  We study the convergence of a class of asymptotic preserving numerical
schemes initially proposed by F. Filbet & S. Jin \cite{filb1} and G. Dimarco &
L. Pareschi \cite{DimarcoP} in the context of nonlinear and stiff kinetic
equations. Here, our analysis is devoted to the approximation of a system of
transport equations with a nonlinear source term, for which the asymptotic
limit is given by a conservation laws. We investigate the convergence of the
approximate solution $(\ueps_h,\veps_h)$ to a nonlinear relaxation system,
where $\eps>0$ is a physical parameter and $h$ represents the discretization
parameter. Uniform convergence with respect to $\eps$ and $h$ is proven and
error estimates are also obtained. Finally, several numerical tests are
performed to illustrate the accuracy and efficiency of such a scheme.
