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18
The sharpinterface approach for fluids with phase change: Riemann problems and ghost fluid techniques
 M2AN Math. Model. Numer. Anal
, 2007
"... Abstract. Systems of mixed hyperbolicelliptic conservation laws can serve as models for the evolution of a liquidvapor
uid with possible sharp dynamical phase changes. We focus on the equations of ideal hydrodynamics in the isothermal case and introduce a thermodynamically consistent solution of ..."
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Abstract. Systems of mixed hyperbolicelliptic conservation laws can serve as models for the evolution of a liquidvapor
uid with possible sharp dynamical phase changes. We focus on the equations of ideal hydrodynamics in the isothermal case and introduce a thermodynamically consistent solution of the Riemann problem in one space dimension. This result is the basis for an algorithm of ghost
uid type to solve the sharpinterface model numerically. In particular the approach allows to resolve phase transitions sharply, i. e., without articial smearing in the physically irrelevant elliptic region. Numerical experiments demonstrate the reliability of the method.
General constrained conservation laws. Application to pedestrian flow modeling
, 2012
"... (Communicated by Benedetto Piccoli) Abstract. We extend the results on conservation laws with local flux constraint obtained in [2, 12] to general (nonconcave) flux functions and nonclassical solutions arising in pedestrian flow modeling [15]. We first provide a wellposedness result based on wave ..."
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Cited by 9 (2 self)
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(Communicated by Benedetto Piccoli) Abstract. We extend the results on conservation laws with local flux constraint obtained in [2, 12] to general (nonconcave) flux functions and nonclassical solutions arising in pedestrian flow modeling [15]. We first provide a wellposedness result based on wavefront tracking approximations and the Kruˇzhkov doubling of variable technique for a general conservation law with constrained flux. This provides a sound basis for dealing with nonclassical 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 “fronttracking ” finite volume scheme allowing to sharply capture classical and nonclassical discontinuities. Numerical simulations illustrating the Braess paradox are presented as validation of the method. 1. Introduction. Several
A convergent and conservative schemes for nonclassical solutions based on kinetic relations
 I. Interfaces and Free Bound
, 2008
"... AbstractWe propose a new numerical approach to compute nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keep nonclassical shock waves as sharp interfaces, contrary to standard finite difference schemes. The main ..."
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Cited by 8 (4 self)
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AbstractWe propose a new numerical approach to compute nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keep nonclassical shock waves as sharp interfaces, contrary to standard finite difference schemes. The main challenge is to achieve, at the discretization level, a consistency property with respect to a prescribed kinetic relation. The latter is required for the selection of physically meaningful nonclassical shocks. Our method is based on a reconstruction technique performed in each computational cell that may contain a nonclassical shock. To validate this approach, we establish several consistency and stability properties, and we perform careful numerical experiments. The convergence of the algorithm toward the physically meaningful solutions selected by a kinetic relation is demonstrated numerically for several test cases, including concaveconvex as well as convexconcave fluxfunctions. Résume ́ Nous proposons un nouvel algorithme pour approcher les solutions non classiques de lois de conservation hyperboliques. Le schéma aux différences finies présente ́ ici est conservatif et transporte de manière exacte les chocs non classiques, a ̀ la différences des algorithmes standard. La principale difficulte ́ est de garantir, au niveau discret, la consistance avec une re
The AwRascle traffic model with locally constrained flow
, 2010
"... We consider solutions of the AwRascle model for traffic flow fulfilling a constraint on the flux at x = 0. Two different kinds of solutions are proposed: at x = 0 the first one conserves both the number of vehicles and the generalized momentum, while the second one conserves only the number of cars ..."
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We consider solutions of the AwRascle model for traffic flow fulfilling a constraint on the flux at x = 0. Two different kinds of solutions are proposed: at x = 0 the first one conserves both the number of vehicles and the generalized momentum, while the second one conserves only the number of cars. We study the invariant domains for these solutions and we compare the two Riemann solvers in terms of total variation of relevant quantities. Finally we construct ad hoc finite volume numerical schemes to compute these solutions.
Conservation laws with unilateral constraints in traffic modeling
 Transport Management and LandUse Effects in Presence of Unusual Demand”, Atti del convegno SIDT 2009
, 2009
"... Macroscopic models for both vehicular and pedestrian traffic are based on conservation laws. The mathematical description of toll gates along roads or of the escape dynamics for crowds needs the introduction of unilateral constraints on the observable flow. This note presents a rigorous approach to ..."
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Cited by 6 (3 self)
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Macroscopic models for both vehicular and pedestrian traffic are based on conservation laws. The mathematical description of toll gates along roads or of the escape dynamics for crowds needs the introduction of unilateral constraints on the observable flow. This note presents a rigorous approach to these constraints, and numerical integrations of the resulting models are included to show their practical usability. 1.
Comparative model accuracy of a datafitted generalized AwRascleZhang model
, 2013
"... Abstract. The AwRascleZhang (ARZ) model can be interpreted as a generalization of the LighthillWhithamRichards (LWR) model, possessing a family of fundamental diagram curves, each of which represents a class of drivers with a different empty road velocity. A weakness of this approach is that dif ..."
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Cited by 5 (4 self)
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Abstract. The AwRascleZhang (ARZ) model can be interpreted as a generalization of the LighthillWhithamRichards (LWR) model, possessing a family of fundamental diagram curves, each of which represents a class of drivers with a different empty road velocity. A weakness of this approach is that different drivers possess vastly different densities at which traffic flow stagnates. This drawback can be overcome by modifying the pressure relation in the ARZ model, leading to the generalized AwRascleZhang (GARZ) model. We present an approach to determine the parameter functions of the GARZ model from fundamental diagram measurement data. The predictive accuracy of the resulting datafitted GARZ model is compared to other traffic models by means of a threedetector test setup, employing two types of data: vehicle trajectory data, and sensor data. This work also considers the extension of the ARZ and the GARZ models to models with a relaxation term, and conducts an investigation of the optimal relaxation time. 1.
A comparison of datafitted first order traffic models and their second order generalizations via trajectory and sensor data
 93RD ANNUAL MEETING OF TRANSPORTATION RESEARCH BOARD, WASHINGTON DC, 2013. PAPER NUMBER 13–4853
, 2013
"... The AwRascleZhang (ARZ) model can be interpreted as a generalization of the first order LighthillWhithamRichards (LWR) model, possessing a family of fundamental diagram curves, rather than a single one. We investigate to which extent this generalization increases the predictive accuracy of the m ..."
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Cited by 4 (3 self)
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The AwRascleZhang (ARZ) model can be interpreted as a generalization of the first order LighthillWhithamRichards (LWR) model, possessing a family of fundamental diagram curves, rather than a single one. We investigate to which extent this generalization increases the predictive accuracy of the models. To that end, a systematic comparison of two types of datafitted LWR models and their second order ARZ counterparts is conducted, via a version of the threedetector problem test. The parameter functions of the models are constructed using historic fundamental diagram data. The model comparisons are then carried out using timedependent data, of two very different types: vehicle trajectory data, and singleloop sensor data. The study of these PDE models is carried out in a macroscopic sense, i.e., continuous field quantities are constructed from the discrete data, and discretization effects are kept negligibly small.
A class of multiphase traffic theories for microscopic, kinetic and continuum traffic models
 Comp. Math. Appl
, 2012
"... Abstract. In the present paper a review and numerical comparison of a special class of multiphase traffic theories based on microscopic, kinetic and macroscopic traffic models is given. Macroscopic traffic equations with multivalued fundamental diagrams are derived from different microscopic and ..."
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Abstract. In the present paper a review and numerical comparison of a special class of multiphase traffic theories based on microscopic, kinetic and macroscopic traffic models is given. Macroscopic traffic equations with multivalued fundamental diagrams are derived from different microscopic and kinetic models. Numerical experiments show similarities and differences of the models, in particular, for the appearance and structure of stop and go waves for highway traffic in dense situations. For all models, but one, phase transitions can appear near bottlenecks depending on the local density and velocity of the flow. 1.
OpenCL numerical simulations of twofluid compressible flows with a 2D random choice method
, 2012
"... Abstract. In this paper, we propose a new very simple numerical method for solving liquidgas compressible flows. Such flows are difficult to simulate because classical conservative finite volume schemes generate pressure oscillations at the liquidgas interface. We extend to several dimensions the r ..."
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Abstract. In this paper, we propose a new very simple numerical method for solving liquidgas compressible flows. Such flows are difficult to simulate because classical conservative finite volume schemes generate pressure oscillations at the liquidgas interface. We extend to several dimensions the random choice scheme that we have constructed in [13]. The extension is performed through Strang dimensional splitting. The resulting scheme exhibits interesting conservation and stability properties. For achieving high performance, the scheme is tested on recent muulticore processors and GPU, using the OpenCL environment. hal00759135, version 1 30 Nov 2012 1.