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