On High-Resolution Entropy-Consistent Flux with Slope Limiter for Hyperbolic Conservation Laws

This paper proposes a new version of the high-resolution entropy-consistent (EC-Limited) flux for hyperbolic conservation laws based on a new minmod-type slope limiter. Firstly, we identify the numerical entropy production, a third-order differential term deduced from the previous work of Ismail and Roe [11]. The corresponding dissipation term is added to the original Roe flux to achieve entropy consistency. The new, resultant entropy-consistent (EC) flux has a general and explicit analytical form without any corrective factor, making it easy to compute and a less-expensive method. The inequality constraints are imposed on the standard piece-wise quadratic reconstruction to enforce the pointwise values of bounded-type numerical solutions. We design the new minmod slope limiter as combining two separate limiters for left and right states. We propose the EC-Limited flux by adding this reconstruction data method to the primitive variables rather than to the conservative variables of the EC flux to preserve the equilibrium of the primitive variables. These resulting fluxes are easily applied to general hyperbolic conservation laws while having attractive features: entropy-stable, robust, and non-oscillatory. To illustrate the potential of these proposed fluxes, we show the applications to the Burgers equation and the Euler equations.