in the universe [6,7].…”

Section: Introductionmentioning

confidence: 99%

Vanishing viscosity limit for Riemann solutions to zero-pressure gas dynamics with flux perturbation

Zhang

Junchao

2018

Bound Value Probl

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In this paper, by the viscosity vanishing approach, we consider the Riemann problem for zero-pressure gas dynamics with flux perturbation. The Riemann solutions involve parameterized delta shock wave and constant density state. For the parameterized delta-shock solution, its generalized Rankine-Hugoniot and entropy condition are clarified. While for the constant density solution, the formation of it is rigorously analyzed. Moreover, all of their existence, uniqueness, and stability to reasonable viscous perturbations are shown.

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Sticky Particles and Scalar Conservation Laws

Abstract: One-dimensional scalar conservation laws with nondecreasing initial conditions and general fluxes are shown to be the appropriate equations to describe large systems of free particles on the real line, which stick under collision with conservation of mass and momentum.

Cited by 305 publications

References 10 publications

Some stochastic particle methods for nonlinear parabolic PDEs

Some stochastic particle methods for nonlinear parabolic PDEs

A high order moment method simulating evaporation and advection of a polydisperse liquid spray

Vanishing viscosity limit for Riemann solutions to zero-pressure gas dynamics with flux perturbation

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