  High order numerical methods for networks of hyperbolic conservation laws
have recently gained increasing popularity. Here, the crucial part is to treat
the boundaries of the single (one-dimensional) computational domains in such a
way that the desired convergence rate is achieved in the smooth case but also
stability criterions are fulfilled, in particular in the presence of
discontinuities. Most of the recently proposed methods rely on a WENO
extrapolation technique introduced by Tan and Shu in [\emph{J.\ Comput.\ Phys.}
229, pp.\ 8144--8166 (2010)]. Within this work, we refine and in a sense
generalize these results for the case of a third order scheme. Numerical
evidence for the analytically found parameter bounds is given as well as
results for a complete third order scheme based on the proposed boundary
treatment.
