Convergence of relaxation schemes for initial boundary value problems for conservation laws

We consider a scalar conservation law with stiff source term in the quarter plan. This equation is relaxed in a quasi-linear hyperbolic system. The relaxed system is approximated using an upwind scheme converging for fixed relaxation time and vanishing discretization mesh. Further, we use Chapman-Enskog expansion to obtain a viscous scheme for the equilibrium law and prove its convergence to the physical solution. Boundary information is carefully handled in an appropriate inequality linking the entropy numerical flux and the flux function. An extension to general conservative schemes is also investigated.