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

High order numerical methods for networks of hyperbolic conservation laws coupled with ODEs and lumped parameter models