Fully well balanced entropy controlled DGSEM for shallow water flows: global flux quadrature and cell entropy correction

In this paper we propose a high order DGSEM formulation for balance laws which embeds a general well balanced criterion agnostic of the exact steady state. The construction proposed exploits the idea of a global flux formulation to infer an ad-hoc quadrature strategy called here "global flux quadrature". This quadrature approach allows to establish a one to one correspondence, for a given local set of data on a given stencil, between the discretization of a non-local integral operator, and the local steady differential problem. This equivalence is a discrete well balanced notion which allows to construct balanced schemes without explicit knowledge of the steady state, and in particular without the need of solving a local Cauchy problem. The use of Gauss-Lobatto DGSEM allows a natural connection to continuous collocation methods for integral equations. This allows to fully characterize the discrete steady solution with a superconvergence result. The notion of entropy control is also included in the construction via appropriately designed cell artificial viscosity corrections. The accuracy and equilibrium preservation of these corrections are characterized theoretically and numerically. In particular, thorough numerical benchmarking, we confirm all the theoretical expectations showing improvements in accuracy on steady states of one or two orders of magnitude, with a simple modification of a given DGSEM implementation. Robustness on more complex cases is also proved. Preliminary tests on multidimensional problems shows improvements on the preservation of vortex like solutions with important error reductions, despite of the fact that no genuine 2D balancing criterion is embedded.