We consider a scalar conservation law with zero-flux boundary conditions imposed on the boundary of a rectangular multidimensional domain. We study monotone schemes applied to this problem. For the Godunov version of the scheme, we simply set the boundary flux equal to zero. For other monotone schemes, we additionally apply a simple modification to the numerical flux. We show that the approximate solutions produced by these schemes converge to the unique entropy solution, in the sense of [7], of the conservation law. Our convergence result relies on a BV bound on the approximate numerical solution. In addition, we show that a certain functional that is closely related to the total variation is nonincreasing from one time level to the next. We extend our scheme to handle degenerate convection-diffusion equations and for the one-dimensional case we prove convergence to the unique entropy solution.

Traffic flow is modeled by a conservation law describing the density of cars. It isassumed that each driver chooses his own departure time in order to minimize the sum of adeparture and an arrival cost. There are N groups of drivers, Thei-th group consists of κidrivers, sharing the same departure and arrival costsϕi(t),ψi(t).For any given population sizesκ1,...,κn,we prove the existence of a Nash equilibrium solution, where no driver can lower his owntotal cost by choosing a different departure time. The possible non-uniqueness, and acharacterization of this Nash equilibrium solution, are also discussed.