  We present the derivation of hydrodynamical equations for a perfect fluid in
General Relativity, within the 3+1 decomposition of spacetime framework, using
only primitive variables. Primitive variables are opposed to conserved
variables, as defined in the widely used Valencia formulation of the same
hydrodynamical equations. The equations are derived in a covariant way, so that
they can be used to describe any configuration of the perfect fluid. Once
derived, the equations are tested numerically. We implement them in an
evolution code for spherically symmetric self-gravitating compact objects. The
code uses pseudospectral methods for both the metric and the hydrodynamics.
First, convergence tests are performed, then the frequencies of radial modes of
polytropes are recovered with and without the Cowling approximation, and
finally the performance of our code in the black hole collapse and migration
tests are described. The results of the tests and the comparison with a
reference corecollapse and neutron star oscillations code suggests that not
only our code can handle very strong gravitational fields, but also that this
new formulation helps gaining a significant amount of computational time in
hydrodynamical simulations of smooth flows in General Relativity.
