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Section: New Results

Birkhoff normal form for splitting methods applied semi linear Hamiltonian PDEs. Part II: Abstract splitting

Participant : Erwan Faou.

This work [26] extends the results of the previous paper to the case where no space discretization is made in the splitting methods applied to Hamiltonian PDEs that can be split into a linear unbounded operator and a regular non linear part. Obtaining results for the abstract splitting method is equivalent to obtaining bounds in classical Birkhoff normal form results that are independent of the dimension of the phase space. Using techniques recently developed to prove conservation results for the exact solution of Hamiltonian PDEs, we prove a normal form result for the corresponding discrete flow under generic non resonance conditions on the frequencies of the linear operator and on the step size. This result implies the conservation of the regularity of the numerical solution associated with the splitting method over arbitrary long time, provided the initial data is small enough. This result holds for numerical schemes controlling the round-off error at each step to avoid possible high frequency energy drift.


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