| 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 |
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.
Citation: |
Table 1.
| 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.
| 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.
| | | | |
| 4.46e-02 | 6.85e-03 | 9.90e-04 | 1.60e-04 |
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