  The present work investigates the numerical evolution of linearized
oscillations of non-rotating, spherically symmetric neutron stars within the
framework of general relativity. We derive the appropriate equations using the
(3+1)-formalism. We first focus on the evolution of radial oscillations, which
do not emit gravitational waves. We demonstrate how to handle a numerical
instability that also occurs in the non-radial case, when the stellar model is
constructed based on a realistic equation of state. We devise a coordinate
transformation that not only removes this instability but also provides much
more accurate results. [...] The main part deals with the evolution of
non-radial oscillations (l >= 2) of neutron stars. Here, we compare different
formulations of the equations and discuss how they have to be numerically dealt
with in order to avoid instabilities at the origin. We present results for
various polytropic stellar models and different initial data. [...] In the last
part of this thesis we consider a physical mechanism for exciting oscillations
of neutron stars. We use the time dependent gravitational field of a small
point mass \mu that orbits the neutron star to induce stellar oscillations.
With this particle we have a physical means which removes the arbitrariness in
choosing the initial data. [...] By sampling various orbital parameters of the
particle we show that in general the particle is not able to excite any
w-modes. It is only for speeds very close to the speed of light that the w-mode
is a significant part of the wave signal.
