An extended range of energy stable flux reconstruction methods on triangles

TL;DR

This paper develops an extended set of energy-stable flux reconstruction methods on triangular elements using a summation-by-parts framework, enabling wider stability bounds and analyzing their relation to spectral difference schemes.

Contribution

It introduces a new extended range of stable flux reconstruction schemes on triangles, expanding the stability analysis and connecting to existing spectral difference methods.

Findings

Extended stable FR schemes include single parameter schemes of prior work.

Wider stability bounds are established for these schemes.

Only the first order spectral difference scheme is recovered within this set.

Abstract

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 \citet{Castonguay2011} 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.