  Four different approximate approaches used to model the stressing of coronal
magnetic fields due to an imposed photospheric motion are compared with each
other and the results from a full time-dependent magnetohydrodynamic (MHD)
code. The assumptions used for each of the approximate methods are tested by
considering large photospheric footpoint displacements.
  We consider a simple model problem, comparing the full nonlinear
magnetohydrodynamic evolution, determined with the Lare2D numerical code, with
four approximate approaches. Two of these, magneto-frictional relaxation and a
quasi-1D Grad-Shafranov approach, assume sequences of equilibria, whilst the
other two methods, a second-order linearisation of the MHD equations and
Reduced MHD, are time-dependent.
  The relaxation method is very accurate compared to full MHD for force-free
equilibria for all footpoint displacements but has significant errors when the
plasma $\beta_0$ is of order unity. The 1D approach gives an extremely accurate
description of the equilibria away from the photospheric boundary layers, and
agrees well with Lare2D for all parameter values tested. The linearised MHD
equations correctly predict the existence of photospheric boundary layers that
are present in the full MHD results. As soon as the footpoint displacement
becomes a significant fraction of the loop length, the RMHD method fails to
model the sequences of equilibria correctly. The full numerical solution is
interesting in its own right, and care must be taken for low $\beta_0$ plasmas
if the viscosity is too large.
