A Unifying Algebraic Framework for Discontinuous Galerkin and Flux Reconstruction Methods Based on the Summation-by-Parts Property

Abstract

We propose a unifying framework for the matrix-based formulation and analysis of discontinuous Galerkin (DG) and flux reconstruction (FR) methods for conservation laws on general unstructured grids. Within such an algebraic framework, the multidimensional summation-by-parts (SBP) property is used to establish the discrete equivalence of strong and weak formulations, as well as the conservation and energy stability properties of a broad class of DG and FR schemes. Specifically, the analysis enables the extension of the equivalence between the strong and weak forms of the discontinuous Galerkin collocation spectral-element method demonstrated by Kopriva and Gassner (J Sci Comput 44:136–155, 2010) to more general nodal and modal DG formulations, as well as to the Vincent-Castonguay-Jameson-Huynh (VCJH) family of FR methods. Moreover, new algebraic proofs of conservation and energy stability for DG and VCJH schemes with respect to suitable quadrature rules and discrete norms are presented in which the SBP property serves as a unifying mechanism for establishing such results. Numerical experiments are provided for the two-dimensional linear advection and Euler equations, highlighting the design choices afforded for methods within the proposed framework and corroborating the theoretical analysis.

Appendices

Appendix

1.1 Appendix A Polynomial Bases

Although the theoretical analysis in this work is independent of the choice of basis $\mathcal {{B}}$ employed for a given discretization, the concrete implementation of a DG or FR method nevertheless requires a basis to be chosen. Two families of such bases, both of which are used for the computations in Sect. 5, are thus described in this appendix.

1.2 A.1 Modal ($L^2$-Orthogonal) Basis

A basis $\mathcal {{B}}_0 {:}{=}\{\phi _0^{(1)}, \ldots ,\phi _0^{(N^*)} \}$ is said to be orthogonal with respect to the $L^2$ inner product on the reference element if any two basis functions $\phi _0^{(i)}, \phi _0^{(j)}\in \mathcal {{B}}$ satisfy

$$\begin{aligned} \int _{{\hat{\varOmega }}}\phi _0^{(i)}(\varvec{\xi })\phi _0^{(j)}(\varvec{\xi })\, \text {d} \varvec{\xi } = 0, \quad \forall \, i \ne j. \end{aligned}$$

(A.1)

Analytical expressions for such bases are well known for certain spaces and element types; for example, an orthogonal basis for the space ${\mathbb {P}}_p(\hat{\mathcal {{T}}}^2)$ was introduced by Proriol [59] (see also Koornwinder [60,  Sect. 3.3] and Dubiner [61,  Sect. 5]) as a triangular analogue of the Legendre polynomial system. Defining ${P}_r^{(a,b)} \in {\mathbb {P}}_r([-1,1])$ as the degree $r \in {\mathbb {N}}_0$ Jacobi polynomial satisfying

$$\begin{aligned} \int _{-1}^1 {P}_q^{(a,b)}(\xi ){P}_r^{(a,b)}(\xi )(1-\xi )^a(1+\xi )^b \, \text {d} \xi = 0, \quad \forall \, q \ne r \end{aligned}$$

(A.2)

the normalized Proriol-Koornwinder-Dubiner (PKD) basis functions are given by

$$\begin{aligned} \phi _0^{(\sigma (\alpha ))}(\varvec{\chi }(\varvec{\eta })) {:}{=}\sqrt{\big (\alpha _1 + \textstyle {\frac{1}{2}}\big )\big (\alpha _1 + \alpha _2 + 1\big )} {P}_{\alpha _1}^{(0,0)}(\eta _1)\Big (\frac{1-\eta _2}{2}\Big )^{\alpha _1}{P}_{\alpha _2}^{(2\alpha _1+1,0)}(\eta _2), \end{aligned}$$

(A.3)

where $\sigma : \mathcal {{N}} \rightarrow \{1,\ldots , (p+1)(p+2)/2\}$ defines an ordering¹¹ of the multi-index set $\mathcal {{N}} {:}{=}\{\varvec{\alpha } \in {\mathbb {N}}_{0}^2 : \alpha _1 + \alpha _2 \le p \}$, and the mapping $\varvec{\chi } : [-1,1]^2 \rightarrow \hat{\mathcal {{T}}}^2$ is given by

$$\begin{aligned} \varvec{\chi }(\varvec{\eta }) {:}{=}\left[ \begin{array}{c} \frac{1}{2}(1+\eta _1)(1-\eta _2) - 1 \\ \eta _2\end{array}\right] . \end{aligned}$$

(A.4)

Similar bases are described for quadrilateral, hexahedral, tetrahedral, prismatic, and pyramidal elements in [39,  Ch. 3].

1.3 A.2 Nodal (Lagrange) Basis

While it is possible to directly employ a modal basis to represent the numerical solution such that $\mathcal {{B}} = \mathcal {{B}}_0$, it can often be more computationally efficient to use a Lagrange basis $\mathcal {{B}} {:}{=}\{\phi ^{(1)}, \ldots , \phi ^{(N^*)}\}$ associated with a nodal set $\tilde{\mathcal {{S}}} {:}{=}\{\tilde{\varvec{\xi }}{}^{(1)}, \ldots ,\tilde{\varvec{\xi }}{}^{(N^*)} \} \subset {\hat{\varOmega }}$ such that $\phi ^{(i)}({\tilde{\varvec{\xi }}}{}^{(j)}) = \delta _{ij}$. Although the corresponding basis functions are given explicitly by (2.17) for $d=1$, analytical expressions do not necessarily exist for nodal bases on arbitrary non-tensorial nodal sets in the case of $d \ge 2$. We therefore take a more general approach by constructing a matrix $\tilde{\underline{\underline{V}}{}} \in {\mathbb {R}}^{N^*\times N^*}$ with entries

$$\begin{aligned} {\tilde{V}}_{ij} {:}{=}\phi _0^{(j)}(\tilde{\varvec{\xi }}{}^{(i)}) \end{aligned}$$

(A.5)

for a modal basis $\mathcal {{B}}_0$ as described in Appendix A.1, which is invertible when $\tilde{\mathcal {{S}}}$ is unisolvent for the space ${\mathbb {P}}_{\mathcal {{N}}}({\hat{\varOmega }})$. The Lagrange polynomials constituting the basis $\mathcal {{B}}$ may then be obtained in terms of the modal basis as

$$\begin{aligned} \phi ^{(i)}(\varvec{\xi }) {:}{=}\sum _{j=1}^{N^*}\Big [\tilde{\underline{\underline{V}}{}}^{-\mathrm {T}}\Big ]_{ij}\phi _0^{(j)}(\varvec{\xi }). \end{aligned}$$

(A.6)

The entries of the corresponding derivative matrix satisfying (3.10) are therefore

$$\begin{aligned} D_{ij}^{(m)} = \frac{\partial \phi ^{(j)}}{\partial \xi _m}(\tilde{\varvec{\xi }}{}^{(i)}) = \sum _{k=1}^{N^*}\Big [\tilde{\underline{\underline{V}}{}}^{-\mathrm {T}}\Big ]_{jk}\frac{\partial \phi _0^{(k)}}{\partial \xi _m}(\tilde{\varvec{\xi }}{}^{(i)}), \quad \forall \, i,j \in \{1,\ldots , N^*\}, \end{aligned}$$

(A.7)

defining a nodal SBP operator on $\tilde{\mathcal {{S}}}$ in the sense of [21,  Definition 2.1] in the case of discrete inner products satisfying Assumptions 3.1 and 3.2.

Remark A.1

In the case of $d \ge 2$, unisolvency is not guaranteed for arbitrary nodal sets $\tilde{\mathcal {{S}}}$ containing $N^*$ distinct nodes. Special care is therefore needed to ensure that $\tilde{\underline{\underline{V}}{}}$ is invertible, where we refer to Marchildon and Zingg [62] for the derivation of necessary conditions for obtaining unisolvent symmetrical nodal sets for total-degree polynomial spaces on triangles and tetrahedra. Further details regarding nodal bases for high-order methods may be found, for example, in [38,  Sect. 3.1, Sect. 6.1, Sect. 10.1].

Appendix B Discrete Inner Products

Implementations of the DG and FR methods presented in Sect.  2.5 can often be characterized by the techniques employed for numerical integration or projection, which may be formalized in terms of discrete inner products defined as in (3.1) and (3.2). In this appendix, we situate the standard quadrature-based integration approach commonly employed for DG methods (e.g. those in [4,5,6,7]) as well as the collocation-based approach employed for the nodal DG formulations described in [38] and the FR schemes in [8,9,10,11,12] within the general context of the present framework.

1.1 B.1 Quadrature-Based Approximation

Considering quadrature rules on $\mathcal {{S}}$ and $\mathcal {{S}}^{(\zeta )}$ with positive weights given by $\{\omega ^{(i)}\}_{i=1}^{N}$ and $\{\omega ^{(\zeta ,i)}\}_{i=1}^{N_\zeta }$, respectively, we may define the diagonal weight matrices

$$\begin{aligned} \underline{\underline{W}}{} {:}{=}{\text {diag}}\big (\omega ^{(1)}, \ldots , \omega ^{(N)}\big ) \quad \text {and} \quad \underline{\underline{B}}{}^{(\zeta )} {:}{=}{\text {diag}}\big (\omega ^{(\zeta ,1)}, \ldots , \omega ^{(\zeta ,N_\zeta )}\big ), \end{aligned}$$

(B.1)

such that the double sums in (3.1) and (3.2) reduce to

$$\begin{aligned} \big \langle {u},{v}\big \rangle _{W} {:}{=}\sum _{i=1}^{N}{u}(\varvec{\xi }^{(i)}) {v}(\varvec{\xi }^{(i)})\,\omega ^{(i)} \end{aligned}$$

(B.2)

and

$$\begin{aligned} \big \langle {u},{v}\big \rangle _{B^{(\zeta )}} {:}{=}\sum _{i=1}^{N_\zeta }{u}(\varvec{\xi }^{(\zeta ,i)}) {v}(\varvec{\xi }^{(\zeta ,i)})\,\omega ^{(\zeta ,i)}, \end{aligned}$$

(B.3)

respectively. The following lemma provides sufficient conditions on such quadrature rules for Assumption 3.1 to be satisfied.

Lemma B.1

Assumption 3.1 is satisfied with ${\mathbb {P}}_{\mathcal {{N}}}({\hat{\varOmega }})={\mathbb {P}}_p({\hat{\varOmega }})$ for any $p \in {\mathbb {N}}_0$, on any polytopal reference element, if the discrete inner products are computed as in (B.2) and (B.3) using quadrature rules of at least total degree $2p-1$ and 2p, respectively.

Proof

For any ${u},{v} \in {\mathbb {P}}_p({\hat{\varOmega }})$, the volume integrals in (3.3) contain functions in ${\mathbb {P}}_{2p-1}({\hat{\varOmega }})$, while the facet integrals contain functions in ${\mathbb {P}}_{2p}({\hat{\varGamma }}^{(\zeta )})$. For volume and facet quadrature rules of total degree $2p-1$ or greater and 2p or greater, respectively, all terms in (3.3) are computed exactly, and hence (3.4) holds for $m \in \{1,\ldots , d\}$. $\square$

Remark B.1

In addition to the positive-definiteness of $\underline{\underline{W}}{}$ and $\underline{\underline{B}}{}^{(\zeta )}$, which is clearly the case if and only if the volume and facet quadrature weights are strictly positive, Assumption 3.2 requires $\underline{\underline{V}}{}$ to be of rank $N^*$, which is difficult to ensure theoretically, particularly in the case of $N > N^*$. We have nevertheless numerically verified that such a property is satisfied for all schemes employed for the computations in Sect. 5.

1.2 B.2 Collocation-Based Approximation

If $\mathcal {{S}}$ is unisolvent for a space ${\mathbb {P}}_{\mathcal {{K}}}({\hat{\varOmega }}) \supseteq {\mathbb {P}}_{\mathcal {{N}}}({\hat{\varOmega }})$ of dimension $N\ge N^*$, we may define a nodal basis $\{ \ell ^{(1)}, \ldots , \ell ^{(N)}\}$ for ${\mathbb {P}}_{\mathcal {{K}}}({\hat{\varOmega }})$ satisfying $\ell ^{(i)}(\varvec{\xi }^{(j)}) = \delta _{ij}$ as in Appendix A.2. The collocation projection ${I}_{\mathcal {{K}}}{u} \in {\mathbb {P}}_{\mathcal {{K}}}({\hat{\varOmega }})$ of a function ${u} : {\hat{\varOmega }} \rightarrow {\mathbb {R}}$ is then given by

$$\begin{aligned} ({I}_{\mathcal {{K}}}{u})(\varvec{\xi }) {:}{=}\sum _{i=1}^{N}{u}(\varvec{\xi }^{(i)})\ell ^{(i)}(\varvec{\xi }), \end{aligned}$$

(B.4)

recovering the projection in (2.13) when ${\mathbb {P}}_{\mathcal {{K}}}({\hat{\varOmega }}) = {\mathbb {P}}_{\mathcal {{N}}}({\hat{\varOmega }})$. Similarly, if $\mathcal {{S}}^{(\zeta )}$ is unisolvent for a polynomial space on ${\hat{\varGamma }}^{(\zeta )} \subset \partial {\hat{\varOmega }}$ which is of dimension $N_\zeta \ge N_\zeta ^*$ and contains ${\mathbb {P}}_{\mathcal {{N}}}({\hat{\varGamma }}^{(\zeta )})$, we may define a projection operator analogously to (B.4) as

$$\begin{aligned} ({I}^{(\zeta )}{u})(\varvec{\xi }) {:}{=}\sum _{i=1}^{N_\zeta }{u}(\varvec{\xi }^{(\zeta ,i)})\ell ^{(\zeta ,i)}(\varvec{\xi }), \end{aligned}$$

(B.5)

where the nodal basis functions $\{ \ell ^{(\zeta ,1)}, \ldots , \ell ^{(\zeta ,N_\zeta )}\}$ satisfy $\ell ^{(\zeta ,i)}(\varvec{\xi }^{(\zeta ,j)}) = \delta _{ij}$. Integrating the products of such projections, we obtain discrete inner products given by

$$\begin{aligned} \big \langle {u},{v}\big \rangle _{W} {:}{=}\int _{{\hat{\varOmega }}} ({I}_{\mathcal {{K}}}{u})(\varvec{\xi })({I}_{\mathcal {{K}}}{v})(\varvec{\xi }) \, \text {d} \varvec{\xi } \end{aligned}$$

(B.6)

and

$$\begin{aligned} \big \langle {u},{v}\big \rangle _{B^{(\zeta )}} {:}{=}\int _{{\hat{\varGamma }}^{(\zeta )}} ({I}^{(\zeta )}{u})(\varvec{\xi })({I}^{(\zeta )}{v})(\varvec{\xi }) \, \text {d} {\hat{s}}, \end{aligned}$$

(B.7)

which take the forms in (3.1) and (3.2), respectively, with

$$\begin{aligned} W_{ij} {:}{=}\int _{{\hat{\varOmega }}}\ell ^{(i)}(\varvec{\xi })\ell ^{(j)}(\varvec{\xi })\, \text {d} \varvec{\xi } \quad \text {and} \quad B_{ij}^{(\zeta )} {:}{=}\int _{{\hat{\varGamma }}^{(\zeta )}}\ell ^{(\zeta ,i)}(\varvec{\xi })\ell ^{(\zeta ,j)}(\varvec{\xi })\, \text {d} {\hat{s}}, \end{aligned}$$

(B.8)

where the above integrals may be evaluated as part of a preprocessing stage.

Remark B.2

Noting that the matrices $\underline{\underline{W}}{}$ and $\underline{\underline{B}}{}^{(\zeta )}$ defined in (B.8) are generally dense, and are SPD due to the linear independence of the nodal bases used to define the projection operators (see, for example, [44,  Theorem 7.2.10]), Assumptions 3.1 and 3.2 (as well as Assumption 3.3 for meshes satisfying Assumption 2.1, at least in the case of ${\mathbb {P}}_{\mathcal {{K}}}({\hat{\varOmega }}) = {\mathbb {P}}_{\mathcal {{N}}}({\hat{\varOmega }})$ considered for the computations in Sect. 5) are therefore satisfied by construction. Such a quadrature-free formulation enables the construction of conservative and energy-stable schemes on arbitrary unisolvent nodal sets for which the quadrature accuracy requirements of Lemma B.1 may not be met.

Remark B.3

Recalling (4.15), the weights of the interpolatory quadrature rule on the abscissae $\mathcal {{S}}$ under which conservation may be proven for collocation-based approximations may be expressed in terms of the Lagrange basis functions as

$$\begin{aligned} \omega ^{(i)} = \sum _{j=1}^{N} W_{ij} = \int _{{\hat{\varOmega }}}\ell ^{(i)}(\varvec{\xi })\, \text {d} \varvec{\xi }, \quad \forall \, i \in \{1,\ldots , N\}, \end{aligned}$$

(B.9)

where such a quadrature is exact (at least) for all functions in ${\mathbb {P}}_{\mathcal {{K}}}({\hat{\varOmega }})$. If, however, the nodes in $\mathcal {{S}}$ are chosen such that the quadrature rule with weights given as in (B.9) is exact for all products of two functions in ${\mathbb {P}}_{\mathcal {{K}}}({\hat{\varOmega }})$,¹² it can be shown (see, for example, [37]) that $\underline{\underline{W}}{}$ reduces to a diagonal matrix of quadrature weights, recovering an approximation identical to that in Appendix B.1. An analogous equivalence holds for the matrices $\underline{\underline{B}}{}^{(\zeta )}$ and the associated facet quadrature rules.

Appendix C Parametrization of the $\underline{\underline{K}}{}$ Matrix

The following lemma establishes suitable choices of multi-index sets $\mathcal {{M}} \subset \mathcal {{N}}$ satisfying the first part of Assumption 3.4 for schemes employing total-degree polynomial spaces (the case of tensor-product elements is discussed by Cicchino and Nadarajah in [63]).

Lemma C.1

Taking ${\mathbb {P}}_{\mathcal {{N}}}({\hat{\varOmega }}) = {\mathbb {P}}_p({\hat{\varOmega }})$ for any $p \in {\mathbb {N}}_0$, the choice of $\mathcal {{M}} {:}{=}\{\varvec{\alpha } \in {\mathbb {N}}_0^d : |\varvec{\alpha } |= p\}$ in (3.26) ensures that $\underline{\underline{K}}{}\underline{\underline{D}}{}^{(m)} = \underline{\underline{0}}$ is satisfied for all $m \in \{1,\ldots , d\}$.

Proof

Noting from (3.26) that $\underline{\underline{K}}{}\underline{\underline{D}}{}^{(m)} = \underline{\underline{0}}$ is satisfied for a given $m \in \{1,\ldots , d\}$ if $\underline{\underline{D}}{}^{\varvec{\alpha }}\underline{\underline{D}}{}^{(m)} = \underline{\underline{0}}$ for all $\varvec{\alpha } \in \mathcal {{M}}$, it is sufficient to require the image of any ${v} \in {\mathbb {P}}_p({\hat{\varOmega }})$ under the corresponding differential operator $\partial ^{{|{\varvec{\alpha }}|}+1}/\partial \xi _1^{\alpha _1} \cdots \partial \xi _m^{\alpha _m + 1} \cdots \partial \xi _d^{\alpha _d}$ to be the zero function. Since differentiating ${v} \in {\mathbb {P}}_p({\hat{\varOmega }})$ a total of $p+1$ times always yields the zero function, and the action of the above operator consists of differentiating a total of $|\varvec{\alpha } |+ 1$ times, constraining $\mathcal {{M}}$ to satisfy $|\varvec{\alpha }|= p$ leads to the desired result. $\square$

Based on symmetry considerations discussed in [11,  Sect. 5.3] and [12,  Sect. 4.1], the set of multi-indices $\mathcal {{M}}$ defined in Lemma C.1 may be parametrized in terms of $d-1$ scalar indices. The matrix $\underline{\underline{K}}{}$ in (3.26) is then given in two and three space dimensions by

$$\begin{aligned} \underline{\underline{K}}{} {:}{=}\frac{c}{|{\hat{\varOmega }}|}\sum _{q=0}^p\left( \begin{array}{c}p \\ q\end{array} \right) \big (\underline{\underline{D}}{}^{(p-q,q)}\big )^\mathrm {T}\underline{\underline{M}}{}\underline{\underline{D}}{}^{(p-q,q)} \end{aligned}$$

(C.1)

and

$$\begin{aligned} \underline{\underline{K}}{} {:}{=}\frac{c}{|{\hat{\varOmega }}|}\sum _{q_1=0}^p\sum _{q_2=0}^{q_1}\left( \begin{array}{c}p \\ q_1\end{array} \right) \left( \begin{array}{c}q_1 \\ q_2\end{array} \right) \big (\underline{\underline{D}}{}^{(p-q_1,q_1-q_2,q_2)}\big )^\mathrm {T}\underline{\underline{M}}{}\underline{\underline{D}}{}^{(p-q_1,q_1-q_2,q_2)}, \end{aligned}$$

(C.2)

respectively, recovering (2.27) in the one-dimensional case, where $c \in {\mathbb {R}}$ determines the properties of the resulting schemes. With respect to the formalism of Theorem 3.1, the parametrizations defining $\underline{\underline{K}}{}$ in (C.1) and (C.2) correspond to $c_{\varvec{\alpha }} = c\,\left( {\begin{array}{c}p\\ \alpha _2\end{array}}\right)$ and $c_{\varvec{\alpha }} = c\, \left( {\begin{array}{c}p\\ p-\alpha _1\end{array}}\right) \left( {\begin{array}{c}p-\alpha _1\\ \alpha _3\end{array}}\right)$, respectively, and lead to $\underline{\underline{K}}{}$ being SPSD for all $c \ge 0$, which is sufficient under Assumption 3.2 for $\underline{\underline{M}}{}+\underline{\underline{K}}{}$ to be SPD, implying that the second part of Assumption 3.4 is satisfied under such conditions.

Remark C.1

While $c \ge 0$ is typically assumed for simplicial elements with $d \ge 2$, it has been shown for $d=1$ (see, for example, [10,  Sect. 3.3], [13,  Sect. 3.2], and [23,  Sect. 3.6]) that there exists $c_- < 0$ depending only on the polynomial degree p (and on the volume quadrature rule, if (3.24) is not satisfied) such that $\underline{\underline{M}}{}+\underline{\underline{K}}{}$ is SPD for $c_-< c < \infty$.