The well-known Lighthill-Whitham-Richards (LWR) kinematic model of traffic flow models the evolution of the local density of cars by a nonlinear scalar conservation law. The transition between free and congested flow regimes can be described by a flux or velocity function that has a discontinuity at a determined density. A numerical scheme to handle the resulting LWR model with discontinuous velocity was proposed in [J.D. Towers, A splitting algorithm for LWR traffic models with flux discontinuities in the unknown, J. Comput. Phys., 421 (2020), article 109722]. A similar scheme is constructed by decomposing the discontinuous velocity function into a Lipschitz continuous function plus a Heaviside function and designing a corresponding splitting scheme. The part of the scheme related to the discontinuous flux is handled by a semi-implicit step that does, however, not involve the solution of systems of linear or nonlinear equations. It is proved that the whole scheme converges to a weak solution in the scalar case. The scheme can in a straightforward manner be extended to the multiclass LWR (MCLWR) model, which is defined by a hyperbolic system of $ N $ conservation laws for $ N $ driver classes that are distinguished by their preferential velocities. It is shown that the multiclass scheme satisfies an invariant region principle, that is, all densities are nonnegative and their sum does not exceed a maximum value. In the scalar and multiclass cases no flux regularization or Riemann solver is involved, and the CFL condition is not more restrictive than for an explicit scheme for the continuous part of the flux. Numerical tests for the scalar and multiclass cases are presented.

中文翻译：

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具有不连续速度函数的多类Lighthill-Whitham-Richards交通模型

著名的Lighthill-Whitham-Richards（LWR）运动流动力学模型通过非线性标量守恒定律来模拟汽车局部密度的变化。自由和拥挤流态之间的过渡可以通过在确定的密度下具有不连续性的通量或速度函数来描述。在[JD Towers，J。Comput。，2004年，在通量不连续的LWR交通模型的分裂算法中，提出了一种处理所得的具有不连续速度的LWR模型的数值方案。物理 ，421（2020），第109722条]。通过将不连续速度函数分解为Lipschitz连续函数加Heaviside函数并设计相应的拆分方案，可以构造类似的方案。方案中与不连续磁通有关的部分由半隐式步骤处理，但是该步骤不涉及线性或非线性方程组的求解。证明了在标量情况下整个方案收敛到一个弱解。该方案可以以直接的方式扩展到多类LWR（MCLWR）模型，该模型由针对N $驾驶员类的N $守恒律的双曲线系统定义，该N $驾驶员类通过其优先速度来区分。结果表明，多类方案满足不变区域原理，即 所有密度均为非负数，总和不超过最大值。在标量和多类情况下，不涉及通量正则化或Riemann求解器，并且CFL条件并不比通量连续部分的显式方案更具限制性。给出了标量和多类情况的数值测试。