Stiff Systems of Hyperbolic Conservation Laws: Convergence and Error Estimates

Abstract

We are concerned with $2\times2$ nonlinear relaxation systems of conservation laws of the form $u_t+f(u)_x=-\frac{1}{\delta}S(u,v), v_t=\frac{1}{\delta}S(u,v)$ which are coupled through the stiff source term $\frac{1}{\delta}S(u,v)$. Such systems arise as prototype models for combustion, adsorption, etc. Here we study the convergence of $(u,v)\equiv(\ud,\vd)$ to its equilibrium state, $(\bar{u},\bar{v})$, governed by the limiting equations, $\bar{u}_t+\bar{v}_t+ f(\bar{u})_x=0, S(\bar{u},\bar{v})=0$. In particular, we provide sharp convergence rate estimates as the relaxation parameter $\delta \downarrow 0$. The novelty 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 ${||S(u_0^\delta,v_0^\delta)||}_{L^1}$, together with accumulated relaxation error of order ${\cal O}(\delta)$. The sharpness of our results is found to be in complete agreement with the numerical ex...