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The discontinuous Galerkin finite element method (DGFEM) developed byRhebergen et al. (2008) offers a robust method for solving systems ofnonconservative hyperbolic partial differential equations but, as we show here,does not satisfactorily deal with topography in shallow water flows at lowestorder (so-called DG0, or equivalently finite volume). In particular, numericalsolutions of the space-DG0 discretised one-dimensional shallow water equationsover varying topography are not truly `well-balanced'. A numerical scheme iswell-balanced if trivial steady states are satisfied in the numerical solution;in the case of the shallow water equations, initialised rest flow should remainat rest for all times. Whilst the free-surface height and momentum remainconstant and zero, respectively, suggesting that the scheme is indeedwell-balanced, the fluid depth and topography evolve in time. This is bothundesirable and unphysical, leading to incorrect numerical solutions for thefluid depth, and is thus a concern from a predictive modelling perspective. Weexpose this unsatisfactory issue, both analytically and numerically, andindicate a solution that combines the DGFEM formulation for nonconservativeproducts with a fast and stable well-balanced finite-volume method. Thiscombined scheme bypasses the offending issue and successfully integratesnonconservative hyperbolic shallow water-type models with varying topography atlowest order. We briefly discuss implications for the definition of awell-balanced scheme, and highlight applications when higher-order schemes maynot be desired, which give further value to our finding beyond its exposurealone.

Ensuring 'well-balanced' shallow water flows via a discontinuous Galerkin finite element method: issues at lowest order