  We consider an open interacting particle system on a finite lattice. The
particles perform asymmetric simple exclusion and are randomly created or
destroyed at all sites, with rates that grow rapidly near the boundaries. We
study the hydrodynamic limit for the particle density at the hyperbolic
space-time scale and obtain the entropy solution to a boundary-driven
quasilinear conservation law with a relaxation term. Different from the usual
boundary conditions introduced in [Bardos, Roux, and Nedelec, (1979), Comm.
Part. Diff. Equ], discontinuity (boundary layer) does not formulate at the
boundaries due to the strong relaxation scheme.
