An Extended Range of Energy Stable Flux Reconstruction Methods on Triangles

We present an extended range of stable flux reconstruction (FR) methods on triangles through the development and application of the summation-by-parts framework in two-dimensions. This extended range of stable schemes is then shown to contain the single parameter schemes of Castonguay et al. (J Sci Comput 51:224–256, 2011) on triangles, and our definition enables wider stability bounds to be developed for those single parameter families. Stable upwinded spectral difference (SD) schemes on triangular elements have previously been found using Fourier analysis. We used our extended range of FR schemes to investigate the linear stability of SD methods on triangles, and it was found that a only first order SD scheme could be recovered within this set of FR methods.