The discontinuous Galerkin spectral element methods for compressible flows on two-dimensional mixed grids

doi:10.1016/j.jcp.2018.03.001

Journal of Computational Physics

Volume 364, 1 July 2018, Pages 314-346

https://doi.org/10.1016/j.jcp.2018.03.001 Get rights and content

Highlights

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The discontinuous Galerkin spectral element (DGSEM) is generalized to mixed triangular and quadrilateral grids.
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The influence of insufficient SCP quadrature to DGSEM is analyzed.
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The dispersion and dissipation property of DGSEM on triangular grids is given and compared with that of nodal DG method.
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A problem independent limiting procedure containing a new high-resolution WENO limiter is proposed for capturing shock waves.
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The DGSEM scheme is applied to solve the two dimensional compressible flows on mixed grids.

Abstract

This paper presents a practical algorithm for constructing high order discontinuous Galerkin spectral element methods (DGSEM) on mixed triangular and quadrilateral grids. The traditional DGSEM belongs to the collocation-type nodal discontinuous Galerkin method which is computationally efficient on one-dimensional and tensor-product grids. This work generalizes DGSEM to triangular grids using symmetric quadrature points in sphere close packed arrangement. The dispersion and dissipation analysis shows that the scheme is stable for linear problems on triangles. A problem independent limiting procedure is proposed for capturing the shock waves. The main components are a modified KXRCF shock detector, a smoothness indicator and a high-resolution WENO limiter based on candidate polynomials constructed by newly developed projection approach. The implementation of the DGSEM on the mixed grids is also presented. Numerical examples on triangular and mixed grids verify the scheme's high order accuracy and robustness in computing inviscid and viscous compressible flows.

Introduction

The discontinuous Galerkin (DG) method is widely studied for its great potentials. It combines the merits of both the finite element method and finite volume method, and shows lots of attractive features, such as high order accuracy, high resolution and the flexibility in hp adaptive mesh refinement. To improve the computational efficiency, the modal discontinuous Galerkin method would use the hierarchical orthogonal bases on the iso-parametric element, e.g., the Dubiner bases on iso-parametric triangles [1], [2]. This could bring convenience to perform the p-adaptivity or p-multigrid. Comparing with the modal schemes, the nodal high order schemes are preferred by certain researchers in high-order community for the clear physical meaning of the degrees of freedom (DOF) and less effort to obtain the values at the face and volume quadrature points. Hesthaven [3] used a set of nodal points to represent DOFs inside the cells and constructed the nodal DG method using the strong form of DG formulation. Huynh [4], [5] constructed the nodal-type high order flux reconstruction (FR) schemes on one-dimensional or tensor-product multi-dimensional grids, which was considered as the unification of the discontinuous Galerkin method, spectral difference (SD) method and spectral volume (SV) method by choosing different correct functions. The extension of FR to triangular grids, which was named as lifting collocation penalty (LCP) scheme, was given by Wang and Gao [6] in solving the Euler equations. Additional effort has been devoted to generalize the LCP scheme to mixed 3D grids in solving Navier–Stokes equations [8] and to make the scheme conservative [7]. Such schemes are renamed as the Correction Procedure via Reconstruction (CPR) method by combining the FR and LCP schemes. Vincent et al. [9], [10], [11] developed another family of energy stable FR (ESFR) method which was parameterized by a single scalar quantity. This family, also known as the VCJH scheme, can recover various well known high-order methods by choosing different scalar quantities. The VCJH scheme has been generalized to the mixed grids in solving advection–diffusion problems.

The DGSEM has been used in solving the convection–diffusion equation and compressible turbulence flows [19], [20], [21], [22]. It is easy to generalize to arbitrary high order accuracy on tensor-product grids by choosing different orders of Gauss–Legendre (GL) quadrature points as the solution points. Gassner [12] also proved that the DGSEM with GL points was equivalent to one of the most prominent members of the Hyunh's FR scheme with correction function ‘ $g_{2}$ ’ or ‘ $g_{L u m p, L o}$ ’. Whereas due to the lack of Gaussian quadrature points with exact number $N_{p} = \frac{(p + 1) (p + 2)}{2}$ (p is the order of polynomial for the approximate solution) on the triangular grids, DGSEM on triangular grids cannot work in the same way as on tensor-product grids.

The weak form instead of strong form is adopted in this paper for better efficiency which is explained in [24]. As noted in [13], [14], for the integral appearing in the weak formulation of a $(p + 1)$ -th order DG scheme, a sufficient condition is that the quadrature rules over faces is exact for polynomials of degree $2 p + 1$ , and over the element is exact for polynomials of degree 2p. This paper investigates the feasibility of using the optimal quadrature order with $N_{p}$ quadrature points for a $(p + 1)$ -th order DGSEM scheme on triangular grids. Such quadrature rules on tetrahedron have been derived by L. Shunn and F. Ham [17] termed as the cubic close-packed (CCP) lattice arrangement using $N_{p}^{3} = \frac{(p + 1) (p + 2) (p + 3)}{6}$ quadrature points. The corresponding quadrature rules on triangle were given by Williams et al. [18] with sphere close packed (SCP) lattice arrangements of $N_{p}$ points. The quadrature rules with SCP arranged points, which is named as “SCP quadrature” for short, integrate a polynomial of the highest possible order and minimize the magnitude of the truncation error term. These quadrature points are adopted as the solution points in the DGSEM scheme. This quadrature accuracy would affect the evaluation of mass matrix and volume integration. Thus an analysis is required to study the behavior of SCP quadrature points on DGSEM schemes.

This paper analyzes the dispersion and dissipation property of different orders of DGSEM schemes on triangular grids. Hu et al. [26] described a wave propagation analysis for the one-dimensional and two-dimensional DG methods. Gassner et al. [20] studied the dispersion and dissipation properties of the Gauss–Legendre and Gauss–Lobatto DGSEM for linear wave propagation problems. With similar methodology, Abeele et al. [27] analyzed the stability of spectral volume (SV) method and found optimized SV partitions for lower dispersion and diffusion errors. Abeele et al. [28] also proved higher than second-order 1D spectral difference (SD) schemes using the Chebyshev–Gauss–Lobatto nodes as the flux points exhibited a weak instability and they identified new flux points for stable SD schemes. This paper will apply the dispersion and diffusion analysis to DGSEM on triangular grids to show its mathematical properties.

The developed DGSEM scheme on triangles also belongs to the spectral collocation-type discontinuous Galerkin method. The diagonal mass matrix and the possibility to compute the fluxes directly from the solution points give rise to higher efficiency comparing with the standard nodal DG methods [3], [35], [46]. The nodal DG in strong form [3] and the CPR scheme [6], [7] on triangular grids require the evaluation of flux divergence. If the nonlinear fluxes in Euler/Navier–Stokes equations are reconstructed using Lagrangian interpolation on the solution points, there is an accuracy loss of half to one order. The nodal DG scheme in strong form could recover the order of accuracy by reconstructing the polynomial of nonlinear fluxes using both the solution and face flux points. The chain-rule method is adopted for CPR to evaluate the flux divergence for better accuracy, however the resulting scheme would not be conservative. On the other hand, the DGSEM scheme avoids the evaluation of flux divergence and is always conservative.

The rest of this article is organized as follows. Section 2 describes the framework of DGSEM on quadrilateral and triangular grids. The influence of the SCP quadrature points and the dispersion and dissipation errors for second to sixth order DGSEM schemes on triangular grids are analyzed in Section 3. A problem independent limiting procedure is designed in Section 4 for DGSEM schemes to capture shock waves. Then the scheme is generalized to mixed high order triangular and quadrilateral grids in Section 5. Numerical examples are given in Section 6 to validate the efficiency of the schemes for computing compressible flows on different kinds of grids.

Section snippets

Governing equations

The Navier–Stokes equations for the two-dimensional compressible unsteady flows are used for the governing equations and the conservation form is written as $\frac{\partial U}{\partial t} + \frac{\partial (F - F_{v})}{\partial x} + \frac{\partial (G - G_{v})}{\partial y} = 0,$ where the conservative variables U and convective flux vectors F, G are given by $U = (\begin{matrix} ρ \\ ρ u \\ ρ v \\ E \end{matrix}), F = (\begin{matrix} ρ u \\ ρ u^{2} + p \\ ρ u v \\ u (E + p) \end{matrix}), G = (\begin{matrix} ρ u \\ ρ u v \\ ρ v^{2} + p \\ v (E + p) \end{matrix}) .$ Here, $ρ, u, v, p$ and E are the density, the velocity components in x and y directions, the pressure and the total energy respectively. The diffusive flux vectors $F_{v}$ , $G_{v}$ are given by $\begin{matrix} F_{v} = μ \end{matrix}$

Comparisons with other nodal DG methods in weak form on triangular grids

The DGSEM scheme derived on triangular grids in last section shares some similarities with other types of nodal schemes. The standard nodal DG method in weak form usually adopts two sets of nodal points to store the DOFs and perform the volume integration respectively. The location of solution points in Dumbser et al. [35] and Persson&Peraire [46] used the equidistant points and Hesthaven [3] used the α-optimized LGL points to reduce the Lebesgue constant. In these methods, a quadrature rule

A problem independent limiting procedure for capturing shock waves

The developed DGSEM schemes can solve smooth compressible flows on mixed 2D grids. However, the schemes would produce numerical oscillations near the discontinuities or even nonlinear instabilities for strong shock waves. There has been an abundance of work on shock capturing techniques for DG methods over the past two decades. The limiters as one technique usually contain two steps. The first step is to find the troubled cells with discontinuities using the shock detector, e.g., the TVB [13]

DGSEM on mixed grids

There are different ways to treat the mixed elements with different DOFs in the domain. This paper stores the variables on the solution points in a list and uses indicators to specify the location of solution points in a given cell as shown in Fig. 14. The grids are generated using the GMSH open source software [29] and are output with the high order interpolation. This high order grids are not only for the internal cells, but also in the boundary cells to represent the curved faces, as shown

Numerical examples

In this section, we apply the DGSEM scheme derived in the previous sections to solve a number of 2D test cases. The contours for all the cases are output with ${(p + 1)}^{2}$ uniform sub-cells for a ( $p + 1$ )-th order scheme. Fig. 16 shows an example of the sub-cells division for a p3 DGSEM scheme.

Conclusions

In the present paper, the DGSEM schemes on mixed quadrilateral and triangular grids are proposed. The points in SCP arrangement on triangles are adopted as the quadrature points for the volume integration and solution points of the DGSEM schemes. The insufficient SCP quadrature accuracy would not affect the DGSEM-SCP scheme's order of accuracy according to our analysis. The dispersion and diffusion errors show that the DGSEM-SCP is quite close to the standard nodal DG scheme. The p1 to p5

Acknowledgements

The work is financially supported by the National Natural Science Foundation of China (No. 11402313 and 91752114) and SYSU project (No. 14lgjc10).

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The discontinuous Galerkin spectral element methods for compressible flows on two-dimensional mixed grids

Highlights

Abstract

Introduction

Section snippets

Governing equations

Comparisons with other nodal DG methods in weak form on triangular grids

A problem independent limiting procedure for capturing shock waves

DGSEM on mixed grids

Numerical examples

Conclusions

Acknowledgements

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