  (Abridged) H2 is the most abundant molecule in the Universe. Thanks to its
widely spaced energy levels, it predominantly lights up in warm gas, T > 100 K,
such as shocked regions, and it is one of the key targets of JWST observations.
These include shocks from protostellar outflows, all the way up to starburst
galaxies and AGN. Shock models are able to simulate H2 emission. We aim to
explore H2 excitation using such models, and to test over which parameter space
distinct signatures are produced in H2 emission. We present simulated H2
emission using the Paris-Durham shock code over an extensive grid of 14,000
plane-parallel stationary shock models, a large subset of which are exposed to
an external UV radiation field. The grid samples 6 input parameters: preshock
density, shock velocity, transverse magnetic field strength, UV radiation field
strength, cosmic-ray-ionization rate, and PAH abundance. Physical quantities,
such as temperature, density, and width, have been extracted along with H2
integrated line intensities. The strength of the transverse magnetic field, set
by the scaling factor, b, plays a key role in the excitation of H2. At low
values of b (<~ 0.3, J-type shocks), H2 excitation is dominated by
vibrationally excited lines; at higher values (b >~ 1, C-type shocks),
rotational lines dominate the spectrum for shocks with an external radiation
field comparable to (or lower than) the solar neighborhood. Shocks with b >= 1
can be spatially resolved with JWST for nearby objects. When the input kinetic
energy flux increases, the excitation and integrated intensity of H2 increases
similarly. An external UV field mainly serves to increase the excitation,
particularly for shocks where the input radiation energy is comparable to the
input kinetic energy flux. These results provide an overview of the energetic
reprocessing of input kinetic energy flux and the resulting H2 line emission.
