  At finite temperature, the hydrodynamic collective modes of superfluid
trapped Fermi gases are coupled to the motion of the normal component, which in
the BCS limit behaves like a collisionless normal Fermi gas. The coupling
between the superfluid and the normal components is treated in the framework of
a semiclassical transport theory for the quasiparticle distribution function,
combined with a hydrodynamic equation for the collective motion of the
superfluid component. We develop a numerical test-particle method for solving
these equations in the linear response regime. As a first application we study
the temperature dependence of the collective quadrupole mode of a Fermi gas in
a spherical trap. The coupling between the superfluid collective motion and the
quasiparticles leads to a rather strong damping of the hydrodynamic mode
already at very low temperatures. At higher temperatures the spectrum has a
two-peak structure, the second peak corresponding to the quadrupole mode in the
normal phase.
