An analysis of solution point coordinates for flux reconstruction schemes on triangular elements

Abstract

The flux reconstruction approach offers an efficient route to high-order accuracy on unstructured grids. The location of the solution points plays an important role in determining the stability and accuracy of FR schemes on triangular elements. In particular, it is desirable that a solution point set (i) defines a well conditioned nodal basis for representing the solution, (ii) is symmetric, (iii) has a triangular number of points and, (iv) minimises aliasing errors when constructing a polynomial representation of the flux. In this paper we propose a methodology for generating solution points for triangular elements. Using this methodology several thousand point sets are generated and analysed. Numerical performance is assessed through an Euler vortex test case. It is found that the Lebesgue constant and quadrature strength of the points are strong indicators of stability and performance. Further, at polynomial orders ℘ = 4, 6, 7 solution points with superior performance to those tabulated in literature are discovered.