  This paper presents a systematic methodology for the discretization and
reduction of a class of one-dimensional Partial Differential Equations (PDEs)
with inputs and outputs collocated at the spatial boundaries. The class of
system that we consider is known as Boundary-Controlled Port-Hamiltonian
Systems (BC-PHSs) and covers a wide class of Hyperbolic PDEs with a large type
of boundary inputs and outputs. This is, for instance, the case of waves and
beams with Neumann, Dirichlet, or mixed boundary conditions. Based on a
Partitioned Finite Element Method (PFEM), we develop a numerical scheme for the
structure-preserving spatial discretization for the class of one-dimensional
BC-PHSs. We show that if the initial PDE is passive (or impedance energy
preserving), the discretized model also is. In addition and since the
discretized model or Full Order Model (FOM) can be of large dimension, we
recall the standard Loewner framework for the Model Order Reduction (MOR) using
frequency domain interpolation. We recall the main steps to produce a Reduced
Order Model (ROM) that approaches the FOM in a given range of frequencies. We
summarize the steps to follow in order to obtain a ROM that preserves the
passive structure as well. Finally, we provide a constructive way to build a
projector that allows to recover the physical meaning of the state variables
from the ROM to the FOM. We use the one-dimensional wave equation and the
Timoshenko beam as examples to show the versatility of the proposed approach.
