2025/09/30 by De Nitti, Nicola, Huang, Kuang · 4 citations
#35L65 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2510.00221
We consider a class of nonlocal conservation laws modeling traffic flows, given by ∂t ρε + ∂x(V(ρε ∗ γε) ρε) = 0 with a suitable convex kernel γε , 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 Wε := ρε ∗ γε 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 ρε , along with an L1 -contraction property for Wε . 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.