  We consider partial differential equations on networks with a small parameter
$\epsilon$, which are hyperbolic for $\epsilon>0$ and parabolic for
$\epsilon=0$. With a combination of an $\epsilon$-expansion and Runge-Kutta
schemes for constrained systems of parabolic type, we derive a new class of
time discretization schemes for hyperbolic systems on networks, which are
constrained due to interconnection conditions. For the analysis we consider the
coupled system equations as partial differential-algebraic equations based on
the variational formulation of the problem. We discuss well-posedness of the
resulting systems and estimate the error caused by the $\epsilon$-expansion.
