In this paper we present a non-local numerical scheme based on the Local Discontinuous Galerkin method for a non-local diffusive partial differential equation with application to traffic flow. In this model, the velocity is determined by both the average of the traffic density as well as the changes in the traffic density at a neighborhood of each point. We discuss nonphysical behaviors that can arise when including diffusion, and our measures to prevent them in our model. The numerical results suggest that this is an accurate method for solving this type of equation and that the model can capture desired traffic flow behavior. We show that computation of the non-local convolution results in $\mathcal{O}(n^2)$ complexity, but the increased computation time can be mitigated with high-order schemes like the one proposed.

翻译：本文提出了一种基于局部间断Galerkin方法的非局部数值格式，用于求解一类具有交通流应用背景的非局部扩散偏微分方程。在该模型中，速度同时由交通密度的平均值以及各点邻域内交通密度的变化共同决定。我们讨论了引入扩散时可能产生的非物理行为，并提出了模型中避免这些行为的措施。数值结果表明，该方法能精确求解此类方程，且模型可捕捉期望的交通流行为。我们指出，非局部卷积的计算复杂度为$\mathcal{O}(n^2)$，但通过采用本文提出的高阶格式可有效缓解计算时间的增加。