Energy stable flux reconstruction schemes for advection–diffusion problems

High-order methods for unstructured grids provide a promising option for solving challenging problems in computational fluid dynamics. Flux reconstruction (FR) is a framework which unifies a number of these high-order methods, such as the spectral difference (SD) and collocation-based nodal discontinuous Galerkin (DG) methods, allowing for their more concise and flexible implementation. Additionally, the FR approach can be used to facilitate development of new numerical methods that offer arbitrary orders of accuracy on unstructured grids. In previous work, it has been shown that a particular range of FR schemes, referred to as Vincent–Castonguay–Jameson–Huynh (VCJH) schemes, are guaranteed to be stable for linear advection problems for all orders of accuracy. There have remained questions, however, regarding the stability of FR schemes for advection–diffusion problems. In this study a new class of VCJH schemes is developed for solving one-dimensional advection–diffusion problems. For the first time, it is shown that the schemes are linearly stable for linear advection–diffusion problems for all orders of accuracy on nonuniform grids. Linear and nonlinear numerical experiments are performed in 1D and 2D to investigate the accuracy and stability properties of the new schemes. The results indicate that certain VCJH schemes for advection–diffusion problems possess significantly higher explicit time-step limits than discontinuous Galerkin schemes, while still maintaining the expected order of accuracy.