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 ...

This paper concerns 2 2 systems of conservation laws with quadratic uxes corresponding to Case II in the classiication of such problems. Aided by a computer program , we have constructed the solution that satisses the viscous proole criterion for shock admissibility. Our solution diiers from that obtained using the Lax admissibility criterion, even though solutions exist and are unique for both...

We develop and analyze an improved version of the Multi-Stage (MUSTA) approach to the construction of upwind Godunov-type fluxes whereby the solution of the Riemann problem, approximate or exact, is not required. The new MUSTA schemes improve upon the original schemes in terms of monotonicity properties, accuracy and stability in multiple space dimensions. We incorporate the MUSTA technology in...

It is perhaps surprising for a shock wave to exist in the solution of a rarefaction Riemann problem for the compressible Euler equations in two space dimensions. We present numerical evidence and generalized characteristic analysis to establish the existence of a shock wave in such a 2D Riemann problem, defined by the interaction of four rarefaction waves. We consider both the customary configu...

Here, U ≡ (v,ux,uy,e), superscript T stands for transpose, r is mass density, v ≡ 1/ρ, ux and uy are the components of flow velocity, e is the specific total energy, and Fx and Fy are fluxes in the xand ydirections respectively. A 2-D Riemann problem is Eq.(1) with a set of constant states for each region surrounding a point; for example, four constant states in the four quadrants in a structur...