A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function

Raimund Bürger; Christophe Chalons; Rafael Ordoñez; Luis Miguel Villada

doi:10.3934/nhm.2021004

Article Contents

2021, Volume 16, Issue 2: 187-219. Doi: 10.3934/nhm.2021004

This issue Previous Article Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph Next Article Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model

A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function

1.
CI²MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
2.
Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 78035 Versailles, France
3.
GIMNAP-Departamento de Matemáticas, Universidad del Bío-Bío, Concepción, Chile, CI²MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile

^* Corresponding author: R. Ordoñez
^* Corresponding author: R. Ordoñez

Date: January 11, 2021.

Received: September 30, 2020

Revised: November 30, 2020

Early access: January 2021

Published: June 2021

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Abstract

The well-known Lighthill-Whitham-Richards (LWR) kinematic model of traffic flow models the evolution of the local density of cars by a nonlinear scalar conservation law. The transition between free and congested flow regimes can be described by a flux or velocity function that has a discontinuity at a determined density. A numerical scheme to handle the resulting LWR model with discontinuous velocity was proposed in [J.D. Towers, A splitting algorithm for LWR traffic models with flux discontinuities in the unknown, J. Comput. Phys., 421 (2020), article 109722]. A similar scheme is constructed by decomposing the discontinuous velocity function into a Lipschitz continuous function plus a Heaviside function and designing a corresponding splitting scheme. The part of the scheme related to the discontinuous flux is handled by a semi-implicit step that does, however, not involve the solution of systems of linear or nonlinear equations. It is proved that the whole scheme converges to a weak solution in the scalar case. The scheme can in a straightforward manner be extended to the multiclass LWR (MCLWR) model, which is defined by a hyperbolic system of $ N $ conservation laws for $ N $ driver classes that are distinguished by their preferential velocities. It is shown that the multiclass scheme satisfies an invariant region principle, that is, all densities are nonnegative and their sum does not exceed a maximum value. In the scalar and multiclass cases no flux regularization or Riemann solver is involved, and the CFL condition is not more restrictive than for an explicit scheme for the continuous part of the flux. Numerical tests for the scalar and multiclass cases are presented.

Keywords:

Lighthill-Whitham-Richards traffic model,
multiclass traffic model,
system of conservation laws,
discontinuous flux,
invariant region principle.

Mathematics Subject Classification: Primary: 65M06; Secondary: 35L45, 35L65, 76A99.

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Figure 1. (a) Piecewise continuous velocity function $ V(\phi) $ with discontinuity at $ \phi = \phi^* $, (b) continuous and discontinuous portions $ p_V(\phi) $ (solid line) and $ g_V(\phi) $ (dashed line)

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Figure 2. (a) function $ z \mapsto \smash{\tilde{G}_V}(z;\phi) $ given by (2.9a) with $ \lambda v^{\max} = 1/2 $, $ \alpha_V = 0.3 $, and $ \phi = 0.8 $, (b) its inverse $ z \mapsto \smash{\tilde{G}_V^{-1}} (z;\phi) $ given by (2.9b)

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Figure 3. Example 1: numerical solution with $ M = 800 $ and comparison with the exact solution of the Riemann problem (a) with $ \phi_{\mathrm{L}} = 0.3 $ and $ \phi_{\mathrm{R}} = 0.9 $ at simulated time $ T = 1.8 $, (b) with $ \phi_{\mathrm{L}} = 0.9 $ and $ \phi_{\mathrm{R}} = 0.3 $ at simulated time $ T = 1.5 $. Here and in Figures 4 and 5 we label with 'Towers scheme' the scheme (1.7) proposed in [29] and by 'BCOV scheme' the scheme of Algorithm 2.1 advanced in the present work

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Figure 4. Example 2: numerical solutions for $ M = 100 $ at simulated times (a) $ T = 0.1 $, (b) $ T = 0.3 $

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Figure 5. Example 3: numerical solutions depending on the boundary conditions $ \mathcal{F}(t)\in\smash{\tilde{f}}(\phi^*) $ with $ M = 1600 $ at simulated time $ T = 0.5 $, with (a) $ \smash{\mathcal{F}(t)\in\tilde{f}}(\phi^*-) $ (free flow), (b) $ \mathcal{F}(t)\in\smash{\tilde{f}} (\phi^*+) $ (congested flow)

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Figure 6. Example 4: density profiles simulated with $ M = 1600 $ at (a) $ T = 0.2 $, (b) $ T = 0.4 $, (c) $ T = 0.6 $

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Figure 7. Example 5: numerical solution for a free-flow regime ($ \mathcal{G}(t) = \alpha_V $): (a) initial condition, (b, c) density profiles with $ M = 1600 $ at simulated times (b) $ T = 0.1, $ (c) $ T = 0.2 $

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Figure 8. Example 5: simulated total density computed with BCOV scheme with $ N = 3 $ and $ M = 1600 $: (a) free flow ($ \mathcal{G}(t) = \alpha_V $), (b) congested flow ($ \mathcal{G}(t) = 0 $)

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Figure 9. Example 5: numerical solution for a congested flow regime ($ \mathcal{G}(t) = 0 $): density profiles with $ M = 1600 $ at simulated times (a) $ T = 0.1, $ (b) $ T = 0.2 $. The initial condition is the same as in Figure 7(a)

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Figure 10. Example 6: numerical solutions obtained with BCOV scheme with $ N = 5 $ and $ M = 1600 $ at simulated times (a) $ T = 0.02 $, (b) $ T = 0.12 $

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Figure 11. Example 6: simulated total density obtained with BCOV scheme with $ N = 5 $ and $ M = 1600 $: (a) discontinuous problem, (b) continuous problem

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Figure 12. Example 6: comparison of reference solution ($ M_{\text{ref}} = 12800 $) with approximate solutions computed by BCOV scheme with $ M = 100 $ at simulated time $ T = 0.02 $

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Figure 13. Example 6: comparison of reference solution ($ M_{\text{ref}} = 12800 $) with approximate solutions computed by BCOV scheme with $ M = 100 $ at simulated time $ T = 0.02 $

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Figure 14. Example 7: numerical solution computed with BCOV scheme with $ N = 5 $ and $ M = 12800 $ at simulated times (a) $ T = 0.1 $, (b) $ T = 0.2 $ and (c) $ T = 0.3 $

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Figure 15. Example 7: simulated total density computed with BCOV scheme with $ N = 5 $ and $ M = 1600 $

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Table 1. Example 2: approximate $ L^1 $ errors $ e_{M}(u) $ with $ \Delta x = 2/M $

	$ T=0.1 $		$ T=0.3 $
	Towers	BCOV	Towers	BCOV
$ M $	$ e_{M}(\phi^{\Delta}) $	$ e_{M}(\phi^{\Delta}) $	$ e_{M}(\phi^{\Delta}) $	$ e_{M}(\phi^{\Delta}) $
100	1.32e-2	1.76e-2	1.63e-2	2.39e-2
200	6.55e-3	9.22e-3	8.59e-3	1.31e-2
400	3.29e-3	4.46e-3	4.25e-3	6.46e-3
800	1.72e-3	2.403-3	2.12e-3	3.31e-3
1600	8.00e-4	1.18e-3	9.29e-4	1.563-3

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Table 2. Example 6: approximate $ L^1 $ errors $ e_{M}(u) $ with $ \Delta x = 2/M $

	$ T=0.02 $	$ T=0.12 $
$ M $	$ e_{M}(\phi^{\Delta}) $	$ e_{M}(\phi^{\Delta}) $
100	1.39e-2	3.87e-2
200	7.90e-3	2.47e-2
400	4.20e-3	1.55e-2
800	2.00e-3	9.20e-3
1600	1.00e-3	5.10e-3

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Table 3. Example 7: Approximate $ L^1 $ errors $ e_{M}(u) $ with $ \Delta x = 5/M $

	$ T=0.1 $	$ T=0.2 $	$ T=0.3 $
$ M $	$ e_{M}(\phi^{\Delta}) $	$ e_{M}(\phi^{\Delta}) $	$ e_{M}(\phi^{\Delta}) $
100	7.42e-2	9.50e-2	1.06e-1
200	4.12e-2	5.50e-2	6.49e-2
400	2.27e-2	3.34e-2	3.88e-2
800	1.24e-2	1.97-2	2.35e-2
1600	6.50e-3	1.10e-2	1.35e-2

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