Stability of numerical schemes for the initial-boundary value problem of relaxation balance laws

In many industrial applications, models are based on hyperbolic partial differential equations which involve source terms. One of the main features of these models is related to the notion of dissipation, leading to smooth solutions and asymptotic stability. The most classical model is the Jin-Xin system and we consider here the associated initial-boudary value problem (IBVP) in the quarter plane. This problem has been addressed by Xin and Xu who derive a necessary and sufficient condition for stiff stability, i.e. stability uniformly with respect to the stiffness of the relaxation term. Our aim is to study the discrete case in the context of finite difference. We study for instance the relationship between the stiff stability of numerical solutions and the Stiff Kreiss Condition (SKC) for the boundary problem. Due to the effects of the boundary layer and the interactions of the bound- ary and initial layer, numerical schemes have to be properly designed in order to provide accurate approximations and consistant behaviors. The asymptotic stability and boundary layer behavior are studied by discrete L2 analysis and energy estimate .