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

  • * Corresponding author: R. Ordoñez

    * Corresponding author: R. Ordoñez 

Date: January 11, 2021.

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  • 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.

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

    Citation:

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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)

    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)

    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

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

    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)

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

    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 $

    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 $)

    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)

    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 $

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

    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 $

    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 $

    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 $

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

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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