Positivity-preserving discontinuous spectral element methods for compressible multi-species flows

TL;DR

This paper presents a new positivity-preserving stabilization method for high-order spectral element simulations of compressible multi-species flows, improving accuracy and robustness near interfaces and shocks.

Contribution

It extends an adaptive entropy filtering approach with an adaptive, parameter-free switch for better performance in multi-species flow simulations.

Findings

Retains high-order accuracy at smooth extrema

Ensures positivity of species density and pressure near shocks

Accurately captures small-scale flow features

Abstract

We introduce a novel positivity-preserving, parameter-free numerical stabilisation approach for high-order discontinuous spectral element approximations of compressible multi-species flows. The underlying stabilisation method is the adaptive entropy filtering approach (Dzanic and Witherden, J. Comput. Phys., 468, 2022), which is extended to the conservative formulation of the multi-species flow equations. We show that the straightforward enforcement of entropy constraints in the filter yields poor results around species interfaces and propose an adaptive, parameter-free switch for the entropy bounds based on the convergence properties of the pressure field which drastically improves its performance for multi-species flows. The proposed approach is shown in a variety of numerical experiments applied to the multi-species Euler and Navier--Stokes equations computed on unstructured grids, ranging from shock-fluid interaction problems to three-dimensional viscous flow instabilities. We demonstrate that the approach can retain the high-order accuracy of the underlying numerical scheme even at smooth extrema, ensure the positivity of the species density and pressure in the vicinity of shocks and contact discontinuities, and accurately predict small-scale flow features with minimal numerical dissipation.