A Legendre spectral viscosity (LSV) method applied to shock capturing for high-order flux-reconstruction schemes

TL;DR

This paper introduces a Legendre spectral viscosity method for shock capturing in high-order flux reconstruction schemes, enabling accurate and robust simulations of shock discontinuities with minimal degrees of freedom.

Contribution

The novel LSV approach uses Legendre polynomials to dynamically activate dissipation only near shocks, improving resolution and robustness in high-order schemes.

Findings

Accurately captures shocks with high resolution and few degrees of freedom.

Demonstrates robustness in 1D and 2D shock problems.

Achieves stable shock capturing with spectral accuracy.

Abstract

A novel approach to shock capturing for high-order flux reconstruction schemes is derived based on the mathematical formalism of the filtered governing equations. While the latter perspective is only typically used for turbulence modeling in the context of Large-Eddy Simulations (LES), the novel Legendre Spectral Viscosity (LSV) sub-filter scale (SFS) closure model is capable of performing simulations in the presence of shock-discontinuities. The LSV method exploits the set of hierarchical basis functions formed by the Legendre polynomials to extract the information on the energy content near the resolution limit and estimate the overall magnitude of the required SFS dissipative terms, resulting in a scheme that dynamically activates only in cells where nonlinear behavior is important. Additionally, the modulation of such terms in the Legendre spectral space allows for the concentration of the dissipative action at small scales. The proposed method is tested in canonical shock-dominated flow setups in both one and two dimensions. These include the 1D Burgers' problem, a 1D shock tube, a 1D shock-entropy wave interaction, a 2D inviscid shock-vortex interaction and a 2D double Mach reflection. Results showcase a high-degree of resolution power, achieving accurate results with a small number of degrees of freedom, and robustness, being able to capture shocks associated with the Burgers' equation and the 1D shock tube within a single cell with orders 120 and higher.