A LWR model with constraints at moving interfaces

Sylla, Abraham

doi:10.1051/m2an/2022030

Cited by 5 publications

(22 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In addition, the integral weak formulation for the approximate solution follows from the scheme's conservativity. We use the same compactness argument as in [22,Sect. 3.4].…”

Section: 2mentioning

confidence: 99%

“…The discretization (A.1c)-(A.1d) of the ξ interface is detailled in [22] Section 3.1 where it is required to construct the adapted mesh. For any n, we denote j n the unique element of −J, J such that ξ n ∈ [x jn , x jn+1 ).…”

mentioning

confidence: 99%

“…5.1] for the case of open-end condition at exits. We are now in a position to start the proof of convergence, which merely assembles with the help of the partition-of-unity technique of [22,6] the arguments from [6] (for the inner interface situated at x = ξ(t) and [7] (for the constraints set at x = ±1).…”

mentioning

confidence: 99%

See 2 more Smart Citations

Existence of solutions to a class of one-dimensional models for pedestrian evacuations

Andreianov,

Girard

2023

Preprint

View full text Add to dashboard Cite

In the framework inspired by R. L. Hughes model (Transp. Res. B, 2002) for pedestrian evacuation in a corridor, we establish existence of a solution by a topological fixed point argument. This argument applies to a class of models where the dynamics of the pedestrian density ρ (governed by a discontinuous-flux Lighthill,Whitham and Richards model ρt + (sign(x − ξ(t))ρv(ρ)) x = 0 ) is coupled via an abstract operator to the computation of a Lipschitz continuous "turning curve" ξ. We illustrate this construction by several examples, including the standard Hughes' model with affine cost, and either with open-end conditions or with conditions corresponding to panic behaviour with capacity drop at exits. Other examples put forward versions of the Hughes model with inertial dynamics of the turning curve and general costs.

show abstract

“…In addition, the integral weak formulation for the approximate solution follows from the scheme's conservativity. We use the same compactness argument as in [22,Sect. 3.4].…”

Section: 2mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Existence of solutions to a class of one-dimensional models for pedestrian evacuations

Andreianov,

Girard

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…The theory of (2.1),(2.2) is well understood for the case of isolated discontinuities in ω (cf. [31,48]) but the case of interest, in the context of our model, requires much deeper investigation. The goal of this section is to sketch the existence theory, via convergence of the splitting approximations, based upon the propagation of the BV regularity of the initial datum w 0 .…”

Section: Variants Of the Modelmentioning

confidence: 99%

“…Last but not least, the numerical strategy developed in Section 6 below for the spatially non-local problem of Section 5.1 is applicable also to the local problem of Section 5.2, provided consistent discretization of (2.1),(2.2) is used taking into account the possible sharp discontinuities in the expression of the ux function (cf. [48]).…”

Section: Page 16mentioning

confidence: 99%

Existence analysis and numerical approximation for a second-order model of traffic with orderliness marker

Andreïanov

Sylla

2022

Math. Models Methods Appl. Sci.

Self Cite

View full text Add to dashboard Cite

We propose a toy model for self-organized road traffic taking into account the state of orderliness in drivers’ behavior. The model is reminiscent of the wide family of generalized second-order models (GSOM) of road traffic. It can also be seen as a phase-transition model. The orderliness marker is evolved along vehicles’ trajectories and it influences the fundamental diagram of the traffic flow. The coupling we have in mind is non-local, leading to a kind of “weak decoupling” of the resulting [Formula: see text] system; this makes the mathematical analysis similar to the analysis of the classical Keyfitz–Kranzer system. Taking advantage of the theory of weak and renormalized solutions of one-dimensional transport equations [Panov, 2008], which we further develop on this occasion in the Appendix, we prove the existence of admissible solutions defined via a mixture of the Kruzhkov and the Panov approaches; note that this approach to admissibility does not rely upon the classical hyperbolic structure for [Formula: see text] systems. First, approximate solutions are obtained via a splitting strategy; compactification effects proper to the notion of solution we rely upon are carefully exploited, under general assumptions on the data. Second, we also address fully discrete approximation of the system, constructing a [Formula: see text]-stable Finite Volume numerical scheme and proving its convergence under the no-vacuum assumption and for data of bounded variation. As a byproduct of our approach, an original treatment of local GSOM-like models in the [Formula: see text] setting is briefly discussed, in relation to discontinuous-flux LWR models.

show abstract

Finite volume approximation and well-posedness of conservation laws with moving interfaces under abstract coupling conditions

Andreïanov

Sylla

2023

Nonlinear Differ. Equ. Appl.

Self Cite

View full text Add to dashboard Cite

Scalar conservation law ∂ t ρ(t, x) + ∂ x (f (t, x, ρ)) = 0 with a ux C 1 in the state variable ρ, piecewise C 1 in the (t, x)-plane admits innitely many consistent notions of solution which dier by the choice of interface coupling. Only the case of the so-called vanishing viscosity solutions received full attention, while dierent choice of coupling is relevant in modeling situations that appear, e.g., in road trac and in porous medium applications. In this paper, existence of solutions for a wide set of coupling conditions is established under some restrictions on f , via a nite volume approximation strategy adapted to slanted interfaces and to the presence of interface crossings. The notion of solution, restated under the form of an adapted entropy formulation which is consistently approximated by the numerical scheme, implies uniqueness and stability of solutions. Numerical simulations are presented to illustrate the reliability of the scheme.

show abstract

A LWR model with constraints at moving interfaces

Cited by 5 publications

References 26 publications

Existence of solutions to a class of one-dimensional models for pedestrian evacuations

Existence of solutions to a class of one-dimensional models for pedestrian evacuations

Existence analysis and numerical approximation for a second-order model of traffic with orderliness marker

Finite volume approximation and well-posedness of conservation laws with moving interfaces under abstract coupling conditions

Contact Info

Product

Resources

About