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

Abstract

This work frames the problem of constructing non-oscillatory entropy stable fluxes as a least square optimization problem. A flux sign stability condition is defined for a pair of entropy conservative flux ($F^*$) and a non-oscillatory flux ($F^s$). This novel approach paves a way to construct non-oscillatory entropy stable flux ($\hat{F}$) as a simple combination of $(F^*$ and $F^s)$ which inherently optimize the numerical diffusion in the entropy stable flux ($\hat{F}$) such that it reduces to the underlying non-oscillatory flux ($F^s$) in the flux sign stable region. This robust approach is (i) agnostic to the choice of flux pair $(F^*,F^s)$, (ii) does not require the computation of costly dissipation operator and high order reconstruction of scaled entropy variable to construct the diffusion term. Various non-oscillatory entropy stable fluxes are constructed and exhaustive computational results for standard test problems are given which show that these entropy stable schemes completely remove spurious oscillations in approximating the discontinuities compared to the non-oscillatory schemes using underlying fluxes ($F^s$) only. Moreover, these entropy stable schemes maintain the formal order of accuracy of the lower order flux in the pair.