Stability of Energy Stable Flux Reconstruction for the Diffusion Problem using Compact Numerical Fluxes on Quadratic Elements

摘要

The flux reconstruction method has gained popularity in the research community as it recovers promising high-order methods through modally Altered correction fields, such as the Discontinuous Galerkin (DG) method, on unstructured grids over complex geometries. The attraction of the method follows with its stability proofs for the linear advection problem, under a class of energy stable flux reconstruction (ESFR) schemes also known as Vincent-Castonguay-Jameson-Huynh (VCJH) schemes. The proof has later been developed for the diffusion probtem on triangular elements for Local Discontinuous Galerkin (LDG) and 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. For the diffusion problem, on Cartesian meshes, the proof has been extended for the LDG numerical flux. This paper expands the proof for compact numerical fluxes, and demonstrates the stability's independence on the correction parameter in the auxiliary equation for the IP and BR2 numerical fluxes. The conditions for stability restrict the values of the penalty term of the different schemes. These stability conditions are valid for any ESFR schemes including DG and are much sharper than previously known criteria.