Two finite difference schemes, one using entropy-correction artificial viscosity and one using knapsack limiting, satisfy discrete entropy inequalities; the knapsack variant also provably preserves positivity for the compressible Euler and Navier-Stokes equations.
High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains
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Entropy stable finite difference methods via entropy correction artificial viscosity and knapsack limiting
Two finite difference schemes, one using entropy-correction artificial viscosity and one using knapsack limiting, satisfy discrete entropy inequalities; the knapsack variant also provably preserves positivity for the compressible Euler and Navier-Stokes equations.