  We report the development of a discontinuous spectral element flow solver
that includes the implementation of both spectral difference and flux
reconstruction formulations. With this high order framework, we have
constructed a foundation upon which to provide a fair and accurate assessment
of these two schemes in terms of accuracy, stability, and performance with
special attention to the true spectral difference scheme and the modified
spectral difference scheme recovered via the flux reconstruction formulation.
Building on previous analysis of the spectral difference and flux
reconstruction schemes, we provide a novel nonlinear stability analysis of the
spectral difference scheme. Through various numerical experiments, we
demonstrate the additional stability afforded by the true, baseline spectral
difference scheme without explicit filtering or de-aliasing due to its inherent
feature of staggered flux points. This arrangement leads to favorable
suppression of aliasing errors and improves stability needed for under-resolved
simulations of turbulent flows.
