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

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## 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 \(\wp = 4,6,7\) solution points with superior performance to those tabulated in literature are discovered.

## Keywords

Flux reconstruction High-order methods Nodal discontinuous Galerkin method Computational fluid dynamics## 1 Introduction

Theoretical studies and numerical experiments suggest that unstructured high-order methods can provide solutions to otherwise intractable fluid flow problems within the vicinity of complex geometries. In 2007 Huynh proposed the flux reconstruction (FR) approach [1], a unifying framework for high-order schemes that encompasses several existing methods while simultaneously admitting an efficient implementation. Using FR it is possible to recover nodal discontinuous Galerkin (DG) methods of the type described by Hesthaven and Warburton [2], and the spectral difference schemes, original proposed by Kopriva and Kolias in 1996 [3] and later popularised by Sun et al. [4]. Unlike traditional DG methods, which are based on a weak formulation, and hence require integration, the FR approach is based on the differential form of the governing system. As a consequence, implementations of FR forego having to perform numerical quadrature within each element. This not only reduces complexity, but can also lead to decreased computational cost.

One problem with the FR approach is that for a non-linear flux function aliasing driven instabilities may develop. The severity of these instabilities depends upon the degree to which the flux is under resolved within each element. This is particularly problematic within the context of the Euler and Navier–Stokes equations. Here the flux is not only non-linear with respect to the solution but also non-polynomial. It has been demonstrated, both theoretically [5] and empirically [6] that the degree of aliasing driven instabilities depends upon the location of the solution points inside each element. Specifically, it has been found that placing points at the abscissa of strong Gaussian quadrature rules has a positive impact on their performance.

In this paper we describe a methodology for generating sets of candidate solution points for triangular elements. With this methodology we obtain a range of point sets for polynomial orders \(\wp = 3,4,5,6,7\). The performance of these points is then analysed, both theoretically and experimentally, and compared against those in literature. The paper is set out as follows. In Sect. 2 we give a short overview of previous work on the subject. Existing point sets proposed by Hesthaven and Warburton [2] and Williams and Jameson [7] are discussed. We also take the opportunity to elaborate on the constraints that candidate points must fulfil. The methodology used to derive our point sets is described in Sect. 3. Based on the approach of Zhang et al. [8], we show how it can be employed to generate quadrature rules in triangular elements. In Sect. 4 mathematical tools for analysing the rules that are produced will be presented. Potential issues regarding the lack of unisolvency are discussed. Candidate solution point sets are presented in Sect. 5. A two dimensional Euler vortex test case is introduced in Sect. 6. This test case, along with the metrics of Sect. 4 are then used to evaluate, compare and contrast the various rules with the results being presented in Sect. 7. Finally, in Sect. 8 conclusions are drawn and guidance on the choice of points is given.

## 2 Solution Point Requirements

Relationship between polynomial order, \(\wp \), and solution point count, \(N_p(\wp )\), in triangular elements

\(\wp \) | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

\(N_p(\wp )\) | 3 | 6 | 10 | 15 | 21 | 28 | 36 |

In 2011 Castonguay et al. [6] used the FR approach to solve the Euler equations on triangular grids. While evaluating the performance of the schemes on a Euler vortex test problem the \(\alpha \)-optimised points were found to exhibit extremely poor performance. However, similar test cases on quadrilateral meshes—with nodes placed at a tensor product of either Gauss–Legendre or Gauss–Legendre–Lobatto points—were found to be stable. One difference between the tensor produced points used in quadrilaterals and the \(\alpha \)-optimised points in triangles is that the latter do not correspond to the abscissa of good quadrature rules. Noting this the authors proceeded to use the abscissa of the (mildly asymmetric) quadrature rules presented in [9] as the solution points. Unlike the \(\alpha \)-optimised points which are designed for polynomial *interpolation* these points are designed for polynomial *integration*. By using good quadrature points as the solution points a marked improvement in the stability of the Euler vortex simulation was observed. Also in 2011 Jameson et al. [5] published a technical note investigating the non-linear stability of FR schemes in one dimension. In this paper a connection between solution point placement and non-linear stability was derived. It was found that instabilities associated with the projection (at the solution points) of a non-linear flux onto the polynomial space can be minimised by placing solution points at the abscissa of strong quadrature rules. This is in accordance with the empirical findings of Castonguay et al. [6].

In summary, for FR we seek solution points which fulfil dual roles: those of polynomial interpolation and of non-linear flux projection. Hence, we require solution points which satisfy four criteria. In addition to fulfilling the three requirements of Hesthaven and Warburton they must also be located at the abscissa of strong Gaussian quadrature rules.

## 3 Rule Derivation

### 3.1 Background

There is an extremely large body of literature on the derivation of Gaussian quadrature rules for triangular domains. Along with the aforementioned paper by Taylor et al. [9] a collection of near-optimal symmetric rules were discovered by Dunavant [10]. When deriving pure quadrature rules the objective is usually to obtain a rule which utilises a minimal number of points for a given integration strength. However, seldom are these minima coincident with the triangular number of points required by nodal schemes. A consequence of this is that few of the rules presented by Dunavant feature a triangular number of points. It is therefore not possible to repurpose these rules as FR solution points. An additional complication arises on account of the fact that basis functions on a triangle are not *unisolvent* [11]. This means that there exists configurations of distinct points inside of a triangle for which the associated basis set is linearly dependent. While this does not affect the suitability of a given set of points for numerical quadrature it does prevent them from being used for polynomial interpolation. Hence, such nodes are unsuitable candidates for solution points. These three constraints: symmetry, unisolvency and a triangular number of points serve to exclude almost all existing rules.

More recently Shunn and Ham [12] presented a set of quadrature rules for tetrahedra derived using a cubic close packed (CCP) lattice arrangement. One advantage of the CCP methodology employed is that all of the rules have, by construction, a tetrahedral number of points. By applying this approach to triangles Williams [13] was able to generate a suitable set of solution points for \(1 \le \wp \le 11\). Following Williams we will refer to these herein as the Williams–Shunn (WS) points. Further, in [7] Williams and Jameson proceeded to analyse the points by Shunn and Ham [12] for solving unsteady flow problems on tetrahedral meshes. (In these experiments the corresponding set of WS points were used on the faces of the tetrahedra.) It was found that the symmetric quadrature points outperformed the \(\alpha \)-optimised points in all of the test cases.

### 3.2 Methodology

To identify solution points in this paper we will employ the methodology of Zhang et al. [8] which provides a simple means of generating symmetric quadrature rules on triangles with a prescribed number of points. This will give us a means of generating candidate solution point sets which fulfil three of the four requirements described in Sect. 2. Namely those of being symmetric, having a triangular number of points, and with abscissa that correspond to a quadrature rule.

*symmetry orbits*as defined in Table 2. A point where \(\lambda _1 = \lambda _2 = \lambda _3 = \tfrac{1}{3}\) gives rise to an \(S_3\) orbit and one degree of freedom, the weight. Similarly, a point where any two barycentric components are identical corresponds to an \(S_{21}\) orbit which has three unique permutations and two degrees of freedom: the repeated component \(\alpha \) and a weight. Any symmetric set of points can be written as a combination of these orbits. From this it follows that \(N_p = n_3 + 3n_{21} + 6n_{111}\) where \(n_3 \in \{0,1\}\) and \((n_{21},n_{111}) \ge 0\). The above expression is simply a linear Diophantine equation. Hence, by solving it we can decompose an \(N_p\) into orbits. For example in the case of \(N_p = 15\) we obtain

Symmetry orbits inside of a triangle

Orbit | \(n_{\mathrm{pts}}\) | \(n_{\mathrm{dof}}\) | Barycentric coordinates |
---|---|---|---|

\(S_3(\alpha =\tfrac{1}{3})\) | 1 | 1 | \({{\mathrm{Perm}}}(\alpha ,\alpha ,\alpha )\) |

\(S_{21}(\alpha )\) | 3 | 2 | \({{\mathrm{Perm}}}(\alpha ,\alpha ,1-2\alpha )\) |

\(S_{111}(\alpha ,\beta )\) | 6 | 3 | \({{\mathrm{Perm}}}(\alpha ,\beta ,1-\alpha -\beta )\) |

*dependent*variables. As a consequence the weights are not included as degrees of freedom in the non-linear least squares problem. Rather, they are computed dynamically at each iteration of the non-linear solve. This is accomplished by taking the abscissa as given and then finding a set of weights which minimise the integration error. Since the only free parameters are the weights this is a linear least squares problem. This reduces the size of the non-linear least squares problem from \(n_1 + 2n_2 + 3n_3\) unknowns to \(n_2 + 2n_3\) unknowns [8]. This modification has been found to increase the probability that a given initial guess will converge to a solution. A description of the algorithm can be found in Algorithm 1.

*less*than the number of polynomials in Eq. 2. Specifically, by exploiting symmetries in the basis the number of equations can be reduced from

## 4 Metrics

### 4.1 Background

By employing the algorithm of the previous section it is possible to readily generate a large number of quadrature rules. However, while the methodology guarantees that the rules will be symmetric, of a given power and with a prescribed (triangular) number of points, it does not ensure that they will be suitable for polynomial interpolation. In this section we will introduce two mathematical constants that can be used a priori to compare the generated solution point candidates. The first of these, the Lebesgue constant, will allow us to assess how well suited a set of solution points are to the task of polynomial interpolation. The second quantity, an approximation of the truncation error, will permit us to evaluate the performance of a quadrature rule when it is used to integrate a basis of order \(\varphi ^{+} > \varphi \).

*normalised*Jacobi polynomial as specified in §18.3 of [15], \(i + j \le N_p\), and \(n\) is a bijective mapping onto \((i,j)\). Using the orthogonality of the Jacobi polynomials it can be readily shown that

### 4.2 Lebesgue constant

### 4.3 Truncation error

## 5 Candidate Point Sets

Number of rules, \(N_r\), discovered at each polynomial order, \(\wp \), and the associated quadrature strengths, \(\varphi \)

\(\wp \) | 3 | 4 | 5 | 5 | 6 | 6 | 7 |

\(\varphi \) | 5 | 7 | 8 | 9 | 10 | 11 | 12 |

\(\varphi ^{+}\) | 6 | 9 | 10 | 10 | 12 | 12 | 14 |

\(N_r\) | 95 | 66 | 722 | 473 | 412 | 12 | 136 |

\(N_r(\det \mathcal {V} \ne 0)\) | 24 | 64 | 452 | 2 | 236 | 0 | 100 |

## 6 Numerical Experiments

### 6.1 Euler Equations

### 6.2 Error Estimation

To completely specify the proposed numerical experiment it is also necessary to specify the time-marching algorithm/time-step, the approximate Riemann solver, and the choice of *flux points* along each edge. Here we choose to use a classical fourth order Runge–Kutta (RK4) scheme with \(\Delta t = 0.0005\). For computing the numerical fluxes at element interfaces we use a Rusanov type Riemann solver, as presented in [4]. Finally, at the edges of the triangles, we take the flux points to be at Gauss–Legendre points.

## 7 Results and Discussion

Starting with the Lebesgue-truncation plots, we note that for all orders except \(\wp = 4\) the \(\sigma \)- and \(\langle \sigma \rangle \)-optimal point sets—along with those of [13]—feature both low Lebesgue constants *and* low truncation errors. At higher orders it is evident that points with either high Lebesgue constants or high truncation errors are more likely to either become unstable before \(t = 100\) or perform poorly. A good example of this is the \(\Lambda \)-optimal points at orders \(\wp = 3,5,6\). At these orders the \(\Lambda \)-optimal points all have truncation errors within the upper-quartile and exhibit markedly worse performance than the \(\sigma \)-optimal or WS points. Conversely, at \(\wp = 7\) (when the \(\Lambda \)-optimal point set has a truncation error which lies in the lower-quartile of the distribution) the performance of the set is extremely good. These three results all serve to reaffirm the dual role that solution points play in FR schemes.

From the error-time plots it is observed that for all polynomial orders the performance of the \(\alpha \)-optimised points is significantly worse than those which are good quadrature rules. This is in good agreement with [6]. It is also observed from Table 4 that at orders \(\wp = 4, 6, 7\) the \(\sigma \)-optimal rule sets outperform the WS points by 73, 33, and 34 %, respectively.

\(L^2\) errors at \(t = 100\) for \(\Lambda \)-optimal, \(\xi \)-optimal, \(\sigma \)-optimal, \(\langle \sigma \rangle \)-optimal, and WS point sets

\(\wp \) | \(\sigma (t = 100)\) |
---|---|

3 | |

\(\alpha \)-opt | DNF \(t = 81.40\) |

\(\Lambda \)-opt | \(2.58 \times 10^{-2}\) |

\(\xi \)-opt | \(8.20 \times 10^{-3}\) |

\(\sigma \)-opt | \(8.20 \times 10^{-3}\) |

\(\langle \sigma \rangle \)-opt | \(8.61 \times 10^{-3}\) |

WS | \(8.27 \times 10^{-3}\) |

4 | |

\(\alpha \)-opt | DNF \(t = 13.30\) |

\(\Lambda \)-opt | \(1.20 \times 10^{-3}\) |

\(\xi \)-opt | \(2.09 \times 10^{-3}\) |

\(\sigma \)-opt | \(6.59 \times 10^{-4}\) |

\(\langle \sigma \rangle \)-opt | \(6.76 \times 10^{-4}\) |

WS | \(1.15 \times 10^{-3}\) |

5 | |

\(\alpha \)-opt | DNF \(t = 18.95\) |

\(\Lambda \)-opt | \(2.64 \times 10^{-3}\) |

\(\xi \)-opt | \(1.36 \times 10^{-4}\) |

\(\sigma \)-opt | \(9.69 \times 10^{-5}\) |

\(\langle \sigma \rangle \)-opt | \(9.69 \times 10^{-5}\) |

WS | \(6.92 \times 10^{-5}\) |

6 | |

\(\alpha \)-opt | DNF \(t = 22.24\) |

\(\Lambda \)-opt | \(2.02 \times 10^{-4}\) |

\(\xi \)-opt | \(4.16 \times 10^{-5}\) |

\(\sigma \)-opt | \(2.38 \times 10^{-5}\) |

\(\langle \sigma \rangle \)-opt | \(2.42 \times 10^{-5}\) |

WS | \(3.16 \times 10^{-5}\) |

7 | |

\(\alpha \)-opt | DNF \(t = 12.50\) |

\(\Lambda \)-opt | \(5.95 \times 10^{-6}\) |

\(\xi \)-opt | \(1.10 \times 10^{-5}\) |

\(\sigma \)-opt | \(5.95 \times 10^{-6}\) |

\(\langle \sigma \rangle \)-opt | \(5.95 \times 10^{-6}\) |

WS | \(8.00 \times 10^{-6}\) |

## 8 Conclusion

In this paper we have investigated how solution point placement contributes to the stability and accuracy of FR schemes on triangular elements. To begin we described a simple methodology for generating candiate solution point sets. After analysing the performance of these points for an Euler vortex test problem the main findings can be summarised as follows. Firstly, the requirement that points be at the abscissa of a quadrature rule appears to be necessary but not sufficient for a good numerical scheme. As a corollary to this we have reaffirmed previous findings that the \(\alpha \)-optimised points are unsuitable for non-linear problems. Secondly, both the Lebesgue constant and truncation error associated with a point set are very strong indicators of stability. Finally, we have demonstrated that existing quadrature-optimised solution points are not necessarily optimal. Specifically, at orders \(\wp = 4, 6, 7\) we have discovered rules which outperform those of Williams [13].

## Notes

### Acknowledgments

The authors would like to thank the Engineering and Physical Sciences Research Council for their support via a Doctoral Training Grant and an Early Career Fellowship (EP/K027379/1).

## Supplementary material

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