An extended range of stable flux reconstruction schemes on quadrilaterals for various polynomial bases

TL;DR

This paper develops an extended set of energy stable flux reconstruction schemes on quadrilaterals using various polynomial bases, introducing new correction functions and analyzing their stability and efficiency.

Contribution

It introduces new correction functions for flux reconstruction on quadrilaterals, including for Euclidean order bases, expanding the stability and efficiency of these schemes.

Findings

New correction functions enable stable schemes for various polynomial bases.

Approximate Euclidean order basis achieves similar accuracy with fewer points.

Schemes are stable and applicable to quadrilaterals with different polynomial bases.

Abstract

An extended range of energy stable flux reconstruction schemes, developed using a summation-by-parts approach, is presented on quadrilateral elements for various sets of polynomial bases. For the maximal order bases, a new set of correction functions which result in stable schemes is found. However, for a range of orders it is shown that only a single correction function can be cast as a tensor-product. Subsequently, correction functions are identified using a generalised analytic framework that results in stable schemes for total order and approximate Euclidean order polynomial bases on quadrilaterals -- which have not previously been explored in the context of flux reconstruction. It is shown that the approximate Euclidean order basis can provide similar numerical accuracy as the maximal order basis but with fewer points per element, and thus lower cost.