STIFF SYSTEMS OF HYPERBOLIC CONSERVATION-LAWS - CONVERGENCE AND ERROR-ESTIMATES

We are concerned with 2 x 2 nonlinear relaxation systems of conservati on laws of the for u(t) + f(u)(x) = -1/8S(u,v), u(t) = 1/8S(u,v) which are coupled through the stiff source term 1/8 S(u, v). Such systems a rise as prototype models for combustion, adsorption, etc. Here we stud y the convergence of (u, v) = (u(delta), v(delta)) to its equilibrium state, ((u) over bar,(v) over bar), governed by the limiting equations , (u) over bar(t) + (v) over bar(t) + f((u) over bar(x) = 0, S((u) ove r bar,(v) over bar = 0. In particular, we provide sharp convergence ra te estimates as the relaxation parameter delta down arrow 0; The novel ty of our approach is the use of a weak W-1(L-1)-measure of the error, which allows us to obtain sharp error estimates. It is shown that the error consists of an initial contribution of size parallel to S(u(0)( delta),v(0)(delta))parallel to(L)(1); together with accumulated relaxa tion error of order phi(delta). The sharpness of our results is found to be in complete agreement with the numerical experiments reported in [Schfoll, Tveito, and Winther, SIAM J. numer. Anal., 34 (1997), pp. 1 152-1166].