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The methodology proposed in this paper bridges the gap between entropy stableand positivity-preserving discontinuous Galerkin (DG) methods for nonlinearhyperbolic problems. The entropy stability property and, optionally,preservation of local bounds for the cell averages are enforced using fluxlimiters based on entropy conditions and discrete maximum principles,respectively. Entropy production by the (limited) gradients of thepiecewise-linear DG approximation is constrained using Rusanov-type entropyviscosity, as proposed by Abgrall in the context of nodal finite elementapproximations. We cast his algebraic entropy fix into a form suitable forarbitrary polynomial bases and, in particular, for modal DG approaches. TheTaylor basis representation of the entropy stabilization term reveals that itpenalizes the solution gradients in a manner similar to slope limiting andrequires semi-implicit treatment to achieve the desired effect. The implicitTaylor basis version of the Rusanov entropy fix preserves the sparsity patternof the element mass matrix. Hence, no linear systems need to be solved if theTaylor basis is orthogonal and an explicit treatment of the remaining terms isadopted. The optional application of a vertex-based slope limiter constrainsthe piecewise-linear DG solution to be bounded by local maxima and minima ofthe cell averages. The combination of entropy stabilization with flux and slopelimiting leads to constrained approximations that possess all desiredproperties. Numerical studies of the new limiting techniques and entropycorrection procedures are performed for two scalar two-dimensional testproblems with nonlinear and nonconvex flux functions.

Entropy stabilization and property-preserving limiters for discontinuous Galerkin discretizations of nonlinear hyperbolic equations