Boundary conditions for hyperbolic systems with stiff source terms

This work is concerned with boundary conditions for multi-dimensional first -order hyperbolic systems with stiff source terms (also called relaxation). It is observed that usual relaxation stability conditions and the uniform Kreiss condition are not enough for the existence of the zero relaxation li mit. To remedy this, we propose a so-called generalized Kreiss condition fo r initial-boundary value problems (henceforth, IBVPs) of the relaxation sys tems. By assuming that the relaxation system admits the quasi-stability con dition and the prescribed boundary condition satisfies the generalized Krei ss condition, we derive a reduced boundary condition, for the corresponding equilibrium system, satisfying the uniform Kreiss condition and show the e xistence of boundary-layers. Moreover, if the relaxation system admits a mo re restrictive relaxation stability condition, then Friedrichs' strictly di ssipative boundary conditions, which induce certain uniform stability estim ates, are shown to satisfy the generalized Kreiss condition. The present results are expected to be used as theoretical criteria to cons truct relaxation approximations for IBVPs of conservation laws, which are o f practical interest.