Quantum mechanics

Compressibility

Algorithm

Classification of discontinuities

Flux limiter

Applied mathematics

Tangent

Discontinuity (linguistics)

Mathematical optimization

Realizability

Statistical physics

Dissipation

Thermodynamics

Physics

Nonlinear system

Mathematics

Compressible flow

Smoothness

Hyperbola

Mathematical analysis

Geometry

Infimum and supremum

Mechanics

Journal of Computational Physics

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.

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