Nonlocal reaction traffic flow model with on-off ramps

Felisia Angela Chiarello; Harold Deivi Contreras; Luis Miguel Villada

doi:10.3934/nhm.2022003

Article Contents

2022, Volume 17, Issue 2: 203-226. Doi: 10.3934/nhm.2022003

This issue Previous Article Homogenization of stiff inclusions through network approximation Next Article Stochastic two-scale convergence and Young measures

Nonlocal reaction traffic flow model with on-off ramps

1.
Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
2.
GIMNAP-Departamento de Matemáticas, Universidad del Bío-Bío, Concepción, Chile
3.
GIMNAP-Departamento de Matemáticas, Universidad del Bío-Bío, Concepción, Chile, and CI²MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile

^* Corresponding author: Luis Miguel Villada
^* Corresponding author: Luis Miguel Villada

Received: July 31, 2021

Revised: November 30, 2021

Early access: February 2022

Published: April 2022

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Abstract

We present a non-local version of a scalar balance law modeling traffic flow with on-ramps and off-ramps. The source term is used to describe the inflow and output flow over the on-ramp and off-ramps respectively. We approximate the problem using an upwind-type numerical scheme and we provide $ \mathbf{L^{\infty}} $ and $ \mathbf{BV} $ estimates for the sequence of approximate solutions. Together with a discrete entropy inequality, we also show the well-posedness of the considered class of scalar balance laws. Some numerical simulations illustrate the behaviour of solutions in sample cases.

Keywords:

Lighthill-Whitham-Richards traffic model,
nonlocal conservation laws,
ramps,
upwind scheme,
well-posedness.

Mathematics Subject Classification: Primary: 65M08; Secondary: 35L45, 90B20.

Citation:

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Figure 1. Illustration of the model setting

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Figure 2. Example 1. Numerical approximations of the problem (2.1). Dynamic of Model 1 vs. Model 2 at (a)$ T = 0.5 $, (b)$ T = 2 $, (c)$ T = 5 $, (d)$ T = 7 $

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Figure 3. Example 2. Numerical approximations of the problem (2.1) at $ T = 5 $. Comparison of local and non-local versions of the model (2.1) with $ \delta = 0 $ and different values for $ \eta $

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Figure 4. Example 3. Numerical approximation at time $ T = 0.3 $. (a) Model 1, Model 2 satisfying a maximum principle and Model 0 not satisfying a maximum principle. (b) Zoom of a part of (a)

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Figure 5. Example 4. Dynamic of the model (2.1). Behavior of the numerical solution computed with Algorithm 3.1 by means of Model 1 and Model 2 at time (a)$ T = 1 $, (b)$ T = 2 $, (c)$ T = 5 $, (d)$ T = 7 $

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Table 1. Example 2. $ \mathbf{L^{1}} $ distance between the approximate solutions to the non-local problem and the local problem for different values of $ \eta $ at $ T = 5 $ with $ \Delta x = 1/1000 $

$ \eta $ 0.1 0.05 0.01 0.004

$ \mathbf{L^{1}} $ distance 2.8e-1 1.6e-1 3.6e-2 1.1e-2

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