A well posed conservation law with a variable unilateral constraint (original) (raw)

Conservation laws with discontinuous flux

2007

Abstract We consider an hyperbolic conservation law with discontinuous flux. Such partial differential equation arises in different applicative problems, in particular we are motivated by a model of traffic flow. We provide a new formulation in terms of Riemann Solvers. Moreover, we determine the class of Riemann Solvers which provide existence and uniqueness of the corresponding weak entropic solutions. Key-words: Conservation laws–discontinuous flux–Riemann Solvers–front tracking–traffic flow

Nonclassical shocks and the Cauchy problem: general conservation laws

Contemporary Mathematics, 1999

In this paper we establish the existence of nonclassical entropy solutions for the Cauchy problem associated with a conservation law having a nonconvex flux-function. Instead of the classical Oleinik entropy criterion, we use a single entropy inequality supplemented with a kinetic relation. We prove that these two conditions characterize a unique nonclassical Riemann solver. Then we apply the wave-front tracking method to the Cauchy problem. By introducing a new total variation functional, we can prove that the corresponding approximate solutions converge strongly to a nonclassical entropy solution.

Nonclassical shocks and the Cauchy problem for nonconvex conservation laws

1999

The Riemann problem for a conservation law with a nonconvex (cubic) flux can be solved in a class of admissible nonclassical solutions that may violate the Oleinik entropy condition but satisfy a single entropy inequality and a kinetic relation. We use such a nonclassical Riemann solver in a front tracking algorithm, and prove that the approximate solutions remain bounded in the total variation norm.

Mathematical and numerical study of a system of conservation laws

Journal of Evolution Equations, 2007

The system of equations (f (u))t − (a(u)v + b(u)) x = 0 and ut − (c(u)v + d(u)) x = 0, where the unknowns u and v are functions depending on (x, t) ∈ R × R+, arises within the study of some physical model of the flow of miscible fluids in a porous medium. We give a definition for a weak entropy solution (u, v), inspired by the Liu condition for admissible shocks and by Krushkov entropy pairs. We then prove, in the case of a natural generalization of the Riemann problem, the existence of a weak entropy solution only depending on x/t. This property results from the proof of the existence, by passing to the limit on some approximations, of a function g such that u is the classical entropy solution of ut − ((cg + d)(u))x = 0 and simultaneously w = f (u) is the entropy solution of wt − ((ag + b)(f (−1) (w)))x = 0. We then take v = g(u), and the proof that (u, v) is a weak entropy solution of the coupled problem follows from a linear combination of the weak entropy inequalities satisfied by u and f (u). We then show the existence of an entropy weak solution for a general class of data, thanks to the convergence proof of a coupled finite volume scheme. The principle of this scheme is to compute the Godunov numerical flux with some interface functions ensuring the symmetry of the finite volume scheme with respect to both conservation equations.

Non classical solution of a conservation law arising in vehicular traffic modelling

ESAIM, 2016

We are interested in this paper in the modelling and numerical simulation of some phenomena that are observed in the context of car dynamics, in particular the appearance of persistent jams upstream critical points with no real cause of flux limitation. We shall consider the case of a stable jam on a freeway upstream an accident that took place on the opposite lane. This situation is not properly handled by most models, either micro-or macroscopic ones, since it corresponds to a phenomenon that does not have a counterpart in gas dynamics, for which only entropy solution are usually considered as physically feasible. The approach we propose consists in accounting for the very behaviour of agents in the neighbourhood of the discontinuity, and makes it possible to numerically recover in a robust way steady traffic jams. Résumé. Nous nous intéressons dans cet article à la modélisation et à la simulation numérique de phénomènes particuliers observés dans le contexte du trafic routier, plus particulièrement le phénomène d'apparition et de persistance de bouchons en amont de points critiques sans cause objective de ralentissement. Nous considérerons notamment le cas d'un bouchon persistant sur une autoroute en amont d'un accident qui s'est produit sur la voie d'en face. Cette situation n'est pas reproduite par la plupart des modèles, qu'ils soient microscopiques ou macroscopiques, car elle correspond à un phénomène qui n'a pas d'équivalent en dynamique des gaz, pour lesquels seules les solutions dites entropiques sont en général considérées comme correspondant à un comportement observable dans la réalité. L'approche que nous proposons, aux niveaux microscopique et macroscopique, consiste à prendre en compte le comportement particulier des agents humains au voisinage de la discontinuité, et permet de retrouver numériquement de façon robuste des bouchons stables.

Well-Posedness for Scalar Conservation Laws with Moving Flux Constraints

SIAM Journal on Applied Mathematics, 2019

We consider a strongly coupled ODE-PDE system representing moving bottlenecks immersed in vehicular traffic. The PDE consists of a scalar conservation law modeling the traffic flow evolution and the ODE models the trajectory of a slow moving vehicle. The moving bottleneck influences the bulk traffic flow via a point flux constraint, which is given by an inequality on the flux at the slow vehicle position. We prove uniqueness and continuous dependence of solutions with respect to initial data of bounded variation. The proof is based on a new backward in time method established to capture the values of the norm of generalized tangent vectors at every time.