Aims. The interaction of supernova remnants (SNRs) with molecular clouds
gives rise to strong molecular emission in the far-IR and sub-mm wavelength
regimes. The application of MHD shock models in the interpretation of this line
emission can yield valuable information on the energetic and chemical impact of
supernova remnants. Methods. New mapping observations with the APEX telescope
in CO (3-2), (4-3), (6-5), (7-6) and 13CO (3-2) towards two regions in the
supernova remnant W44 are presented. Integrated intensities are extracted on
five different positions, corresponding to local maxima of CO emission. The
integrated intensities are compared to the outputs of a grid of models, which
combine an MHD shock code with a radiative transfer module based on the large
velocity gradient approximation. Results. All extracted spectra show ambient
and line-of-sight components as well as blue- and red-shifted wings indicating
the presence of shocked gas. Basing the shock model fits only on the
highest-lying transitions that unambiguously trace the shock-heated gas, we
find that the observed CO line emission is compatible with non-stationary
shocks and a pre-shock density of 10^4 cm-3. The ages of the modelled shocks
scatter between values of \sim1000 and \sim3000 years. The shock velocities in
W44F are found to lie between 20 and 25 km/s, while in W44E fast shocks (30-35
km/s) as well as slower shocks (\sim20 km/s) are compatible with the observed
spectral line energy diagrams. The pre-shock magnetic field strength components
perpendicular to the line of sight in both regions have values between 100 and
200 \muG. Our best-fitting models allow us to predict the full ladder of CO
transitions, the shocked gas mass in one beam as well as the momentum- and
energy injection.Comment: 20 pages, 13 figures, 13 tables, accepted for publication in
Astronomy and Astrophysic