In this paper we consider numerical approximations of hyperbolic conservation laws in the one-dimensional scalar case, by studying Godunov and van Leer’s methods. Before to present the numerical treatment of hyperbolic conservation laws, a theoretical introduction is given together with the definition of the Riemann problem. Next the numerical schemes are discussed. We also present numerical ex...

The balanced vehicular traffic model is a macroscopic model for vehicular traffic flow. We use this model to study the traffic dynamics at highway bottlenecks either caused by the restriction of the number of lanes or by on-ramps or off-ramps. The coupling conditions for the Riemann problem of the system are applied in order to treat the interface between different road sections consistently. O...

with a well defined notion of trace at x = 0. We allow the boundary x = 0 to have the same speed of one of the characteristic speed of the limiting hyperbolic system. This result is achieved by a new decomposition of the solution into travelling profiles and the non characteristic part of boundary profile, the analysis of the interaction of travelling profiles with the boundary profile, the con...

This paper analyzes the structural properties of the shock and rarefaction wave solutions of a macroscopic, second-order non-local continuum traffic flow model, namely Helbing’s model. We will show that this model has two families of characteristics for the shock wave solutions: one characteristic is slower, and the other one is faster than the average vehicle speed. Corresponding to the slower...

This paper is devoted to the coupling problem of two scalar conservation laws through a fixed interface located for instance at x=0. Each scalar conservation law is associated with its own (smooth) flux function and is posed on a half-space, namely x < 0 or x > 0. At interface x = 0 we impose a coupling condition whose objective is to enforce in a weak sense the continuity of a prescribed varia...

We consider a one-parameter family of nonstrictly hyperbolic systems of conservation laws modeling three-phase flow in a porous medium. For a particular value of the parameter, the model has a shock wave solution that undergoes several bifurcations upon perturbation of its left and right states and the parameter. In this paper we use singularity theory and bifurcation theory of dynamical system...

Using the ‘Riemann Problem with zeros’ method, Ward has constructed exact solutions to a (2 + 1)–dimensional integrable Chiral Model, which exhibit solitons with nontrivial scattering. We give a correspondence between what we conjecture to be all pure soliton solutions and certain holomorphic vector bundles on a compact surface. AMS Classification numbers Primary: 35Q51 Secondary: 58F07

Recently Aw, Rascle and Zhang introduced a second order model (ARZ) that does nor exhibit the usual drawbacks of this family of models, i.e. negative velocities and/or densities. In this paper we analyze the inhomogeous Riemann problem for this model, which is shown to be equivalent to the inhomogeous Riemann problem for a related first order model with modified equilibrium flow density relatio...

1 Introduction 1 2 Parabolic estimates 7 3 Outline of the BV estimates 12 4 A center manifold of viscous travelling waves 15 5 Gradient decomposition 19 6 Bounds on the source terms 23 7 Transversal wave interactions 32 8 Functionals related to shortening curves 35 9 Energy estimates 40 10 Proof of the BV estimates 45 11 Stability estimates 47 12 Propagation speed 55 13 The vanishing viscosity ...