We consider an asymptotic preserving numerical scheme initially proposed by F. Filbet andS. Jin [J. Comput. Phys. 229 (2010)] and G. Dimarco and L.Pareschi [SIAM J. Numer. Anal. 49 (2011) 2057–2077] in thecontext of nonlinear and stiff kinetic equations. Here, we propose a convergence analysisof such a scheme for the approximation of a system of transport equations with a nonlinearsource term, for which the asymptotic limit is given by a conservation law. We investigatethe convergence of the approximate solution (uεh, vεh) to a nonlinear relaxation system, whereε > 0 is a physical parameter andh represents the discretization parameter. Uniform convergence withrespect to ε and h is proved and error estimates arealso obtained. Finally, several numerical tests are performed to illustrate the accuracyand efficiency of such a scheme.
