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Book ChapterDOI

Crowd Dynamics Through Conservation Laws

TL;DR: In this paper, several macroscopic models, based on systems of conservation laws, were considered for the study of crowd dynamics. But none of the models considered here contain nonlocal terms, usually obtained through convolutions with smooth functions used to reproduce the visual horizon of each individual.
Abstract: We consider several macroscopic models, based on systems of conservation laws, for the study of crowd dynamics. All the systems considered here contain nonlocal terms, usually obtained through convolutions with smooth functions, used to reproduce the visual horizon of each individual. We classify the various models according to the physical domain (the whole space \({\mathbb {R}}^N\) or a bounded subset), to the terms affected by the nonlocal operators, and to the number of different populations we aim to describe. For all these systems, we present the basic well posedness and stability results.
Citations
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01 Jan 2017
TL;DR: In this article, the authors give an answer to a question posed in Amorim et al. (ESAIM Math Model Numer Anal 49(1):19-37), which can loosely speaking, be formulated as follows: consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel.
Abstract: We give an answer to a question posed in Amorim et al. (ESAIM Math Model Numer Anal 49(1):19–37, 2015), which can loosely speaking, be formulated as follows: consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law. Can we rigorously justify this formal limit? We exhibit counterexamples showing that, despite numerical evidence suggesting a positive answer, one does not in general have convergence of the solutions. We also show that the answer is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law.

18 citations

Posted Content
TL;DR: In this paper, the problem of approximating a scalar conservation law by a conservation law with nonlocal flux was studied, and it was shown that the (unique) weak solution of the nonlocal problem converges strongly in O(L √ n) to the entropy solution of local conservation law.
Abstract: We deal with the problem of approximating a scalar conservation law by a conservation law with nonlocal flux. As convolution kernel in the nonlocal flux, we consider an exponential-type approximation of the Dirac distribution. This enables us to obtain a total variation bound on the nonlocal term. By using this, we prove that the (unique) weak solution of the nonlocal problem converges strongly in $C(L^{1}_{\text{loc}})$ to the entropy solution of the local conservation law. We conclude with several numerical illustrations which underline the main results and, in particular, the difference between the solution and the nonlocal term.

10 citations

Journal ArticleDOI
TL;DR: In this paper , the existence and uniqueness of weak solutions to conservation laws with nonlocal flux was shown to be true under the condition that the nonlocal term is given by a convolution.
Abstract: Abstract In this note, we extend the known results on the existence and uniqueness of weak solutions to conservation laws with nonlocal flux. In case the nonlocal term is given by a convolution $$\gamma *q$$ γ q , we weaken the standard assumption on the kernel $$\gamma \in L^\infty \big ((0,T); W^{1,\infty }({\mathbb {R}})\big )$$ γ L ( ( 0 , T ) ; W 1 , ( R ) ) to the substantially more general condition $$\gamma \in L^\infty ((0,T); BV({\mathbb {R}}))$$ γ L ( ( 0 , T ) ; B V ( R ) ) , which allows for discontinuities in the kernel.

6 citations

Journal ArticleDOI
TL;DR: In this paper , the problem of approximating a scalar conservation law by a conservation law with nonlocal flux was studied, and it was shown that the (unique) weak solution of the nonlocal problem converges strongly in O(L √ n) to the entropy solution of local conservation law.

5 citations

Posted Content
TL;DR: The well posedness for a class of non local systems of conservation laws in a bounded domain is proved and various stability estimates are provided.
Abstract: The well posedness for a class of non local systems of conservation laws in a bounded domain is proved and various stability estimates are provided. This construction is motivated by the modelling of crowd dynamics, which also leads to define a non local operator adapted to the presence of a boundary. Numerical integrations show that the resulting model provides qualitatively reasonable solutions.

4 citations

References
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Journal ArticleDOI
TL;DR: In this paper, the authors deal with the mathematical modelling of crowd dynamics within the framework of continuum mechanics, using the mass conservation equation closed by phenomenological models linking the local velocity to density and density gradients.
Abstract: This paper deals with the mathematical modelling of crowd dynamics within the framework of continuum mechanics. The method uses the mass conservation equation closed by phenomenological models linking the local velocity to density and density gradients. The closures take into account movement in more than one space dimension, presence of obstacles, pedestrian strategies, and modelling of panic conditions. Numerical simulations of the initial-boundary value problems visualize the ability of the models to predict several interesting phenomena related to the complex system under consideration.

129 citations

Journal ArticleDOI
TL;DR: It is shown that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in L1(R) to within O(N-2) by a piecewise linear function with O(fN) nodes; the nodes are moved according to the method of characteristics.
Abstract: We show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in L1(R) to within O(N-2) by a piecewise linear function with O(fN) nodes; the nodes are moved according to the method of characteristics. We also show that a previous method of Dafermos, which uses piecewise constant approxima- tions, is accurate to O(N-1). These numerical methods for conservation laws are the first to have proven convergence rates of greater than O(fN-1/2). 1. Introduction. It is well-known that the solution of the hyperbolic conservation law,

128 citations

Journal ArticleDOI
TL;DR: In this paper, a synthetic statement of Kružkov-type estimates for multidimensional scalar conservation laws is given, which can be used to obtain various estimates for different approximation problems.
Abstract: We give a synthetic statement of Kružkov-type estimates for multidimensional scalar conservation laws. We apply it to obtain various estimates for different approximation problems. In particular we recover for a model equation the rate of convergence in h1/4 known for finite volume methods on unstructured grids. Les estimations de Kružkov pour les lois de conservation scalaires revisitées Résumé Nous donnons un énoncé synthétique des estimations de type de Kružkov pour les lois de conservation scalaires multidimensionnelles. Nous l’appliquons pour obtenir d’estimations nombreuses pour problèmes différents d’approximation. En particulier, nous retrouvons pour une équation modèle la vitesse de convergence en h1/4 connue pour les méthodes de volumes finis sur des maillages non structurés.

126 citations

Journal ArticleDOI
TL;DR: A model for the simulation of pedestrian flows and crowd dynamics has been developed that performs well for standard benchmarks, and allows for typical crowd dynamics, such as lane forming, overtaking, avoidance of obstacles and panic behaviour.

126 citations

Journal ArticleDOI
TL;DR: In this article, the analytical properties of the solutions to the continuity equation with non-local flow are investigated. But the authors focus on a supply chain model and an equation for the description of pedestrian flows.
Abstract: This paper focuses on the analytical properties of the solutions to the continuity equation with non local flow. Our driving examples are a supply chain model and an equation for the description of pedestrian flows. To this aim, we prove the well posedness of weak entropy solutions in a class of equations comprising these models. Then, under further regularity conditions, we prove the differen- tiability of solutions with respect to the initial datum and characterize this derivative. A necessary condition for the optimality of suitable integral functionals then follows.

117 citations