In this work, we present a semi-discrete scheme to approximate solutions to the scalar LWR traffic model with spatially discontinuous flux, described by the equation $u_t+(k(x)u(1-u))_x=0$. This approach is based on the Lagrangian-Eulerian method proposed by E. Abreu, J. Francois, W. Lambert, and J. Perez [J. Comp. Appl. Math.406 (2022) 114011] for scalar conservation laws. We provide a convergence proof for our scheme, which relies on Kolmogorov-Riesz's $L^1$-compactness theorem. We also derive a non-uniform bound on the growth rate of the total variation for approximate solutions.

翻译：本文提出了一种半离散格式，用于近似求解具有空间间断流通量的标量LWR交通模型，该模型由方程 $u_t+(k(x)u(1-u))_x=0$ 描述。该方法基于E. Abreu、J. Francois、W. Lambert和J. Perez [J. Comp. Appl. Math.406 (2022) 114011] 针对标量守恒律提出的拉格朗日-欧拉方法。我们为所提格式提供了收敛性证明，该证明依赖于Kolmogorov-Riesz的 $L^1$ 紧性定理。我们还推导了近似解总变差增长率的非一致界。