An Analysis of Solution Point Coordinates for Flux Reconstruction Schemes on Tetrahedral Elements

Abstract

The flux reconstruction (FR) approach offers an efficient route to high-order accuracy on unstructured grids. In this work we study the effect of solution point placement on the stability and accuracy of FR schemes on tetrahedral grids. To accomplish this we generate a large number of solution point candidates that satisfy various criteria at polynomial orders ℘ = 3, 4, 5. We then proceed to assess their properties by using them to solve the non-linear Euler equations on both structured and unstructured meshes. The results demonstrate that the location of the solution points is important in terms of both the stability and accuracy. Across a range of cases it is possible to outperform the solution points of Shunn and Ham for specific problems. However, there appears to be a degree of problem-dependence with regards to the optimal point set, and hence overall it is concluded that the Shunn and Ham points offer a good compromise in terms of practical utility.

Temporal Stabilisation of Flux Reconstruction on Linear Problems

“…The positioning of points is of great importance and was investigated for tetrahedra by Witherden. 13 With the domain of the solution established we now introduce both the equation to be solve and the method to solve it.…”

Stable Spectral Difference Approach Using Raviart-Thomas Elements for 3D Computations on Tetrahedral Grids

“…The class of Vincent-Castonguay-Jameson-Huynh (VCJH) [8] schemes were first extended to tetrahedral elements by Williams and Jameson [9]. The effect of the solution point location was studied in [10,11] to determine their effect on the stability and the accuracy of FR schemes on tetrahedral grids. Both works indicate that the best choice in terms of stability and accuracy is to locate the solution points following the Shunn-Ham quadrature rule [12].…”

A unifying algebraic framework for discontinuous Galerkin and flux reconstruction methods based on the summation-by-parts property

“…This is likely the result of aliasing error due to the collocation projection of the (nonlinear) flux components on a set of nodes which do not coincide with the abscissae of a sufficiently accurate quadrature rule. As the standard VCJH methods are recovered from the present framework for schemes of Type 2, substantial research has been conducted in that context to explain this mechanism for instability, from both theoretical and numerical perspectives (see, for example, Jameson et al[79], Williams and Jameson[80], Witherden and Vincent[81,82], and Witherden et al[83]). …”