A Godunov type scheme for a class of LWR traffic flow models with non-local flux

Jan Friedrich; Oliver Kolb; Simone Göttlich

doi:10.3934/nhm.2018024

Article Contents

2018, Volume 13, Issue 4: 531-547. Doi: 10.3934/nhm.2018024

This issue Previous Article Next Article Long time behavior for the visco-elastic damped wave equation in

$\mathbb{R}^n_+$

and the boundary effect

A Godunov type scheme for a class of LWR traffic flow models with non-local flux

University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany

* Corresponding author: Simone Göttlich
* Corresponding author: Simone Göttlich

Received: January 31, 2018

Revised: March 31, 2018

Published: November 2018

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Abstract

We present a Godunov type numerical scheme for a class of scalar conservation laws with non-local flux arising for example in traffic flow models. The proposed scheme delivers more accurate solutions than the widely used Lax-Friedrichs type scheme. In contrast to other approaches, we consider a non-local mean velocity instead of a mean density and provide $L^∞$ and bounded variation 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 conservation laws. The better accuracy of the Godunov type scheme in comparison to Lax-Friedrichs is proved by a variety of numerical examples.

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Mathematics Subject Classification: Primary: 35L65, 90B20; Secondary: 65M12.

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Figure 1. Illustration of a non-local traffic flow model either given by (1)-(3) or (4)-(6)

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Figure 2. Space discretization and downstream kernel $\eta = Nh$ for $N = 2$ in gray

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Figure 3. Comparison of the Godunov and LxF scheme for $v(\rho) = 1-\rho$, $h = 0.01$ at $T = 0.1$

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Figure 4. Comparison of the Godunov and LxF scheme for $v(\rho) = 1-\rho$, $h = 0.01$ at $T = 1$

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Figure 5. Comparison of the Godunov and LxF scheme for $v(\rho) = 1-\rho^5$, $h = 0.01$ at $T = 0.05$

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Figure 6. Approximate solutions at $T = 0.05$ for the two models with non-linear velocity function $v(\rho) = 1-\rho^5$

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Figure 7. Approximate solutions to the LWR and non-local model (4) to (6) for different $\eta$ at $T = 0.05$

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Table 1. $L^1$ errors for $v(\rho) = 1-\rho$ at $T = 0.1$

$n$ Godunov LxF

0 9.38e-03 1.99e-02
1 6.97e-03 1.30e-02
2 4.29e-03 9.31e-03
3 3.00e-03 6.41e-03
4 1.96e-03 4.27e-03
5 1.33e-03 2.71e-03
6 9.05e-04 1.64e-03

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Table 2. $L^1$ errors for $v(\rho) = 1-\rho^5$ at $T = 0.05$

$n$ Godunov LxF

0 1.77e-02 3.13e-02
1 1.24e-02 2.20e-02
2 8.49e-03 1.41e-02
3 5.18e-03 8.67e-03
4 3.29e-03 5.45e-03
5 2.02e-03 3.47e-03
6 1.21e-03 2.06e-03

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Table 3. $L^1$ distances between the approximate solutions to the local LWR model and the non-local model for different $\eta$ at $T = 0.05$

$\eta$ $10^{-1}$ $10^{-2}$ $10^{-3}$ $10^{-4}$

$L^1$ distance 4.46e-02 6.85e-03 9.90e-04 1.60e-04

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