Provably Stable Flux Reconstruction High-Order Methods on Curvilinear Elements

TL;DR

This paper develops provably stable flux reconstruction schemes for partial differential equations on curvilinear elements, emphasizing the importance of split forms for stability and demonstrating optimal convergence.

Contribution

It introduces a novel approach to constructing stable flux reconstruction schemes on curvilinear elements using split forms and metric-dependent correction functions.

Findings

Proves that DG forms differ in curvilinear coordinates, even with exact metrics.

Shows that split forms are essential for stability in curvilinear coordinates.

Demonstrates optimal convergence of the proposed schemes.

Abstract

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