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

Abstract

In this paper, the Spectral Difference approach using Raviart-Thomas elements (SDRT) is formulated for the first time on tetrahedral grids. To determine stable formulations, a Fourier analysis is conducted for different SDRT implementations, i.e. different interior flux points locations. This stability analysis demonstrates that using interior flux points located at the Shunn-Ham quadrature rule points leads to linearly stable SDRT schemes up to the third order. For higher orders of accuracy, a significant impact of the position of flux points located on faces is shown. The Fourier analysis is then extended to the coupled time-space discretization and stability limits are determined. Additionally, a comparison between the number of interior FP required for the SDRT scheme and the Flux Reconstruction method is proposed and shows that the two approaches always differ on the tetrahedron. Unsteady validation test cases include a convergence study using the Euler equations and the simulation of the Taylor-Green vortex.

Appendices

Appendix A. Definition of the Raviart-Thomas (RT) Space

The Raviart-Thomas (RT) finite element spaces were originally introduced by Raviart and Thomas [41] to approximate the Sobolev space $\text {H(div)}$ defined by:

$$\begin{aligned} \text {H(div)} = \{ u \in \left( L^2 (K)\right) ^d, \, \, \nabla \cdot \, u \in L^2(K) \}, \end{aligned}$$

(A.1)

where d is the dimension, K is a bounded open subset of ${\mathbb {R}}^d$ with a Lipshitz continuous boundary, $L^2 (K)$ is the Hilbert space of square-integrable function defined on K. The extension to the three-dimensional case considering K as a tetrahedron or a cube was proposed by Nedelec [42]. The space $RT_p$ spanned by the Raviart-Thomas basis functions of degree p is the smallest polynomial space such that the divergence maps $RT_p$ onto ${\mathbb {P}}_p$, the space of piecewise polynomials of degree $\le p$. Considering the reference tetrahedron ${\mathcal {T}}_e$, the RT space of order p is defined in 3D by:

$$\begin{aligned} RT_p = ({\mathbb {P}}_p)^3 + \begin{pmatrix} x \\ y \\ z \end{pmatrix} \bar{{\mathbb {P}}}_p, \end{aligned}$$

(A.2)

where ${\mathbb {P}}_p$ is the space of polynomials of degree at most p:

$$\begin{aligned} {\mathbb {P}}_p(x,y,z) = span \{ x^i y^j z^k,\; i, j,k \ge 0, i+j+k \le p\}, \end{aligned}$$

(A.3)

$\bar{{\mathbb {P}}}_p$ is the space of polynomials of degree p:

$$\begin{aligned} \bar{{\mathbb {P}}}_p(x,y,z) = span \{ x^i y^j z^k,\; i, j, k \ge 0, i+j+k = p\}, \end{aligned}$$

(A.4)

and $({\mathbb {P}}_p)^3 = ({\mathbb {P}}_p,{\mathbb {P}}_p,{\mathbb {P}}_p)^\top$ is the three dimensional vector space for which each component is a polynomial of degree at most p. The dimension of each space is $\text {dim}\; {\mathbb {P}}_p = \frac{(p+1)(p+2)(p+3)}{6}$, $\text {dim}\; {\mathbb {P}}_p^3 = \frac{(p+1)(p+2)(p+3)}{2}$, $\text {dim}\; \bar{{\mathbb {P}}}_p = \frac{(p+1)(p+2)}{2}$ and thus $\text {dim} \; RT_p = \frac{(p+1)(p+2)(p+4)}{2}$. We denote $\varvec{\phi }_n, n \in [\![1, N_{FP} ]\!]$ the monomials which form a basis in the $RT_p$ space where

$$\begin{aligned} N_{FP} = \frac{(p+1)(p+2)(p+4)}{2}. \end{aligned}$$

(A.5)

Determination of $\varvec{\phi }_n$ for $RT_1$, $N_{FP} = 15$

(A.6)

Appendix B. Matrices Formulation for the Fourier Analysis

The matrices ${\mathbf {M}}^{0,0,0}$, ${\mathbf {M}}^{-1,0,0}$, $\mathbf {M}^{+1,0,0}$, ${\mathbf {M}}^{0,-1,0}$, ${\mathbf {M}}^{0,+1,0}$, ${\mathbf {M}}^{0,0,-1}$ and ${\mathbf {M}}^{0,0,+1}$ involved in the SDRT spatial discretization for the Fourier analysis on tetrahedral elements (Eq. (28)) are detailed in this appendix. Those matrices are given as:

$$\begin{aligned} {\mathbf {M}}^{0,0,0} = \begin{bmatrix} {\mathbf {D}}_{jk} &{} \cdots &{} O_{N_{SP}, N_{FP}} \\ \vdots &{} \ddots &{} \vdots \\ O_{N_{SP}, N_{FP}} &{} \cdots &{} {\mathbf {D}}_{jk} \\ \end{bmatrix} {\mathbf {C}}^{0,0,0} \begin{bmatrix} {\mathbf {T}}_{kj} &{} \cdots &{} O_{N_{FP}, N_{SP}} \\ \vdots &{} \ddots &{} \vdots \\ O_{N_{FP}, N_{SP}} &{} \cdots &{} {\mathbf {T}}_{kj} \\ \end{bmatrix}, \end{aligned}$$

(B.1)

where $O_{m, n}$ is the zero matrix of size $m \times n$. The same goes for ${\mathbf {M}}^{-1,0,0}$, ${\mathbf {M}}^{+1,0,0}$, $\mathbf {M}^{0,-1,0}$, ${\mathbf {M}}^{0,+1,0}$, ${\mathbf {M}}^{0,0,-1}$ and ${\mathbf {M}}^{0,0,+1}$, associated respectively to the velocity matrices ${\mathbf {C}}^{-1,0,0}$, ${\mathbf {C}}^{+1,0,0}$, $\mathbf {C}^{0,-1,0}$, ${\mathbf {C}}^{0,+1,0}$, ${\mathbf {C}}^{0,0,-1}$ and ${\mathbf {C}}^{0,0,+1}$. The transfer matrix is given by Eq. (16):

$$\begin{aligned} {\mathbf {T}}_{kj} = \sum _{m=1}^{N^{tet}_{SP}} \left( \Phi _m(\varvec{\xi }_j) \right) ^{-1} \, \Phi _m(\varvec{\xi }_k), \end{aligned}$$

(B.2)

and the differentiation matrix by Eq. (21):

$$\begin{aligned} {\mathbf {D}}_{jk} = \sum _{n=1}^{N^{tet}_{FP}} \left( \varvec{\phi }_n (\varvec{\xi }_k) \cdot \hat{{\mathbf {n}}}_k\right) ^{-1}\;{\hat{\nabla }} \cdot \varvec{\phi }_n (\varvec{\xi }_j ). \end{aligned}$$

(B.3)

The velocity matrices are given as:

$$\begin{aligned} {\mathbf {C}}^{0,0,0}= & {} \begin{bmatrix} {\mathbf {C}}^{L} &{} {\mathbf {C}}^{T_1, T_2} &{} O_{N_{FP}, N_{FP}}&{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} {\mathbf {C}}^{T_1, T_6}\\ {\mathbf {C}}^{T_2, T_1} &{} {\mathbf {C}}^{L} &{} {\mathbf {C}}^{T_2, T_3} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{}O_{N_{FP}, N_{FP}}\\ O_{N_{FP}, N_{FP}} &{} {\mathbf {C}}^{T_3, T_2} &{} {\mathbf {C}}^{L} &{} {\mathbf {C}}^{T_3, T_4} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}}\\ O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} {\mathbf {C}}^{T_4, T_3} &{} {\mathbf {C}}^{L}&{} {\mathbf {C}}^{T_4, T_5} &{} O_{N_{FP}, N_{FP}} \\ O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} {\mathbf {C}}^{T_5, T_4} &{}{\mathbf {C}}^{L}&{} {\mathbf {C}}^{T_5, T_6} \\ {\mathbf {C}}^{T_6, T_1} &{} O_{N_{FP}, N_{FP}} &{}O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} {\mathbf {C}}^{T_6, T_5} &{}{\mathbf {C}}^{L} \\ \end{bmatrix}, \end{aligned}$$

(B.4)

$$\begin{aligned} {\mathbf {C}}^{-1,0,0}= & {} \begin{bmatrix} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} {\mathbf {C}}^{T_1, T_5} &{} O_{N_{FP}, N_{FP}}\\ O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} {\mathbf {C}}^{T_2, T_4} &{} O_{N_{FP}, N_{FP}} &{}O_{N_{FP}, N_{FP}}\\ &{} &{} \begin{bmatrix} O_{4N_{FP}, 6N_{FP}} \end{bmatrix} \\ \end{bmatrix}, \end{aligned}$$

(B.5)

$$\begin{aligned} {\mathbf {C}}^{+1,0,0}= & {} \begin{bmatrix}&\,&\begin{bmatrix} O_{3N_{FP}, 6N_{FP}} \end{bmatrix} \\ O_{N_{FP}, N_{FP}} &{} {\mathbf {C}}^{T_4, T_2} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}}\\ {\mathbf {C}}^{T_5, T_1}&{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{}O_{N_{FP}, N_{FP}}\\ &{} &{} \begin{bmatrix} O_{N_{FP}, 6N_{FP}} \end{bmatrix} \\ \end{bmatrix}, \end{aligned}$$

(B.6)

$$\begin{aligned} {\mathbf {C}}^{0,-1,0}= & {} \begin{bmatrix}&\,&\begin{bmatrix} O_{2N_{FP}, 6N_{FP}} \end{bmatrix} \\ {\mathbf {C}}^{T_3, T_1} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}}\\ O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}}&{} O_{N_{FP}, N_{FP}}&{} {\mathbf {C}}^{T_4, T_6} \\ &{} &{} \begin{bmatrix} O_{2N_{FP}, 6N_{FP}} \end{bmatrix} \\ \end{bmatrix}, \end{aligned}$$

(B.7)

$$\begin{aligned} {\mathbf {C}}^{0,+1,0}= & {} \begin{bmatrix} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{}{\mathbf {C}}^{T_1, T_3} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{}O_{N_{FP}, N_{FP}}\\ &{} &{} \begin{bmatrix} O_{4N_{FP}, 6N_{FP}} \end{bmatrix} \\ O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} {\mathbf {C}}^{T_6, T_4} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}}\\ \end{bmatrix}, \end{aligned}$$

(B.8)

$$\begin{aligned} {\mathbf {C}}^{0,0,-1}= & {} \begin{bmatrix}&\,&\begin{bmatrix} O_{N_{FP}, 6N_{FP}} \end{bmatrix} \\ O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{}{\mathbf {C}}^{T_2, T_6}\\ O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} {\mathbf {C}}^{T_3, T_5} &{} O_{N_{FP}, N_{FP}}\\ &{} &{} \begin{bmatrix} O_{3N_{FP}, 6N_{FP}} \end{bmatrix} \\ \end{bmatrix}, \end{aligned}$$

(B.9)

$$\begin{aligned} {\mathbf {C}}^{0,0,+1}= & {} \begin{bmatrix}&\,&\begin{bmatrix} O_{4N_{FP}, 6N_{FP}} \end{bmatrix} \\ O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} {\mathbf {C}}^{T_5, T_3} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}}\\ O_{N_{FP}, N_{FP}} &{} {\mathbf {C}}^{T_6, T_2} &{} O_{N_{FP}, N_{FP}}&{} O_{N_{FP}, N_{FP}} &{} O_{N_{FP}, N_{FP}} &{}O_{N_{FP}, N_{FP}}\\ \end{bmatrix},\qquad \qquad \end{aligned}$$

(B.10)

where ${\mathbf {C}}^{L}$ is defined by:

$$\begin{aligned} {\mathbf {C}}^{L} = \begin{bmatrix} \begin{bmatrix} \left( {\mathbf {c}} \cdot {\mathbf {n}} \right) \frac{1+{{\,\mathrm{sign}\,}}({\mathbf {c}} \cdot {\mathbf {n}})}{2} &{} \cdots &{} 0 \\ \vdots &{}\ddots &{}\vdots \\ 0&{} \cdots &{} \left( {\mathbf {c}} \cdot {\mathbf {n}} \right) \frac{1+{{\,\mathrm{sign}\,}}({\mathbf {c}} \cdot {\mathbf {n}})}{2} \\ \end{bmatrix}_{4N_f, 4N_f} &{} O_{4N_f, N_i} \\ O_{N_i, 4N_f} &{} \begin{bmatrix} |J| J^{-1} ({\mathbf {c}} \cdot \hat{{\mathbf {n}}}) &{} \cdots &{} 0 \\ \vdots &{}\ddots &{}\vdots \\ 0&{} \cdots &{} |J| J^{-1} ({\mathbf {c}} \cdot \hat{{\mathbf {n}}}) \\ \end{bmatrix}_{N_i, N_i} \end{bmatrix}. \end{aligned}$$

(B.11)

The matrix ${\mathbf {C}}^{T_i, T_j}$ links the FP between the triangular faces of $T_i$ and $T_j$. Its expression will depend on the local connectivity, i.e. the number and the orientation of the two faces in their respective element. The face number gives the local FP numbering whereas the orientation (how the two faces are facing each other) gives the FP order. An example of its determination is provided below for two arbitrary tetrahedral elements.

Determination of the Matrix $\mathbf {{C}^{T_i, T_j}}$ —Example for $\mathbf {p=1}$ Let us consider two tetrahedron $T_1$ and $T_2$ defined by their four nodes:

$$\begin{aligned}&T_1: \quad N_1, N_2, N_3, N_4 = A,B,C,D, \end{aligned}$$

(B.12)

$$\begin{aligned}&T_2: \quad N_1, N_2, N_3, N_4 = A,C,E,D. \end{aligned}$$

(B.13)

Following the CGNS notations, their faces are defined by:

• $T_1$, Face 1: A, C, B,       $\bullet$ $T_2$, Face 1: A, E, C,

• $T_1$, Face 2: A, B, D,       $\bullet$ $T_2$, Face 2: A, C, D,

• $T_1$, Face 3: B, C, D,       $\bullet$ $T_2$, Face 3: C, E, D,

• $T_1$, Face 4: C, A, D,       $\bullet$ $T_2$, Face 4: E, A, D.

They are sharing a face corresponding to Face 4 in $T_1$ and Face 2 in $T_2$. In the case of $p=1$, this indicates that the FP numbers on ($T_1$, Face 4) are [10, 12] whereas the FP number (Face 2, $T_2$) are [4, 6]. Then, the orientation between the faces needs to be determined to know in which order the FP are facing each other. For two arbitrary faces A and B defined respectively by nodes $(A_1, A_2, A_3)$ and $(B_1,B_2,B_3)$, three cases are possible:

• $A_1 = B_3$, $A_2 = B_2$, $A_3 = B_1$,

• $A_1 = B_2$, $A_2 = B_1$, $A_3 = B_3$,

• $A_1 = B_1$, $A_2 = B_3$, $A_3 = B_2$.

Our example is illustrated in Fig. 6 and corresponds to the second case.

Fig. 6

Illustration of the orientation determination: $T_1$ (on the left) and $T_2$ (on the right)

In this example, the matrix ${\mathbf {C}}^{T_1, T_2}$ will take the following expression:

$$\begin{aligned} {\mathbf {C}}^{T_1, T_2} = \left( {\mathbf {c}} \cdot {\mathbf {n}} \right) \begin{bmatrix} O_{3N_f, N_{FP}} \\ \begin{bmatrix} O_{N_f, N_f} &{} \begin{bmatrix} 0 &{} \frac{1+{{\,\mathrm{sign}\,}}({\mathbf {c}} \cdot {\mathbf {n}})}{2} &{} 0 \\ 0 &{} 0 &{} \frac{1+{{\,\mathrm{sign}\,}}({\mathbf {c}} \cdot {\mathbf {n}})}{2} \\ \frac{1+{{\,\mathrm{sign}\,}}({\mathbf {c}} \cdot {\mathbf {n}})}{2} &{} 0 &{} 0 \\ \end{bmatrix} &{} O_{N_f, 2*N_f+N_i}\\ \end{bmatrix}\\ O_{N_i, N_{FP}} \\ \end{bmatrix}. \end{aligned}$$

(B.14)