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：本文提出一种基于局部间断伽辽金方法的非局部数值格式，用于求解一类非局部扩散偏微分方程及其在交通流中的应用。在该模型中，速度由交通密度的平均值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项可能引发的非物理行为及其在本模型中的规避措施。数值结果表明，该方法能精确求解此类方程，且模型可有效捕捉目标交通流特性。研究表明，非局部卷积计算具有$\mathcal{O}(n^2)$复杂度，但采用本文提出的高阶格式可有效缓解计算耗时增加的问题。