  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.
