Coupling Traffic Flow Networks to Pedestrian Motion

Borsche, Raul; Klar, Axel; Kühn, Sebastian; Meurer, Anne

doi:10.1142/s0218202513400113

Cited by 38 publications

(27 citation statements)

References 42 publications

(43 reference statements)

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“…Rotate left and right data as in (15). Once numerical fluxes for all edges have been calculated, element k can be updated through the finite volume formula (11). This description applies to any two-dimensional finite volume method on a general mesh, assumed here to be unstructured.…”

Section: Rotational Invariance and Numerical Methodsmentioning

confidence: 99%

A numerical method for junctions in networks of shallow-water channels

Bellamoli

Muller

Toro

2018

Applied Mathematics and Computation

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There is growing interest in developing mathematical models and appropriate numerical methods for problems involving networks formed by, essentially, one-dimensional (1D) domains joined by junctions. Examples include hyperbolic equations in networks of gas tubes, water channels and vessel networks for blood and lymph in the human circulatory system. A key point in designing numerical methods for such applications is the treatment of junctions, i.e. points at which two or more 1D domains converge and where the flow exhibits multidimensional behaviour. This paper focuses on the design of methods for networks of water channels. Our methods adopt the finite volume approach to make full use of the two-dimensional shallow water equations on the true physical domain, locally at junctions, while solving the usual one-dimensional shallow water equations away from the junctions. In addition to mass conservation, our methods enforce conservation of momentum at junctions; the latter seems to be the missing element in methods currently available. Apart from simplicity and robustness, the salient feature of the proposed methods is their ability to successfully deal with transcritical and supercritical flows at junctions, a property not enjoyed by existing published methodologies. Systematic assessment of the proposed methods for a variety of flow configurations is carried out. The methods are directly applicable to other systems, provided the multidimensional versions of the 1D equations are available.

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Section: Rotational Invariance and Numerical Methodsmentioning

confidence: 99%

A numerical method for junctions in networks of shallow-water channels

Bellamoli

Muller

Toro

2018

Applied Mathematics and Computation

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“…The hallmarks of Ref. 19 can be followed toward this aim, for instance modeling crowd dynamics 20,34 shows an analogy on the way walkers and cells chose their paths and velocity. The derivation of models needs, for both systems, a deep understanding of the dynamics of interactions.…”

Section: A Micro-macro Approachmentioning

confidence: 99%

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

Bellomo

Bellouquid

Tao

et al. 2015

Math. Models Methods Appl. Sci.

866

810

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This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller-Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions. The presentation is organized in three parts. The first part focuses on a survey of some sample models, namely the original model and some of its developments, such as flux limited models, or models derived according to similar concepts. The second part is devoted to the qualitative analysis of analytic problems, such as the existence of solutions, blow-up and asymptotic behavior. The third part deals with the derivation of macroscopic models from the underlying description, delivered by means of kinetic theory methods. This approach leads to the derivation of classical models as well as that of new models, which might deserve attention as far as 1663

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“…The presented reconstruction procedure is independent of the time variable and it therefore suffices to consider functions u = u(x). Further, we assume an equidistant grid with mesh size h, grid points x j = x 0 + jh and finite volumes 5 ], j ∈ {0.5, . .…”

Section: Reconstruction Proceduresmentioning

confidence: 99%

“…There are several applications where networks of hyperbolic conservation laws are involved, e.g., gas and water supply networks [6,12,23], traffic flow [5,9,15,17] or blood flow [26,27]. Those models mainly consist of one-dimensional domains on the edges of an underlying graph complemented with appropriate coupling and boundary conditions at the nodes of the network.…”

Section: Introductionmentioning

confidence: 99%

On a third order CWENO boundary treatment with application to networks of hyperbolic conservation laws

Naumann

Kolb

Semplice

2018

Applied Mathematics and Computation

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High order numerical methods for networks of hyperbolic conservation laws have recently gained increasing popularity. Here, the crucial part is to treat the boundaries of the single (one-dimensional) computational domains in such a way that the desired convergence rate is achieved in the smooth case but also stability criterions are fulfilled, in particular in the presence of discontinuities. Most of the recently proposed methods rely on a WENO extrapolation technique introduced by Tan and Shu in [J. Comput. Phys. 229, pp. 8144-8166 (2010)]. Within this work, we refine and in a sense generalize these results for the case of a third order scheme. Numerical evidence for the analytically found parameter bounds is given as well as results for a complete third order scheme based on the proposed boundary treatment.

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Coupling Traffic Flow Networks to Pedestrian Motion

Cited by 38 publications

References 42 publications

A numerical method for junctions in networks of shallow-water channels

A numerical method for junctions in networks of shallow-water channels

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

On a third order CWENO boundary treatment with application to networks of hyperbolic conservation laws

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