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

Stability of Energy Stable Flux Reconstruction for the Diffusion Problem using the Interior Penalty and Bassi and Rebay II Numerical Fluxes