Asymptotic problems and numerical schemes for traffic flows with unilateral constraints describing the formation of jams
Florent Berthelin Thierry Goudon Bastien Polizzi Magali Ribot
Networks & Heterogeneous Media 2017, 12(4): 591-617 doi: 10.3934/nhm.2017024

We discuss numerical strategies to deal with PDE systems describing traffic flows, taking into account a density threshold, which restricts the vehicle density in the situation of congestion. These models are obtained through asymptotic arguments. Hence, we are interested in the simulation of approached models that contain stiff terms and large speeds of propagation. We design schemes intended to apply with relaxed stability conditions.

keywords: Traffic flows unilateral constraints finite volume schemes asymptotic preserving schemes
Non--local macroscopic models based on Gaussian closures for the Spizer-Härm regime
Thierry Goudon Martin Parisot
Kinetic & Related Models 2011, 4(3): 735-766 doi: 10.3934/krm.2011.4.735
The Spitzer-Härm regime arising in plasma physics leads asymptotically to a nonlinear diffusion equation for the electron temperature. In this work we propose a hierarchy of models intended to retain more features of the underlying modeling based on kinetic equations. These models are of non--local type. Nevertheless, owing to energy discretization they can lead to coupled systems of diffusion equations. We make the connection between the different models precise and bring out some mathematical properties of the models. A numerical scheme is designed for the approximate models, and simulations validate the proposed approach.
keywords: Plasma physics hydrodynamic limits.
On a generalized Boltzmann equation for non-classical particle transport
Martin Frank Thierry Goudon
Kinetic & Related Models 2010, 3(3): 395-407 doi: 10.3934/krm.2010.3.395
We are interested in non-standard transport equations where the description of the scattering events involves an additional "memory variable''. We establish the well posedness and investigate the diffusion asymptotics of such models. While the questions we address are quite classical the analysis is original since the usual dissipative properties of collisional transport equations is broken by the introduction of the memory terms.
keywords: Transport equations Diffusion asymptotics Radiative transfer.
Low field regime for the relativistic Vlasov-Maxwell-Fokker-Planck system; the one and one half dimensional case
Mihai Bostan Thierry Goudon
Kinetic & Related Models 2008, 1(1): 139-170 doi: 10.3934/krm.2008.1.139
We study the asymptotic regime for the relativistic Vlasov-Maxwell-Fokker-Planck system which corresponds to a mean free path small compared to the Debye length, chosen as an observation length scale, combined to a large thermal velocity assumption. We are led to a convection-diffusion equation, where the convection velocity is obtained by solving a Poisson equation. The analysis is performed in the one and one half dimensional case and the proof combines dissipation mechanisms and finite speed of propagation properties.
keywords: Diffusion approximation. Asymptotic behavior Vlasov-Maxwell-Fokker-Planck system
A model describing the growth and the size distribution of multiple metastatic tumors
Anne Devys Thierry Goudon Pauline Lafitte
Discrete & Continuous Dynamical Systems - B 2009, 12(4): 731-767 doi: 10.3934/dcdsb.2009.12.731
Cancer is one of the greatest killers in the world, particularly in western countries. A lot of the effort of the medical research is devoted to cancer and mathematical modeling must be considered as an additional tool for the physicians and biologists to understand cancer mechanisms and to determine the adapted treatments. Metastases make all the seriousness of cancer. In 2000, Iwata et al. [9] proposed a model which describes the evolution of an untreated metastatic tumors population. We provide here a mathematical analysis of this model which brings us to the determination of a Malthusian rate characterizing the exponential growth of the population. We provide as well a numerical analysis of the PDE given by the model.
keywords: asymptotic behavior McKendrick-Von Foerster equation WENO scheme relative entropy metastatic tumors.
The Lifschitz-Slyozov equation with space-diffusion of monomers
Thierry Goudon Frédéric Lagoutière Léon M. Tine
Kinetic & Related Models 2012, 5(2): 325-355 doi: 10.3934/krm.2012.5.325
The Lifschitz--Slyozov system describes the dynamics of mass exchanges between macro--particles and monomers in the theory of coarsening. We consider a variant of the classical model where monomers are subject to space diffusion. We establish the existence--uniqueness of solutions for a wide class of relevant data and kinetic coefficients. We also derive a numerical scheme to simulate the behavior of the solutions.
keywords: Lifschitz-Slyozov equation theory of coarsening PDEs of hybrid type finite-volume finite-difference schemes. existence--uniqueness of solutions

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