  We consider kinetic and associated macroscopic equations on networks. The
general approach will be explained in this paper for a linear kinetic BGK model
and the corresponding limit for small Knudsen number, which is the wave
equation. Coupling conditions for the macroscopic equations are derived from
the kinetic conditions via an asymptotic analysis near the nodes of the
network. This analysis leads to the consideration of a fixpoint problem
involving the coupled solutions of kinetic half-space problems. A new
approximate method for the solution of kinetic half-space problems is derived
and used for the determination of the coupling conditions. Numerical
comparisons between the solutions of the macroscopic equation with different
coupling conditions and the kinetic solution are presented for the case of
tripod and more complicated networks.
