Stability of Energy Stable Flux Reconstruction for the Diffusion Problem Using Compact Numerical Fluxes

Abstract

Recovering some prominent high-order approaches such as the discontinuous Galerkin or the spectral difference methods, the flux reconstruction framework 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 functions, has been proven stable for the linear advection problem. This proof has then been extended to the diffusion equation using the local discontinuous Galerkin (LDG) method to compute the numerical fluxes. Although the LDG formulation is commonly used, many prefer other compact numerical fluxes such as the interior penalty (IP), the Bassi and Rebay II, the compact discontinuous Galerkin, or the compact discontinuous Galerkin 2 numerical fluxes. Similarly to the LDG proof, this article provides a stability analysis for these compact formulation. In fact, we obtain a theoretical condition on the penalty term to ensure stability. This result is then verified through numerical simulations for the IP approach.