Title: Asymptotically compatible entropy-consistent discretization for a class of nonlocal conservation laws Show Chinese title

Title: 非局部守恒定律一类的渐近相容熵一致离散化

Abstract: 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. Show Chinese abstract

Abstract: 我们考虑一类非局部守恒定律，用于模拟交通流，由$ \partial_t \rho_\varepsilon + \partial_x(V(\rho_\varepsilon \ast \gamma_\varepsilon) \rho_\varepsilon) = 0 $给出，带有合适的凸核$ \gamma_\varepsilon $，以及其Godunov型数值离散化。 我们证明了当非局部参数$ \varepsilon $和网格尺寸$ h $同时趋于零时，$ W_\varepsilon := \rho_\varepsilon \ast \gamma_\varepsilon $的离散逼近$ W_{\varepsilon,h} $收敛于 (局部) 标量守恒律$ \partial_t \rho + \partial_x(V(\rho) \rho) = 0 $的熵解，并给出了一个阶为$ \varepsilon+h+\sqrt{\varepsilon\, t}+\sqrt{h\,t} $的显式收敛速率估计。 特别是，使用指数核，我们建立了离散近似$ \rho_{\varepsilon,h} $对$ \rho_\varepsilon $的相同收敛结果，以及$ \mathrm{L}^1 $-收缩性质对于$ W_\varepsilon $。证明这些结果的关键要素是确保近似解紧性的统一$ \mathrm{L}^\infty $-和$\mathrm{TV}$-估计，以及确保极限解的熵可接受性的离散熵不等式。