Convergence of the Godunov scheme for a scalar conservation law with time and space discontinuities
John D. Towers
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 $x=0$. Time discontinuities occur in the coefficient independently on either side of the interface. This setup applies to the LWR traffic model in the case where different time-varying speed limits are imposed on different segments of a road. We prove that approximate solutions produced by the Godunov scheme converge to the unique entropy solution, as defined in [⭑]. 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.
[⭑] G.M. Coclite and N.H. Risebro, Conservation Laws with time dependent discontinuous coefficients, SIAM J. Math. Anal., 36 (2005), pp. 1293–1309.