Rémi Abgrall, Philipp Öffner, and I have published our new paper Reinterpretation and Extension of Entropy Correction Terms for Residual Distribution and Discontinuous Galerkin Schemes on arXiv.
For the general class of residual distribution (RD) schemes, including many finite element (such as continuous/discontinuous Galerkin) and flux reconstruction methods, an approach to construct entropy conservative semidiscretisations by adding suitable correction terms has been proposed recently by Abgrall (J. Comp. Phys. 372: pp. 640-666, 2018). Here, these correction terms are characterised as solutions of certain optimisation problems and adapted to discontinuous element based schemes such as discontinuous Galerkin and (multi-block) finite difference methods. Novel generalisations to entropy inequalities, multiple constraints, and kinetic energy preservation for the Euler equations are developed and tested in numerical experiments. Finally, the underlying idea to use optimisation problems is applied to grid refinement and coarsening operators, resulting in entropy stable/dissipative grid transfers.