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We consider a class of Hamiltonian PDEs that can be split into a linearunbounded operator and a regular non linear part, and we analyze theirnumerical discretizations by symplectic methods when the initial value is smallin Sobolev norms. The goal of this work is twofold: First we show how standardapproximation methods cannot in general avoid resonances issues, and we givenumerical examples of pathological behavior for the midpoint rule andimplicit-explicit integrators. Such phenomena can be avoided by suitabletruncations of the linear unbounded operator combined with classical splittingmethods. We then give a sharp bound for the cut-off depending on the time step.Using a new normal form result, we show the long time preservation of theactions for such schemes for all values of the time step, provided the initialcontinuous system does not exhibit resonant frequencies.

Resonances in long time integration of semi linear Hamiltonian PDEs