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

TL;DR

This paper identifies and addresses a fundamental issue with the lowest-order discontinuous Galerkin finite element method in modeling shallow water flows over varying topography, proposing a combined scheme to ensure well-balanced solutions.

Contribution

It reveals that the standard DGFEM at lowest order does not produce truly well-balanced solutions for shallow water equations and introduces a combined scheme to fix this problem.

Findings

The standard DGFEM at lowest order fails to maintain steady states in shallow water simulations.

A combined scheme with a finite-volume method ensures well-balanced solutions at lowest order.

The proposed method improves the physical accuracy of shallow water flow modeling over varying topography.

Abstract

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.