Fully well-balanced entropy controlled discontinuous Galerkin spectral element method for shallow water flows: Global flux quadrature and cell entropy correction

We present a novel formulation of the discontinuous Galerkin spectral element method for solving balance laws, with application to the shallow water equations. The scheme proposed is constructed starting from a global flux formulation in which an additional flux term is constructed as the primitive of the source. We show that, in the context of nodal spectral finite elements, this can be translated into a simple modification of the integral of the source term, or equivalently into a modification of a mass matrix. When using Gauss-Lobatto nodal finite elements this modified integration allows to recover at steady state a well known high order Gauss collocation ODE integrator for the flux: the LobattoIIIA method. This method is superconvergent at the collocation points. We thus obtain considerable accuracy enhancements for any steady state, and a characterization via a discrete well-balanced property similar in spirit to the one proposed in [20], albeit not needing the explicit solution of the local Cauchy problem. To control the entropy production, we introduce ad-hoc artificial viscosity corrections at the cell level and incorporate them into the scheme. We provide theoretical and numerical characterizations of the accuracy and equilibrium preservation of these corrections. Through extensive numerical benchmarking we validate all the theoretical predictions, conforming great improvements in accuracy for one dimensional steady states, and robustness for more complex scenarios both in one and two dimensions.