  Close to the origin, the nonlinear Klein--Gordon equations on the circle are
nearly integrable Hamiltonian systems which have infinitely many almost
conserved quantities called harmonic actions or super-actions. We prove that,
at low regularity and with a CFL number of size 1, this property is preserved
if we discretize the nonlinear Klein--Gordon equations with the symplectic
mollified impulse methods. This extends previous results of D. Cohen, E. Hairer
and C. Lubich to non-smooth solutions.
