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We present a novel approach for solving the shallow water equations using adiscontinuous Galerkin spectral element method. The method we propose has threemain features. First, it enjoys a discrete well-balanced property, in a spiritsimilar to the one of e.g. [20]. As in the reference, our scheme does notrequire any a-priori knowledge of the steady equilibrium, moreover it does notinvolve the explicit solution of any local auxiliary problem to approximatesuch equilibrium. The scheme is also arbitrarily high order, and verifies acontinuous in time cell entropy equality. The latter becomes an inequality assoon as additional dissipation is added to the method. The method isconstructed starting from a global flux approach in which an additional fluxterm is constructed as the primitive of the source. We show that, in thecontext of nodal spectral finite elements, this can be translated into a simplemodification of the integral of the source term. We prove that, when usingGauss-Lobatto nodal finite elements this modified integration is equivalent atsteady state to a high order Gauss collocation method applied to an ODE for theflux. This method is superconvergent at the collocation points, thus providinga discrete well-balanced property very similar in spirit to the one proposed in[20], albeit not needing the explicit computation of a local approximation ofthe steady state. To control the entropy production, we introduce artificialviscosity corrections at the cell level and incorporate them into the scheme.We provide theoretical and numerical characterizations of the accuracy andequilibrium preservation of these corrections. Through extensive numericalbenchmarking, we validate our theoretical predictions, with considerableimprovements in accuracy for steady states, as well as enhanced robustness formore complex scenarios

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