  In this paper we construct high order finite volume schemes on networks of
hyperbolic conservation laws with coupling conditions involving ODEs. We
consider two generalized Riemann solvers at the junction, one of Toro-Castro
type and a solver of Harten, Enquist, Osher, Chakravarthy type. The ODE is
treated with a Taylor method or an explicit Runge-Kutta scheme, respectively.
Both resulting high order methods conserve quantities exactly if the
conservation is part of the coupling conditions. Furthermore we present a
technique to incorporate lumped parameter models, which arise from simplifying
parts of a network. The high order convergence and the robust capturing of
shocks is investigated numerically in several test cases.

Numerical Analysis