  Dimensionally split advection schemes are attractive for atmospheric
modelling due to their efficiency and accuracy in each spatial dimension.
Accurate long time-steps can be achieved without significant cost using the
flux-form semi-Lagrangian technique. The dimensionally split scheme used here
is constructed from the one-dimensional Piecewise Parabolic Method and extended
to two dimensions using COSMIC splitting. The dimensionally split scheme is
compared with a genuinely multi-dimensional, method of lines scheme with
implicit time-stepping which is stable for large Courant numbers.
  Two-dimensional advection test cases on Cartesian planes are proposed that
avoid the complexities of a spherical domain or multi-panel meshes. These are
solid body rotation, horizontal advection over orography and deformational
flow. The test cases use distorted meshes either to represent sloping terrain
or to mimic the distortions of a cubed sphere.
  Such mesh distortions are expected to accentuate the errors associated with
dimension splitting, however, the dimensionally split scheme is very accurate
on orthogonal meshes and accuracy decreases only a little in the presence of
large mesh distortions. The dimensionally split scheme also loses some accuracy
when long time-steps are used. The multi-dimensional scheme is almost entirely
insensitive to mesh distortions and asymptotes to second-order accuracy at high
resolution. As is expected for implicit time-stepping, phase errors occur when
using long time-steps but the spatially well resolved features are advected at
the correct speed and the multi-dimensional scheme is always stable.
  An estimate of computational cost reveals that the implicit scheme is the
most expensive, particularly for large Courant numbers. If the
multi-dimensional scheme is used instead with explicit time-stepping, the cost
becomes similar to the dimensionally split scheme.
