Provably stable flux reconstruction high-order methods on curvilinear elements

Provably stable flux reconstruction (FR) schemes are derived for partial differential equations cast in curvilinear coordinates. Specifically, energy (ESFR) considered as they allow design flexibility well stability proofs the linear advection problem on affine elements. Additionally, split forms examined enable development of proofs. The first critical step proves, that coordinates, discontinuous Galerkin (DG) conservative and non-conservative inherently different--even under exact integration analytically metric terms. This analysis demonstrates form is essential to developing provably DG coordinates motivates construction dependent ESFR correction functions each element. Furthermore, FR differ from literature only apply surface terms or form, instead incorporate full equations. It demonstrated scheme divergent when used We numerically verify claims our proposed compare them literature. Lastly, newly shown obtain optimal orders convergence. loses accuracy at equivalent parameter value c one-dimensional scheme.