Theoretical and numerical aspects of the interfacial coupling: The scalar Riemann problem and an application to multiphase flows
Christophe Chalons
Networks & Heterogeneous Media 2010, 5(3): 507-524 doi: 10.3934/nhm.2010.5.507
This paper is devoted to the study of the one dimensional interfacial coupling of two PDE systems at a given fixed interface, say $x=0$. Each system is posed on a half-space, namely $x<0$ and $x>0$. As an interfacial model, a coupling condition whose objective is to enforce the continuity (in a weak sense) of a prescribed variable is generally imposed at $x=0$.
   We first focus on the coupling of two scalar conservation laws and state an existence result for the coupled Riemann problem. Numerical experiments are also proposed. We then consider, both from a theoretical and a numerical point of view, the coupling of two-phase flow models namely a drift-flux model and a two-fluid model. In particular, the link between both models will be addressed using asymptotic expansions.
keywords: Two-phase flow models Riemann problem Discontinuous flux function Drift flux models. Coupling problem
General constrained conservation laws. Application to pedestrian flow modeling
Christophe Chalons Paola Goatin Nicolas Seguin
Networks & Heterogeneous Media 2013, 8(2): 433-463 doi: 10.3934/nhm.2013.8.433
We extend the results on conservation laws with local flux constraint obtained in [2, 12] to general (non-concave) flux functions and non-classical solutions arising in pedestrian flow modeling [15]. We first provide a well-posedness result based on wave-front tracking approximations and the Kružhkov doubling of variable technique for a general conservation law with constrained flux. This provides a sound basis for dealing with non-classical solutions accounting for panic states in the pedestrian flow model introduced by Colombo and Rosini [15]. In particular, flux constraints are used here to model the presence of doors and obstacles. We propose a "front-tracking" finite volume scheme allowing to sharply capture classical and non-classical discontinuities. Numerical simulations illustrating the Braess paradox are presented as validation of the method.
keywords: finite volume schemes macroscopic pedestrian flow models non-classical shocks Braess paradox. Constrained scalar conservation laws
Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model
Stefano Villa Paola Goatin Christophe Chalons
Discrete & Continuous Dynamical Systems - B 2017, 22(10): 3921-3952 doi: 10.3934/dcdsb.2017202

We introduce a second order model for traffic flow with moving bottlenecks. The model consists of the × 2$ Aw-Rascle-Zhang system with a point-wise flow constraint whose trajectory is governed by an ordinary differential equation. We define two Riemann solvers, characterize the corresponding invariant domains and propose numerical strategies, which are effective in capturing the non-classical shocks due to the constraint activation.

keywords: Traffic flow models moving bottlenecks unilateral flux constraints Riemann solvers finite volume schemes

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