  We study the numerical approximation of singularly perturbed
convection-diffusion problems on one-dimensional pipe networks. In the
vanishing diffusion limit, the number and type of boundary conditions and
coupling conditions at network junctions changes, which gives rise to singular
layers at the outflow boundaries of the pipes. A hybrid discontinuous Galerkin
method is proposed, which provides a natural upwind mechanism for the
convection-dominated case. Moreover, the method automatically handles the
variable coupling and boundary conditions in the vanishing diffusion limit,
leading to an asymptotic-preserving scheme. A detailed analysis of the
singularities of the solution and the discretization error is presented, and an
adaptive strategy is proposed, leading to order optimal error estimates that
hold uniformly in the singular perturbation limit. The theoretical results are
confirmed by numerical tests.
