Crowd Dynamics Through Conservation Laws
01 Jan 2020-Vol. 47, Iss: 1, pp 83-110
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.
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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
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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
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
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
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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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TL;DR: It is proved that the numerical solution converges to the entropy weak solution of the continuous problem in $L^p_{loc}$ for every $p\in [ 1, +\infty)$.
Abstract: This paper is devoted to the study of the finite volume methods used in the discretization of conservation laws defined on bounded domains. General assumptions are made on the data: the initial condition and the boundary condition are supposed to be measurable bounded functions. Using a generalized notion of solution to the continuous problem (namely the notion of entropy process solution, see [9]) and a uniqueness result on this solution, we prove that the numerical solution converges to the entropy weak solution of the continuous problem in \(L^p_{loc}\) for every \(p\in [ 1, +\infty)\). This also yields a new proof of the existence of an entropy weak solution.
86 citations
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TL;DR: It is shown that Hughes’ model is incapable of reproducing complex crowd dynamics such as stop-and-go waves and clogging at bottlenecks.
Abstract: We analyze numerically two macroscopic models of crowd dynamics: the classical Hughes model and the second order model being an extension to pedestrian motion of the Payne-Whitham vehicular traffic model. The desired direction of motion is determined by solving an eikonal equation with density dependent running cost, which results in minimization of the travel time and avoidance of congested areas. We apply a mixed finite volume-finite element method to solve the problems and present error analysis for the eikonal solver, gradient computation and the second order model yielding a first order convergence. We show that Hughes' model is incapable of reproducing complex crowd dynamics such as stop-and-go waves and clogging at bottlenecks. Finally, using the second order model, we study numerically the evacuation of pedestrians from a room through a narrow exit.
75 citations
70 citations
TL;DR: In this paper, the authors deal with nonlinear hydrodynamic modelling of traffic flow on roads and with the solution of related nonlinear initial and boundary value problems, and provide a critical analysis and an overview on research perspectives.
70 citations
TL;DR: In this paper, a general class of 1D nonlocal conservation laws from a numerical point of view are studied and an algorithm to numerically integrate them and prove their convergence is presented.
Abstract: We study a rather general class of 1D nonlocal conservation laws from a numerical point of view. First, following [F. Betancourt, R. Burger, K.H. Karlsen and E.M. Tory, On nonlocal conservation laws modelling sedimentation. Nonlinearity 24 (2011) 855–885], we define an algorithm to numerically integrate them and prove its convergence. Then, we use this algorithm to investigate various analytical properties, obtaining evidence that usual properties of standard conservation laws fail in the nonlocal setting. Moreover, on the basis of our numerical integrations, we are led to conjecture the convergence of the nonlocal equation to the local ones, although no analytical results are, to our knowledge, available in this context.
69 citations