  Recovering some prominent high-order approaches such as the discontinuous
Galerkin (DG) or the spectral difference (SD) methods, the flux reconstruction
(FR) approach has been adopted by many individuals in the research community
and is now commonly used to solve problems on unstructured grids over complex
geometries. This approach relies on the use of correction functions to obtain a
differential form for the discrete problem. A class of correction functions,
named energy stable flux reconstruction (ESFR) functions, has been proven
stable for the linear advection problem. This proof has then been extended for
the diffusion equation using the local discontinuous Galerkin (LDG) scheme to
compute the numerical fluxes. Although the LDG scheme is commonly used, many
prefer the interior penalty (IP), as well as the Bassi and Rebay II (BR2)
schemes. Similarly to the LDG proof, this article provides a stability analysis
for the IP and the BR2 numerical fluxes. In fact, we obtain a theoretical
condition on the penalty term to ensure stability. This result is then verified
through numerical simulations. To complete this study, a von Neumann analysis
is conducted to provide a combination of parameters producing the maximal time
step while converging at the correct order. All things considered, this article
has for purpose to provide the community with a stability condition while using
the IP and the BR2 schemes.
