On the behaviour of fully-discrete flux reconstruction schemes

In this study we employ von Neumann analyses to investigate the disper- sion, dissipation, group velocity, and error properties of several fully discrete flux reconstruction (FR) schemes. We consider three FR schemes paired with two explicit Runge-Kutta (ERK) schemes and two singly diagonally implicit RK (SDIRK) schemes. Key insights include the dependence of high-wavenumber numerical dissipation, relied upon for implicit large eddy simulation (ILES), on the choice of temporal scheme and time-step size. Also, the wavespeed characteristics of fully-discrete schemes and the relative dominance of temporal and spatial errors as a function of wavenumber and time-step size are investigated. Salient properties from the aforementioned theoretical analysis are then demonstrated in practice using linear advection test cases. Finally, a Burgers turbulence test case is used to demonstrate the importance of the temporal discretisation when using FR schemes for ILES