Entropy stable methods have become increasingly popular in the field of computational fluid dynamics. They often work by satisfying some form of a discrete entropy inequality: a discrete form of the 2nd law of thermodynamics. Schemes which satisfy a (semi-)discrete entropy inequality typically behave much more robustly, and do so in a way that is hyperparameter free. Recently, a new strategy was introduced to construct entropy stable discontinuous Galerkin methods: knapsack limiting, which blends together a low order, positivity preserving, and entropy stable scheme with a high order accurate scheme, in order to produce a high order accurate, entropy stable, and positivity preserving scheme. Another recent strategy introduces an entropy correction artificial viscosity into a high order scheme, aiming to satisfy a cell entropy inequality. In this work, we introduce the techniques of knapsack limiting and artificial viscosity for finite difference discretizations. The proposed schemes preserve high order accuracy in sufficiently smooth conditions, are entropy stable, and are hyperparameter free. Moreover, the proposed knapsack limiting scheme provably preserves positivity for the compressible Euler and Navier-Stokes equations. Both schemes achieve this goal without significant performance tradeoffs compared to state of the art stabilized schemes.
Christner et al. (Thu,) studied this question.