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 derive a non-uniform bound on the growth rate of the total variation for approximate solutions. Since the total variation can explode only at $x=0$, we can provide a convergence proof for our scheme in $BV_{loc}(\mathbb{R}\setminus \lbrace 0 \rbrace)$ by using Helly's compactness theorem.

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。我们推导了近似解总变差增长率的非一致界。由于总变差仅在 $x=0$ 处可能爆炸，通过应用Helly紧致性定理，我们能够证明该格式在 $BV_{loc}(\mathbb{R}\setminus \lbrace 0 \rbrace)$ 空间中的收敛性。