  We consider singularly perturbed convection-diffusion equations on
one-dimensional networks (metric graphs) as well as the transport problems
arising in the vanishing diffusion limit. Suitable coupling condition at inner
vertices are derived that guarantee conservation of mass as well as dissipation
of a mathematical energy which allows us to prove stability and well-posedness.
For single intervals and appropriately specified initial conditions, it is
well-known that the solutions of the convection-diffusion problem converge to
that of the transport problem with order $O(\sqrt{\epsilon})$ in the
$L^\infty(L^2)$-norm with diffusion $\epsilon \to 0$. In this paper, we prove a
corresponding result for problems on one-dimensional networks. The main
difficulty in the analysis is that the number and type of coupling conditions
changes in the singular limit which gives rise to additional boundary layers at
the interior vertices of the network. Since the values of the solution at these
network junctions are not known a-priori, the asymptotic analysis requires a
delicate choice of boundary layer functions that allows to handle these
interior layers.
