Abstract
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.
Published Version
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