We propose a robust arbitrary-order Discontinuous Galerkin (DG) method for the shallow water equations, based on a reformulation of the Saint-Venant system in terms of the total head H and the discharge q. This change of variables constitutes the core of the approach and allows the standard DG framework to naturally satisfy the well-balanced property, without resorting to hydrostatic reconstruction techniques. The resulting scheme exactly preserves the lake-at-rest equilibrium (C-property) at any polynomial order and on unstructured meshes, simply by indexing the numerical fluxes with the bottom topography. Positivity of the water depth is ensured under a local CFL condition through a conservative linear-scaling limiter, while maintaining high-order accuracy in smooth regimes. The formulation is first analyzed in one space dimension and then extended to two dimensions using a normal-edge DG discretization on unstructured meshes. A comprehensive set of numerical benchmarks, including steady and transient flows, wet/dry interfaces, and dam-break problems, demonstrates the high-order convergence, robustness, and stability of the proposed method.
Ersoy et al. (Mon,) studied this question.