  In this article, we show that prescribing homogeneous Neumann type numerical
boundary conditions at an outflow boundary yields a convergent discretization
in $\ell^\infty$ for transport equations. We show in particular that the
Neumann numerical boundary condition is a stable, local, and absorbing
numerical boundary condition for discretized transport equations. Our main
result is proved for explicit two time level numerical approximations of
transport operators with arbitrarily wide stencils. The proof is based on the
energy method and bypasses any normal mode analysis.
