Initial-boundary value problem to systems of conservation laws with relaxation
In this paper we consider the initial-boundary value problem (IBVP) for the one-dimensional Jin-Xin relaxation model. The main interest is to study the boundary layer behaviors in the solutions to the IBVP of the relaxation system and their asymptotic convergence to solutions of the corresponding hyperbolic conservation laws in the limit of small relaxation rate. First we develop a general expansion theory for the relaxation IBVP using a matched asymptotic analysis. This formal procedure determines a unique equilibrium limit, and also reveals rich initial and boundary layer structures in the solutions of the relaxation system. Arbitrarily accurate solutions to the IBVP of the relaxation system are then constructed by combining the various orders of the equilibrium solutions, the initial and boundary layer solutions. The validity of the initial and boundary layers and the asymptotic convergence results are rigorously justified through a stability analysis for a broad class of boundary conditions in the case when the relaxation system is 2 x 2.