Self-adjusting steepness-based schemes that preserve discontinuous structures in compressible flows

Although conventional total-variation-diminishing schemes can capture discontinuities with essentially non-oscillatory properties and are nominal of second-order accuracy in smooth regions, they cannot generally maintain sharp resolution for various discontinuities (especially contact discontinuity for long-time simulations) owing to excessive numerical dissipation. Designing schemes to overcome this disadvantage is a difficult problem. In this paper, we propose to solve this problem by introducing a novel framework of self-adjusting steepness-based schemes, which are mainly constructed in four steps: (1) design a slope limiter containing a steepness parameter β that provides a mechanism to enable the scheme to accurately solve both smooth and discontinuous problems with proper values of β; (2) determine the infimum of β such that the scheme is order-optimized; (3) determine the supremum of β such that the scheme has a nonlinear stable anti-diffusion/compression effect; (4) calculate the steepness parameter β using an adaptive algorithm in terms of the infimum and supremum to ensure that the final scheme obtains essentially non-oscillatory and sharp resolutions for various discontinuities while maintaining the nominal second-order accuracy for smooth regions. Moreover, such a framework has been implemented on the tangent-of-hyperbola-for-interface-capturing scheme (Xiao et al. (2005) [27]) and the classical harmonic limiter (van Leer (1974) [37]), resulting in two specific self-adjusting steepness-based schemes. Finally, these two schemes were tested using a series of numerical examples in one and two dimensions. It was found that the schemes cannot only obtain second-order accuracy in smooth regions but also preserve discontinuous flow structures, especially contact discontinuities, even after long computation times.