We consider a class of nonlocal conservation laws modeling traffic flows, given by ₜ ρ_ + ₓ (V (ρ_ γ_) ρ_) = 0 with a suitable convex kernel γ_, and its Godunov-type numerical discretization. We prove that, as the nonlocal parameter and mesh size h tend to zero simultaneously, the discrete approximation W, ₇ of W_: = ρ_ γ_ 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 +h+\, t+h\, t. In particular, with an exponential kernel, we establish the same convergence result for the discrete approximation ρ, ₇ of ρ_, along with an L¹ -contraction property for W_. 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.
Nitti et al. (Tue,) studied this question.