Stiff Systems of Hyperbolic Conservation Laws: Convergence and Error Estimates

We are concerned with 2×2 nonlinear relaxation systems of conservation laws of the form uₜ+f(u)ₓ=-1/(u,v), vₜ=1/(u,v) which are coupled through the stiff source term 1/(u,v). Such systems arise as prototype models for combustion, adsorption, etc. Here we study the convergence of (u,v)≡(,) to its equilibrium state, (ū,v̄), governed by the limiting equations, ūₜ+v̄ₜ+ f(ū)ₓ=0, S(ū,v̄)=0. In particular, we provide sharp convergence rate estimates as the relaxation parameter δ ↓ 0. The novelty of our approach is the use of a weak W⁻¹(L¹)-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 ||S(u₀^δ,v₀^δ)||L¹, together with accumulated relaxation error of order O(δ). The sharpness of our results is found to be in complete agreement with the numerical experiments reported in [Schroll, Tveito, and Winther, SIAM J. Numer. Anal., 34 (1997), pp. 1152--1166].