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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.
Lu et al. (Mon,) studied this question.