Entropy stable non-oscillatory fluxes: An optimized wedding of entropy conservative flux with non-oscillatory flux

This work settles the problem of constructing entropy stable non-oscillatory (ESNO) fluxes by framing it as a least square optimization problem. A flux sign stability condition is introduced and utilized to construct arbitrary order entropy stable flux as a convex combination of entropy conservative and non-oscillatory flux. This simple approach is robust which does not explicitly requires the computation of costly dissipation operator and high order reconstruction of scaled entropy variable for constructing the diffusion term. The numerical diffusion is optimized in the sense that entropy stable flux reduces to the underlying non-oscillatory flux. Different non-oscillatory entropy stable fluxes are constructed and used to compute the numerical solution of various standard scalar and systems test problems. Computational results show that entropy stable schemes are comparable in term of non-oscillatory nature of schemes using only the underlying non-oscillatory fluxes. Moreover, these entropy stable schemes maintains the formal order of accuracy of the lower order flux used in the convex combination.