Asymptotically compatible entropy-consistent discretization for a class of nonlocal conservation laws

We consider a class of nonlocal conservation laws modeling traffic flows, given by $\partial_t \rho_\varepsilon + \partial_x(V(\rho_\varepsilon \ast \gamma_\varepsilon) \rho_\varepsilon) = 0$ with a suitable convex kernel $\gamma_\varepsilon$, and its Godunov-type numerical discretization. We prove that, as the nonlocal parameter $\varepsilon$ and mesh size $h$ tend to zero simultaneously, the discrete approximation $W_{\varepsilon,h}$ of $W_\varepsilon := \rho_\varepsilon \ast \gamma_\varepsilon$ 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 convergence result for the discrete approximation $\rho_{\varepsilon,h}$ of $\rho_\varepsilon$, along with an $\mathrm{L}^1$-contraction property for $W_\varepsilon$. The key ingredients in proving these results are uniform $\mathrm{L}^\infty$- and $\mathrm{TV}$-estimates that ensure compactness of approximate solutions, and discrete entropy inequalities that ensure the entropy admissibility of the limit solution.