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We consider {\em discretized} Hamiltonian PDEs associated with a Hamiltonianfunction that can be split into a linear unbounded operator and a regularnonlinear part. We consider splitting methods associated with thisdecomposition. Using a finite dimensional Birkhoff normal form result, we showthe almost preservation of the {\em actions} of the numerical solutionassociated with the splitting method over arbitrary long time, provided theSobolev norms of the initial data is small enough, and for asymptotically largelevel of space approximation. This result holds under {\em generic} nonresonance conditions on the frequencies of the linear operator and on the stepsize. We apply this results to nonlinear Schr\"odinger equations as well as thenonlinear wave equation.}

Birkhoff normal form and splitting methods for semi linear Hamiltonian PDEs. Part I: Finite dimensional discretization