  Gravitational wave and electromagnetic observations can provide new insights
into the nature of matter at supra-nuclear densities inside neutron stars.
Improvements in electromagnetic and gravitational wave sensing instruments
continue to enhance the accuracy with which they can measure the masses, radii,
and tidal deformability of neutron stars. These better measurements place
tighter constraints on the equation of state of cold matter above nuclear
density. In this article, we discuss a complementary approach to get insights
into the structure of neutron stars by providing a model prediction for
non-linear fundamental eigenmodes (f-modes) and their decay over time, which
are thought to be induced by time-dependent tides in neutron star binaries.
Building on pioneering studies that relate the properties of f-modes to the
structure of neutron stars, we systematically study this link in the
non-perturbative regime using models that utilize numerical relativity. Using a
suite of fully relativistic numerical relativity simulations of oscillating TOV
stars, we establish blueprints for the numerical accuracy needed to accurately
compute the frequency and damping times of f-mode oscillations, which we expect
to be a good guide for the requirements in the binary case. We show that the
resulting f-mode frequencies match established results from linear perturbation
theory, but the damping times within numerical errors depart from linear
predictions. This work lays the foundation for upcoming studies aimed at a
comparison of theoretical models of f-mode signatures in gravitational waves,
and their uncertainties with actual gravitational wave data, searching for
neutron star binaries on highly eccentric orbits, and probing neutron star
structure at high densities.
