Convergence of the Godunov scheme for a scalar conservation law with time and space discontinuities

Abstract

We consider the Godunov scheme as applied to a scalar conservation law whose flux has discontinuities in both space and time. The time-and space-dependence of the flux occurs through a positive multiplicative coefficient. That coefficient has a spatial discontinuity along a fixed interface at [Formula: see text]. Time discontinuities occur in the coefficient independently on either side of the interface. This setup applies to the Lighthill–Witham–Richards (LWR) traffic model in the case where different time-varying speed limits are imposed on different segments of a road. We prove that the approximate solutions produced by the Godunov scheme converge to the unique entropy solution, as defined by Coclite and Risebro in 2005. Convergence of the Godunov scheme in the presence of spatial flux discontinuities alone is a well-established fact. The novel aspect of this paper is convergence in the presence of additional temporal flux discontinuities.