  A new set of symmetric correction functions is presented for high-order flux
reconstruction, that expands upon, while incorporating, all previous correction
function sets and opens the possibility for improved performance. By
considering FR applied to the linear advection equation, and through
modification to the Sobolev norm, criteria are presented for a wider set of
correction functions. Legendre polynomials are then used to fulfil these
criterion and realise functions for third to fifth order FR. The sufficient
conditions for the existence of the modified norms are also explored, before
Fourier and Von Neumann analysis are applied to analytically find temporal
stability limits for various Runge-Kutta temporal integration schemes. For all
cases, correction functions are found that extend the temporal stability of FR.
Two application-inspired investigations are performed that aim to explore the
effect of aliasing and non-linear equations. In both cases unique correction
functions could be found that give good performance, compared to previous FR
schemes, while also improving upon the temporal stability limit.
