  Many numerical schemes for hyperbolic systems require a piecewise polynomial
reconstruction of the cell averaged values, and to simulate perturbed steady
states accurately we require a so called 'well balanced' reconstruction scheme.
For the shallow water system this involves reconstructing in surface elevation,
to which modifications must be made as the fluid depth becomes small to ensure
positivity.
  We investigate the scheme proposed in Skevington (2021) though numerical
experiments, demonstrating its ability to resolve steady and near steady states
at high accuracy. We also present a modification to the scheme which enables
the resolution of slowly moving shocks and dam break problems without
compromising the well balanced property.
