  Projection-based model order reduction of dynamical systems usually
introduces an error between the high-fidelity model and its counterpart of
lower dimension. This unknown error can be bounded by residual-based methods,
which are typically known to be highly pessimistic in the sense of largely
overestimating the true error. This work applies two improved error bounding
techniques, namely (a) a hierarchical error bound and (b) an error bound based
on an auxiliary linear problem, to the case of port-Hamiltonian systems. The
approaches rely on a second approximation of (a) the dynamical system and (b)
the error system. In this paper, these methods are for the first time adapted
to port-Hamiltonian systems by exploiting their structure. The mathematical
relationship between the two methods is discussed both, theoretically and
numerically. The effectiveness of the described methods is demonstrated using a
challenging three-dimensional port-Hamiltonian model of a classical guitar with
fluid-structure interaction.
