  The discontinuous Galerkin finite element method (DGFEM) developed by
Rhebergen et al. (2008) offers a robust method for solving systems of
nonconservative hyperbolic partial differential equations but, as we show here,
does not satisfactorily deal with topography in shallow water flows at lowest
order (so-called DG0, or equivalently finite volume). In particular, numerical
solutions of the space-DG0 discretised one-dimensional shallow water equations
over varying topography are not truly `well-balanced'. A numerical scheme is
well-balanced if trivial steady states are satisfied in the numerical solution;
in the case of the shallow water equations, initialised rest flow should remain
at rest for all times. Whilst the free-surface height and momentum remain
constant and zero, respectively, suggesting that the scheme is indeed
well-balanced, the fluid depth and topography evolve in time. This is both
undesirable and unphysical, leading to incorrect numerical solutions for the
fluid depth, and is thus a concern from a predictive modelling perspective. We
expose this unsatisfactory issue, both analytically and numerically, and
indicate a solution that combines the DGFEM formulation for nonconservative
products with a fast and stable well-balanced finite-volume method. This
combined scheme bypasses the offending issue and successfully integrates
nonconservative hyperbolic shallow water-type models with varying topography at
lowest order. We briefly discuss implications for the definition of a
well-balanced scheme, and highlight applications when higher-order schemes may
not be desired, which give further value to our finding beyond its exposure
alone.
