  The vast majority of Pulsar Wind Nebulae (PWNe) present in the Galaxy is
formed by middle-aged systems characterized by a strong interaction of the PWN
itself with the supernova remnant (SNR). Unfortunately, modelling these systems
can be quite complex and numerically expensive, due to the non-linearity of the
PWN-SNR evolution even in the simple 1D / one-zone case when the reverse shock
of the SNR reaches the PWN, and the two begin to interact (and reverberation
starts).
  Here we introduce a new numerical technique that couples the numerical
efficiency of the one-zone thin shell approach with the reliability of a full
``lagrangian'' evolution, able to correctly reproduce the PWN-SNR interaction
during the reverberation and to consistently evolve the particle spectrum
beyond. Based on our previous findings, we show that our novel strategy
resolves many of the uncertainties present in previous approaches, as the
arbitrariness in the SNR structure, and ensure a robust evolution, compatible
with results that can be obtained with more complex 1D dynamical approaches.
Our approach enable us for the first time to provide reliable spectral models
of the later compression phases in the evolution of PWNe. While in general we
found that the compression is less extreme than that obtained without such
detailed dynamical considerations, leading to the formation of less structured
spectral energy distributions, we still find that a non negligible fraction of
PWNe might experience a super-efficient phase, with the optical and/or X-ray
luminosity exceeding the spin-down one.
