  We consider a kinetic relaxation model and an associated macroscopic scalar
nonlinear hyperbolic equation on a network. Coupling conditions for the
macroscopic equations are derived from the kinetic coupling conditions via an
asymptotic analysis near the nodes of the network. This analysis leads to the
combination of kinetic half-space problems with Riemann problems at the
junction. Detailed numerical comparisons between the different models show the
agreement of the coupling conditions for the case of tripod networks.
