On the supremum of the steepness parameter in self-adjusting discontinuity-preserving 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 second-order accuracy with exact steepness infima that are theoretically given 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 demonstrate that any anti-diffusive limiter should be limited by Ultra-bee limiter according to Sweby's total variation diminishing (TVD) condition. 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. Applying this constraint, we derive for the first time two inequalities which the steepness parameters much satisfy. Furthermore, we obtain the analytical expression of the Courant-Friedrichs-Lewy (CFL) number-dependent supremum of the steepness parameter. 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.