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 approach for solving the shallow water equations using a discontinuous Galerkin spectral element method. The method we propose has three main features. First, it enjoys a discrete well-balanced property, in a spirit similar to the one of e.g. [20]. As in the reference, our scheme does not require any a-priori knowledge of the steady equilibrium, moreover it does not involve the explicit solution of any local auxiliary problem to approximate such equilibrium. The scheme is also arbitrarily high order, and verifies a continuous in time cell entropy equality. The latter becomes an inequality as soon as additional dissipation is added to the method. The method is constructed starting from a global flux approach 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. We prove that, when using Gauss-Lobatto nodal finite elements this modified integration is equivalent at steady state to a high order Gauss collocation method applied to an ODE for the flux. This method is superconvergent at the collocation points, thus providing a discrete well-balanced property very similar in spirit to the one proposed in [20], albeit not needing the explicit computation of a local approximation of the steady state. To control the entropy production, we introduce 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 our theoretical predictions, with considerable improvements in accuracy for steady states, as well as enhanced robustness for more complex scenario

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 more orders of magnitude, with a simple modification of a given DGSEM implementation. Robustness on more complex cases involving the propagation of sharp wave fronts 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