Stiff Well-Posedness and Asymptotic Convergence for a Class of Linear Relaxation Systems in a Quarter Plane

In this paper we study the asymptotic equivalence of a general linear system of 1-dimensional conservation laws and the corresponding relaxation model proposed by S. Jin and Z. Xin (1995, Comm. Pure Appl. Math.48, 235–277) in the limit of small relaxation rate. The main interest is this asymptotic equivalence in the presence of physical boundaries. We identify and rigorously justify a necessary and sufficient condition (which we call the Stiff Kreiss Condition, or SKC in short) on the boundary condition to guarantee the uniform well-posedness of the initial boundary value problem for the relaxation system independent of the rate of relaxation. The SKC is derived and simplified by using a normal mode analysis and a conformal mapping theorem. The asymptotic convergence and boundary layer behavior are studied by the Laplace transform and a matched asymptotic analysis. An optimal rate of convergence is obtained.