Steady-state traffic flow on a ring road with up- and down- slopes

This paper studies steady-state traffic flow on a ring road with up- and down- slopes using a semi-discrete model. By exploiting the relations between the semi-discrete and the continuum models, a steady-state solution is uniquely determined for a given total number of vehicles on the ring road. The solution is exact and always stable with respect to the first-order continuum model, whereas it is a good approximation with respect to the semi-discrete model provided that the involved equilibrium constant states are linearly stable. In an otherwise case, the instability of one or more equilibria could trigger stop-and-go waves propagating in certain road sections or throughout the ring road. The indicated results are reasonable and thus physically significant for a better understanding of real traffic flow on an inhomogeneous road

On interface transmission conditions for conservation laws with discontinuous flux of general shape

International audienceConservation laws of the form $\partial_t u+ \partial_x f(x;u)=0$ with space-discontinuous flux $f(x;\cdot)=f_l(\cdot)\mathbf{1}_{x0}$ were deeply investigated in the last ten years, with a particular emphasis in the case where the fluxes are ''bell-shaped". In this paper, we introduce and exploit the idea of transmission maps for the interface condition at the discontinuity, leading to the well-posedness for the Cauchy problem with general shape of $f_{l,r}$. The design and the convergence of monotone Finite Volume schemes based on one-sided approximate Riemann solvers is then assessed. We conclude the paper by illustrating our approach by several examples coming from real-life applications

A phase-by-phase upstream scheme that converges to the vanishing capillarity solution for countercurrent two-phase flow in two-rocks media

International audienceWe discuss the convergence of the upstream phase-by-phase scheme (or upstream mobility scheme) towards the vanishing capillarity solution for immiscible incompressible two-phase flows in porous media made of several rock types. Troubles in the convergence where recently pointed out in [S. Mishra & J. Jaffré, Comput. Geosci., 2010] and [S. Tveit & I. Aavatsmark, Comput. Geosci, 2012]. In this paper, we clarify the notion of vanishing capillarity solution, stressing the fact that the physically relevant notion of solution differs from the one inferred from the results of [E. F. Kaasschieter, Comput. Geosci., 1999]. In particular, we point out that the vanishing capillarity solution de- pends on the formally neglected capillary pressure curves, as it was recently proven in by the authors [B. Andreianov & C. Canc'es, Comput. Geosci., 2013]. Then, we propose a numerical procedure based on the hybridization of the interfaces that converges towards the vanishing capillarity solution. Numerical illustrations are provided

On some partial differential equation models in socio-economic contexts - analysis and numerical simulations

This thesis deals with the analysis and numerical simulation of different partial differential equation models arising in socioeconomic sciences. It is divided into two parts: The first part deals with a mean-field price formation model introduced by Lasry and Lions in 2007. This model describes the dynamic behaviour of the price of a good being traded between a group of buyers and a group of vendors. Existence (locally in time) of smooth solutions is established, and obstructions to proving a global existence result are examined. Also, properties of a regularised version of the model are explored and numerical examples are shown. Furthermore, the possibility of reconstructing the initial datum given a number of observations, regarding the price and the transaction rate, is considered. Using a variational approach, the problem can be expressed as a non-linear constrained minimization problem. We show that the initial datum is uniquely determined by the price (identifiability). Furthermore, a numerical scheme is implemented and a variety of examples are presented. The second part of this thesis treats two different models describing the motion of (large) human crowds. For the first model, introduced by R.L. Hughes in 2002, several regularised versions are considered. Existence and uniqueness of entropy solutions are proven using the technique of vanishing viscosity. In one space dimension, the dynamic behaviour of solutions of the original model is explored for some special cases. These results are compared to numerical simulations. Moreover, we consider a discrete cellular automaton model introduced by A. Kirchner and A. Schadschneider in 2002. By (formally) passing to the continuum limit, we obtain a system of partial differential equations. Some analytical properties, such as linear stability of stationary states, are examined and extensive numerical simulations show capabilities and limitations of the model in both the discrete and continuous setting

A theory of L1L^1-dissipative solvers for scalar conservation laws with discontinuous flux

We propose a general framework for the study of $L^1$ contractive semigroups of solutions to conservation laws with discontinuous flux. Developing the ideas of a number of preceding works we claim that the whole admissibility issue is reduced to the selection of a family of "elementary solutions", which are certain piecewise constant stationary weak solutions. We refer to such a family as a "germ". It is well known that (CL) admits many different $L^1$ contractive semigroups, some of which reflects different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specific attention to the anishing viscosity" germ, which is a way to express the "$\Gamma$-condition" of Diehl. For any given germ, we formulate "germ-based" admissibility conditions in the form of a trace condition on the flux discontinuity line $x=0$ (in the spirit of Vol'pert) and in the form of a family of global entropy inequalities (following Kruzhkov and Carrillo). We characterize those germs that lead to the $L^1$-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes

Analyse mathématique et numérique d'équations aux dérivées partielles issues de la mécanique des fluides : applications aux écoulements en milieux poreux

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