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

• * Corresponding author: Simone Göttlich
• 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.

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

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

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

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

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

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

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

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

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

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

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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Tables(3)

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