Shock excitation of H$_2$ in the James Webb Space Telescope era
(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.