Preprint

Inserted: 13 aug 2025
Last Updated: 30 sep 2025

Year: 2025

Abstract:

We consider a class of nonlocal conservation laws modeling traffic flow, \( \partial_t \rho_\varepsilon + \partial_x(V(\rho_\varepsilon \ast \gamma_\varepsilon) \rho_\varepsilon) = 0\), with a suitable convex kernel $\gamma_\varepsilon$. We introduce a Godunov-type numerical discretization for the model and prove that, as the mesh size $h$ and the nonlocal parameter $\varepsilon$ tend to zero simultaneously, the discretized nonlocal impact $W_\varepsilon := \rho_\varepsilon \ast \gamma_\varepsilon$ converges to the entropy solution of the (local) scalar conservation law \( \partial_t \rho + \partial_x(V(\rho) \rho) = 0\), with an explicit convergence rate estimate of order $\varepsilon+h+\sqrt{\varepsilon\, t}+\sqrt{h\,t}$. In particular, with an exponential kernel, we establish the same result for discretized solution $\rho_\varepsilon$ and an $\mathrm{L}^1$-contraction property. The key ingredients in proving these results are uniform $\mathrm{L}^\infty$- and TV-estimates, and discrete entropy inequalities that ensure the entropy admissibility of the limit solution.

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