  Axisymmetric pulsations of rotating neutron stars can be excited in several
scenarios, such as core-collapse, crust and core-quakes and binary mergers and
could become detectable either in gravitational waves or high-energy radiation.
Here, we present a comprehensive study of all low-order axisymmetric modes of
uniformly and rapidly rotating relativistic stars. Initial stationary
configurations are appropriately perturbed and are numerically evolved using an
axisymmetric, nonlinear relativistic hydrodynamics code, assuming
time-independence of the gravitational field (Cowling approximation). The
simulations are performed using a high-resolution shock-capturing
finite-difference scheme accurate enough to maintain the initial rotation law
for a large number of rotational periods, even for stars at the mass-shedding
limit. Through Fourier transforms of the time evolution of selected fluid
variables, we compute the frequencies of quasi-radial and non-radial modes with
spherical harmonic indices l=0,1,2 and 3, for a sequence of rotating stars from
the non-rotating limit to the mass-shedding limit. The frequencies of the
axisymmetric modes are affected significantly by rotation only when the
rotation rate exceeds about 50% of the maximum allowed. As expected, at large
rotation rates, apparent mode crossings between different modes appear. In
addition to the above modes, several axisymmetric inertial modes are also
excited in our numerical evolutions.
