On the supremum of the steepness parameter in self-adjusting steepness based schemes

Self-adjusting steepness (SAS)-based schemes preserve various structures in the compressible flows. These schemes provide a range of desired behaviors depending on the steepness-adjustable limiters with the steepness measured by a steepness parameter. These properties include either high-order properties with exact steepness parameter values that are theoretically determined or having anti-diffusive/compression properties with a larger steepness parameter. Nevertheless, the supremum of the steepness parameter has not been determined theoretically yet. In this study, we derive a universal method to determine the supremum using total variation diminishing (TVD) condition of Sewby. Two typical steepness-adjustable limiters are analyzed in detail including the tangent of hyperbola for interface capturing (THINC) limiter and the steepness-adjustable harmonic (SAH) limiter. We also obtain the analytical expression of the supremum of the steepness parameter which is dependent on the Courant-Friedrichs-Lewy (CFL) number. Using this solution, we then propose supremum-determined SAS schemes. These schemes are further extended to solve the compressible Euler equations. The results of typical numerical tests confirm our theoretical conclusions and show that the final schemes are capable of sharply capturing contact discontinuities and minimizing numerical oscillations.