We propose a finite difference algorithm for a scalar conservation law associated with the Lighthill-Witham-Richards model of traffic flow for the case where the flux is discontinuous with respect to the unknown. Specifically, the flux has a single decreasing jump, which models a transition from free to congested traffic flow. We write the flux as a Lipschitz continuous flux plus a Heaviside flux, and then construct a splitting based on this decomposition. The portion of the scheme associated with the Heaviside flux is implicit, but does not require the iterative solution of a system of nonlinear equations. The portion associated with the continuous flux is a standard Godunov scheme. The scheme does not employ a flux regularization, nor a Riemann solver for the discontinuous flux problem. Standard hyperbolic time steps are allowed, i.e., the time steps are dictated by the CFL condition associated with the continuous portion of the flux. We prove that the approximate solutions converge, up to extraction of a subsequence, to a weak solution of the conservation law. We propose a shock admissibility criterion, and present some numerical examples.

对于流量相对于未知量不连续的情况，我们提出了一种与Lighthill-Witham-Richards模型相关的标量守恒律的差分算法。具体而言，通量具有单个递减的跳跃，该跳跃模拟了从自由交通流向拥挤交通流的过渡。我们将通量写为Lipschitz连续通量加上Heaviside通量，然后基于该分解构造拆分。该方案中与Heaviside通量相关的部分是隐式的，但不需要非线性方程组的迭代解。与连续通量相关的部分是标准的Godunov方案。该方案没有使用通量正则化，也没有使用Riemann求解器来解决不连续通量问题。允许使用标准双曲线时间步长，即 时间步长由与通量连续部分相关的CFL条件决定。我们证明，近似解收敛到一个子序列的提取，直到守恒律的一个弱解。我们提出了冲击容许性准则，并给出了一些数值示例。