  Physical processes which may lead to solar chromospheric heating are analyzed
using high-resolution 1.5D non-ideal MHD modelling. We demonstrate that it is
possible to heat the chromospheric plasma by direct resistive dissipation of
high-frequency Alfv\'en waves through Pedersen resistivity. However this is
unlikely to be sufficient to balance radiative and conductive losses unless
unrealistic field strengths or photospheric velocities are used. The precise
heating profile is determined by the input driving spectrum since in 1.5D there
is no possibility of Alfv\'en wave turbulence. The inclusion of the Hall term
does not affect the heating rates. If plasma compressibility is taken into
account, shocks are produced through the ponderomotive coupling of Alfv\'en
waves to slow modes and shock heating dominates the resistive dissipation. In
1.5D shock coalescence amplifies the effects of shocks and for compressible
simulations with realistic driver spectra the heating rate exceeds that
required to match radiative and conductive losses. Thus while the heating rates
for these 1.5D simulations are an overestimate they do show that ponderomotive
coupling of Alfv\'en waves to sound waves is more important in chromospheric
heating than Pedersen dissipation through ion-neutral collisions.
