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A New Family of Weighted One-Parameter Flux Reconstruction Schemes

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.