Antidiffusive <scp>L</scp>agrangian‐remap schemes for models of polydisperse sedimentation

Bürger, Raimund; Chalons, Christophe; Villada, Luis Miguel

doi:10.1002/num.22043

Cited by 3 publications

(5 citation statements)

References 37 publications

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“…We extended the L-AR schemes proposed in [5,6] to the non-local multi-class traffic flow model proposed in [10]. We provided some properties of the L-AR scheme and we proved the convergence to weak solutions in the scalar case.…”

Section: Discussionmentioning

confidence: 99%

“…The aim of this paper is to present a generalization to non-local systems of the Lagrangian-Antidiffusive Remap (L-AR) schemes introduced in [5,6], in order to compute approximate solutions of model (1.1). This type of schemes are constructed exploiting the concentrationtimes-velocity form of the fluxes in (1.1).…”

Section: Introductionmentioning

confidence: 99%

“…This type of schemes are constructed exploiting the concentrationtimes-velocity form of the fluxes in (1.1). In [5], one step L-AR schemes were applied to multi-class Lighthill-Whitham-Richards (MCLWR) traffic models and they were extended to polydisperse sedimentation models in [6]. L-AR schemes do not rely on spectral (characteristics) information and their implementation is as easy as that one of first-and second-order of accuracy schemes introduced in [7].…”

Section: Introductionmentioning

confidence: 99%

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Lagrangian-antidiffusive remap schemes for non-local multi-class traffic flow models

2020

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This paper focuses on the numerical approximation of the solutions of a class of nonlocal systems in one space dimension, arising in traffic modeling. We propose alternative simple schemes by splitting the non-local conservation laws into two different equations, namely, the Lagrangian and the remap steps. We provide some properties and estimates recovered by approximating the problem with the Lagrangian-Antidiffusive Remap (L-AR) scheme, and we prove the convergence to weak solutions in the scalar case. Finally, we show some numerical simulations illustrating the efficiency of the L-AR schemes in comparison with classical first and second-order numerical schemes.From the application point of view, ρ i represents the density of the i-th vehicle class, characterized by its maximal velocity v max i and its interaction kernel ω i , which accounts for the reaction to downstream traffic distribution. In particular, η i is proportional to the look-ahead distance of drivers and J i represents the interaction strength. Equations (1.1) are coupled through the velocity function, which depends on an integral evaluation of the total traffic density r. For simplicity, in this work we will consider linear decreasing velocities setting ψ(r) = max{1 − r, 0}.Model (1.1) is a multi-class extension of the one-dimensional scalar conservation law with non-local flux proposed in [3,15]. It can also be seen as the non-local generalization of the "local" multi-population model for traffic flow described in [2]. Indeed, one of the limitations of the standard Lighthill-Whitham-Richards (LWR) traffic flow model [17,18] is the first in first out rule, conversely in multi-class dynamic faster vehicles can overtake slower ones and slower vehicles slow down the faster ones. In our setting, the non-local dependence of the speed functions v i describes the reaction of drivers that adapt their velocity with respect to what happens in front of them in terms of downstream traffic, assigning greater importance to closer vehicles, see also [9].Since solutions to (1.1), (1.4) may be discontinuous, they are intended in the following weak sense.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Lagrangian-antidiffusive remap schemes for non-local multi-class traffic flow models

2020

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“…Remark 6.1. In the case S ≡ 0, system (6.1) has the same structure as the classical Keytz-Kranzer system [35] up to the properties of the ux function f which is monotone in the Keytz-Kranzer case and which is bell-shaped in the case we are concerned with, see also [15]. Discretization of the Keytz-Kranzer system by nite dierence schemes was addressed, in particular, in [36].…”

Section: Motivationmentioning

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.

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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.

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“…General references to models of sedimentation include [6,33]. Models of polydisperse sedimentation in one space dimension similar to the MLB model, and which give rise to strongly coupled systems of nonlinear conservation laws or possibly degenerate convection-diffusion equations were thoroughly studied in recent years including analyses of hyperbolicity [13,23], extensions to flocculated suspensions forming compressible sediments [8], construction of entropy solutions [7,24], development of efficient numerical schemes [10,14,15], and applications in geophysics [25], water resource recovery [16], and others (see also references in the cited works). On the other hand, theoretical and experimental works on polydisperse gravity currents, with which our numerical results could in principle be compared, include [9,21,30].…”

Section: Related Workmentioning

confidence: 99%

A dynamic multilayer shallow water model for polydisperse sedimentation

2019

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A multilayer shallow water approach for the approximate description of polydisperse sedimentation in a viscous fluid is presented. The fluid is assumed to carry finely dispersed solid particles that belong to a finite number of species that differ in density and size. These species segregate and form areas of different composition. In addition, the settling of particles influences the motion of the ambient fluid. A distinct feature of the new approach is the particular definition of the average velocity of the mixture. It takes into account the densities of the solid particles and the fluid and allows us to recover the global mass conservation and linear momentum balance laws of the mixture. This definition motivates a modification of the Masliyah-Lockett-Bassoon (MLB) settling velocities of each species. The multilayer shallow water model allows one to determine the spatial distribution of the solid particles, the velocity field, and the evolution of the free surface of the mixture. The final model can be written as a multilayer model with variable density where the unknowns are the average velocities and concentrations in each layer, the transfer terms across each interface, and the total mass. An explicit formula of the transfer terms leads to a reduced form of the system. Finally, an explicit bound of the minimum and maximum eigenvalues of the transport matrix of the system is utilized to design a Harten-Lax-van Leer (HLL)-type path-conservative numerical method. Numerical simulations illustrate the coupled polydisperse sedimentation and flow fields in various scenarios, including sedimentation in a type of basin that is used in practice in mining industry and in a basin whose bottom topography gives rise to recirculations of the fluid and high solids concentrations.Mathematics Subject Classification. 65N06, 76T20.

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Antidiffusive Lagrangian‐remap schemes for models of polydisperse sedimentation

Cited by 3 publications

References 37 publications

Lagrangian-antidiffusive remap schemes for non-local multi-class traffic flow models

Lagrangian-antidiffusive remap schemes for non-local multi-class traffic flow models

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

A dynamic multilayer shallow water model for polydisperse sedimentation

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