Energy Stable Flux Reconstruction Schemes for Advection–Diffusion Problems on Tetrahedra

Abstract

The flux reconstruction (FR) methodology provides a unifying description of many high-order schemes, including a particular discontinuous Galerkin (DG) scheme and several spectral difference (SD) schemes. In addition, the FR methodology has been used to generate new classes of high-order schemes, including the recently discovered `energy stable' FR schemes. These schemes, which are often referred to as VCJH (Vincent---Castonguay---Jameson---Huynh) schemes, are provably stable for linear advection---diffusion problems in 1D and on triangular elements. The VCJH schemes have been successfully applied to a wide variety of problems in 1D and 2D, ranging from linear advection---diffusion problems, to fluid mechanics problems requiring the solution of the compressible Navier---Stokes equations. Based on the results of these numerical experiments, it has been shown that certain VCJH schemes maintain the expected order of spatial accuracy and possess explicit time-step limits which rival those of the collocation-based nodal DG scheme. However, it remained to be seen whether the VCJH schemes could be extended to 3D on tetrahedral elements, enabling their convenient application to the complex geometries that arise in many real-world problems. For the first time, this article presents an extension of the VCJH schemes to tetrahedral elements. This work provides a formal proof of the stability of the new schemes and assesses their performance via numerical experiments on model problems.