A New Family of Weighted One-Parameter Flux Reconstruction Schemes

TL;DR

This paper introduces a new family of high-order flux reconstruction schemes using weighted Sobolev norms, demonstrating improved stability, reduced dissipation, and higher CFL limits through theoretical analysis and numerical experiments.

Contribution

It proposes a novel family of FR correction functions based on weighted Sobolev norms, enhancing stability and performance over existing schemes.

Findings

Reduced dissipation and dispersion overshoot in new schemes

Higher CFL limits while maintaining convergence rates

Numerical results agree with theoretical stability analysis

Abstract

The flux reconstruction (FR) approach offers a flexible framework for describing a range of high-order numerical schemes; including nodal discontinuous Galerkin and spectral difference schemes. This is accomplished through the use of so-called correction functions. In this study we employ a weighted Sobolev norm to define a new extended family of FR correction functions, the stability of which is affirmed through Fourier analysis. Several of the schemes within this family are found to exhibit reduced dissipation and dispersion overshoot. Moreover, many of the new schemes possess higher CFL limits whilst maintaining the expected rate of convergence. Numerical experiments with homogeneous linear convection and Burgers turbulence are undertaken, and the results observed to be in agreement with the theoretical findings.