  We consider {\em discretized} Hamiltonian PDEs associated with a Hamiltonian
function that can be split into a linear unbounded operator and a regular
nonlinear part. We consider splitting methods associated with this
decomposition. Using a finite dimensional Birkhoff normal form result, we show
the almost preservation of the {\em actions} of the numerical solution
associated with the splitting method over arbitrary long time, provided the
Sobolev norms of the initial data is small enough, and for asymptotically large
level of space approximation. This result holds under {\em generic} non
resonance conditions on the frequencies of the linear operator and on the step
size. We apply this results to nonlinear Schr\"odinger equations as well as the
nonlinear wave equation.}
