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 ∂ₜ ρ_ε + ∂ₓ(V(ρ_ε γ_ε) ρ_ε) = 0 with a suitable convex kernel γ_ε, and its Godunov-type numerical discretization. We prove that, as the nonlocal parameter ε and mesh size h tend to zero simultaneously, the discrete approximation Wε,h of W_ε := ρ_ε γ_ε converges to the entropy solution of the (local) scalar conservation law ∂ₜ ρ+ ∂ₓ(V(ρ) ρ) = 0, with an explicit convergence rate estimate of order ε+h+√ε\, t+√h\,t. In particular, with an exponential kernel, we establish the same convergence result for the discrete approximation ρε,h of ρ_ε, along with an L¹-contraction property for W_ε. The key ingredients in proving these results are uniform L^∞- and TV-estimates that ensure compactness of approximate solutions, and discrete entropy inequalities that ensure the entropy admissibility of the limit solution.