  Aims: This paper presents 2.5D numerical experiments of Alfv\'en wave phase
mixing and aims to assess the effects of nonlinearities on wave behaviour and
dissipation. In addition, this paper aims to quantify how effective the model
presented in this work is at providing energy to the coronal volume.
  Methods: The model is presented and explored through the use of several
numerical experiments which were carried out using the Lare2D code. The
experiments study footpoint driven Alfv\'en waves in the neighbourhood of a
two-dimensional x-type null point with initially uniform density and plasma
pressure. A continuous sinusoidal driver with a constant frequency is used.
Each experiment uses different driver amplitudes to compare weakly nonlinear
experiments with linear experiments.
  Results: We find that the wave trains phase-mix owing to variations in the
length of each field line and variations in the field strength. The
nonlinearities reduce the amount of energy entering the domain, as they reduce
the effectiveness of the driver, but they have relatively little effect on the
damping rate (for the range of amplitudes studied). The nonlinearities produce
density structures which change the natural frequencies of the field lines and
hence cause the resonant locations to move. The shifting of the resonant
location causes the Poynting flux associated with the driver to decrease.
Reducing the magnetic diffusivity increases the energy build-up on the resonant
field lines, however, it has little effect on the total amount of energy
entering the system. From an order of magnitude estimate, we show that the
Poynting flux in our experiments is comparable to the energy requirements of
the quiet Sun corona. However a (possibly unphysically) large amount of
magnetic diffusion was used however and it remains unclear if the model is able
to provide enough energy under actual coronal conditions.
