Dispersion-dissipation condition for finite difference schemes

TL;DR

This paper derives a dispersion-dissipation condition for finite difference schemes to optimize their numerical properties, improving under-resolved turbulence simulations and guiding scheme design.

Contribution

It introduces a general dispersion-dissipation condition based on physical relations, applicable to both linear and non-linear finite difference schemes, and demonstrates its effectiveness in turbulence modeling.

Findings

Improved WENO scheme satisfies the dispersion-dissipation condition.

Enhanced scheme performs well in turbulence transition and decay simulations.

Significant improvement over classical models in LES of Taylor-Green vortex.

Abstract

A general dispersion-dissipation condition for finite difference schemes is derived by analyzing the numerical dispersion and dissipation of explicit finite-difference schemes. The proper dissipation required to damp spurious high-wavenumber waves in the solution is determined from a physically motivated relation between group velocity and dissipation rate. The application to a previously developed low-dissipation weighted essentially non-oscillatory scheme (WENO-CU6-M2) [X. Y. Hu and N. A. Adams; Scale separation for implicit large eddy simulation, J. Comput. Phys. 230 (2011) 7240-7249] demonstrates that this condition can serve as general guideline for optimizing the dispersion and dissipation of linear and non-linear finite-difference schemes. Moreover, the improved WENO-CU6-M2 which satisfies the dispersion-dissipation condition can be used for under-resolved simulations. We demonstrate this capability by considering transition to turbulence and self-similar energy decay of the three-dimensional Taylor-Green vortex. Simulations of the inviscid and the viscous Taylor-Green vortex at Reynolds numbers ranging from $Re=400$ to $Re=3000$ show a significant improvement over the classical dynamic Smagorinsky model and demonstrate competitiveness with state-of-the-art implicit LES models, while preserving shock-capturing properties.