Positivity-Preserving Entropy-Based Adaptive Filtering for Discontinuous Spectral Element Methods

TL;DR

This paper introduces a positivity-preserving, entropy-based adaptive filtering technique for discontinuous spectral element methods, enabling robust shock capturing without problem-specific tuning and applicable to complex flows.

Contribution

It proposes a novel adaptive filtering method that enforces positivity and entropy principles, improving shock resolution in spectral element methods without extensive parameter tuning.

Findings

Effective in resolving strong shocks and discontinuities

Applicable to complex hyperbolic and parabolic conservation laws

Demonstrated robustness in turbulent flow simulations

Abstract

In this work, we present a positivity-preserving entropy-based adaptive filtering method for shock capturing in discontinuous spectral element methods. By adapting the filter strength to enforce positivity and a local discrete minimum entropy principle, the resulting approach can robustly resolve strong discontinuities with sub-element resolution, does not require problem-dependent parameter tuning, and can be easily implemented on general unstructured meshes with relatively low computational cost. The efficacy of the approach is shown in numerical experiments on hyperbolic and mixed hyperbolic-parabolic conservation laws such as the Euler and Navier-Stokes equations for problems including extreme shocks, shock-vortex interactions, and complex compressible turbulent flows.