  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.
