We prove that A-stable symplectic Runge-Kutta time semidiscretizations applied to semilinear wave equations with periodic boundary conditions, analytic nonlinearity and analytic initial data conserve a modified energy p to an exponentially small error. This modified energy is O(h^p)-close to the original energy where p is the order of the method and h the time-stepsize. Standard backward error analysis can not be applied because of the occurrence of unbounded operators in the construction of the modified vectorfield. This loss of regularity in the construction can be taken care of by projecting the PDE to a finite-dimensional space and by coupling the number of excited modes as well as the number of terms in the expansion of the vectorfield with the stepsize. This way we obtain exponential estimates of the form O(\exp(-C/ h^(1/2) )). A similar technique has been used for averaging of rapidly forced Hamiltonian PDEs by [Matthies and Scheel, 2003]. As a side-product, we also provide a convergence analysis of Runge-Kutta methods in Hilbert spaces.
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