Stability Analysis and Error Estimate of the Explicit Single-Step Time-Marching Discontinuous Galerkin Methods with Stage-Dependent Numerical Flux Parameters for a Linear Hyperbolic Equation in One Dimension

Abstract

In this paper, we present the $\hbox {L}^2$-norm stability analysis and error estimate for the explicit single-step time-marching discontinuous Galerkin (DG) methods with stage-dependent numerical flux parameters, when solving a linear constant-coefficient hyperbolic equation in one dimension. Two well-known examples of this method include the Runge–Kutta DG method with the downwind treatment for the negative time marching coefficients, as well as the Lax–Wendroff DG method with arbitrary numerical flux parameters to deal with the auxiliary variables. The stability analysis framework is an extension and an application of the matrix transferring process based on the temporal differences of stage solutions, and a new concept, named as the averaged numerical flux parameter, is proposed to reveal the essential upwind mechanism in the fully discrete status. Distinguished from the traditional analysis, we have to present a novel way to obtain the optimal error estimate in both space and time. The main tool is a series of space–time approximation functions for a given spatial function, which preserve the local structure of the fully discrete schemes and the balance of exact evolution under the control of the partial differential equation. Finally some numerical experiments are given to validate the theoretical results proposed in this paper.

Appendix

In this section we give some supplemental materials for those conclusions unproved in Sect. 3. This process involves many notations and manipulations of matrices.

To do that, we give some elemental notations here. Associated with the multistep number m and the stage number s, we introduce some column vectors and square matrices of size ms, whose component is either 0 or 1. More specifically, we denote $\varvec{1}(m,s)=(1,1,\ldots ,1)^\top$ and let $\varvec{e}_i(m,s)$, for $0\le i\le ms-1$, be the unit vector which has 1 at the i-th position. Let $\underline{\varvec{I}}(m,s)$ be the identity matrix and $\underline{\varvec{E}}(m,s)$ be the shifting matrix which has 1 at the lower second diagonal line. Then we define

$$\begin{aligned} \underline{\varvec{L}}(m,s)=\Big [\underline{\varvec{I}}(m,s)-\underline{\varvec{E}}(m,s)\Big ]^{-1}-\underline{\varvec{I}}(m,s)= \sum _{1\le \kappa \le ms-1}\underline{\varvec{E}}(m,s)^{\kappa }, \end{aligned}$$

(7.1)

which has 1 at the strictly lower region. For simplicity of notations, we would like to denote, for example

$$\begin{aligned} \varvec{1}(m)=\varvec{1}(m,s), \quad \varvec{1}=\varvec{1}(1,s), \quad \hat{\varvec{1}}=\varvec{1}(m,1). \end{aligned}$$

This notation rule will be used throughout the entire section.

1.1 Matrix Description of the Ultimate Spatial Matrix

In this subsection we devote to presenting a matrix description of how to get the ultimate spatial matrix. To do that, we define the ms order matrices

$$\begin{aligned} \underline{\varvec{C}}(m) = \{c_{ij}(m)\}, \quad \underline{\varvec{D}}(m) = \{d_{ij}(m)\}, \quad \underline{\varvec{W}}(m;\vartheta )=\{d_{ij}(m)(\theta _{ij}(m)-\vartheta )\}, \end{aligned}$$

(7.2a)

related to the description of the ESTDG method, and

$$\begin{aligned} \underline{\varvec{\varSigma }}(m)=\{\sigma _{ij}(m)\}, \quad \underline{\varvec{\varPhi }}(m)=\{\phi _{ij}(m)\}, \quad \underline{\varvec{Q}}(m;\vartheta )=\{q_{ij}(m;\vartheta )\}, \end{aligned}$$

(7.2b)

related to the definition of temporal differences of stage solutions. Here all indices i and j are taken from 0 to $ms-1$, and $\vartheta$ is the parameter as mentioned in subsection 3.1.1. We would like to remark that all data in the above matrices are set to be zero, if they are not clearly stated or defined.

1.1.1 Elemental Formula

The ultimate spatial matrix is obtained by running Algorithm 1 for $\ell =1,2,\ldots ,\zeta$, where the crucial calculation is the increment accumulation in Step 2.

To do that, we define a lower triangle matrix $\underline{\varvec{A}}^\star (m)=\{a_{ij}^\star (m)\}_{0\le i,j\le ms-1}$, whose entries are all defined to be zero except

$$\begin{aligned} a_{ij}^\star (m)=(1-\delta _{ij}/2)a_{i+1,j}^{(j)}(m), \quad \text{ for }\quad j\le i\le ms-1\quad \text{ and }\quad 0\le j\le \zeta -1. \end{aligned}$$

Since $\{\tilde{q}_{ij}(m)\}_{0\le i,j\le ms-1}$ is a lower triangle matrix, all summation ranges in Step 2 can be enlarged to $\{0,1,\ldots ,ms-1\}$. Gathering up the related operation till the matrix transferring process stops, we can obtain a unified description for the increment procedure at any fixed position. More specifically, the integrated calculation at every $(i',j')$ position reads (drop** (m) here for convenience)

$$\begin{aligned} g_{i'j'}\leftarrow g_{i'j'} -a_{i'j'}^\star ; \quad g_{i'j'}\leftarrow g_{i'j'} + a_{\kappa ' j'}^\star {\tilde{q}}_{\kappa ' i'}, \quad g_{i'j'}\leftarrow g_{i'j'} + a_{i'\kappa '}^\star \tilde{q}_{\kappa ' j'}, \end{aligned}$$

where $i',j'$ and $\kappa '$ go through $\{0,1,\ldots ,ms-1\}$. As a result, the total increment at Step 2 of Algorithm 1 can be expressed in the matrix form

$$\begin{aligned} \underline{\varvec{G}}(m) = (2\vartheta -1)\underline{\varvec{A}}^\star (m) +\underline{\varvec{Q}}(m;\vartheta )^\top \underline{\varvec{A}}^\star (m) +\underline{\varvec{A}}^\star (m)\underline{\varvec{Q}}(m;\vartheta ), \end{aligned}$$

where the definition (3.16), i.e., $\tilde{q}_{ij}(m)=q_{ij}(m;\vartheta )+\vartheta \delta _{ij}$ is used.

From Step 3 of Algorithm 1, we have the ultimate spatial matrix (below the last row and column is dropped, since they are always zero)

$$\begin{aligned} \begin{aligned} \mathbb {B}(m)=&\; \underline{\varvec{G}}(m)+\underline{\varvec{G}}(m)^\top \\ =&\; \Big (\vartheta -\frac{1}{2}\Big )\underline{\varvec{B}}^{\star }(m) +\frac{1}{2}\Big [\underline{\varvec{B}}^{\star }(m)\underline{\varvec{Q}}(m;\vartheta )+ \underline{\varvec{Q}}(m;\vartheta )^{\top }\underline{\varvec{B}}^{\star }(m)\Big ], \end{aligned} \end{aligned}$$

(7.3)

with the symmetric matrix

$$\begin{aligned} \underline{\varvec{B}}^{\star }(m) =2\underline{\varvec{A}}^\star (m)+2\underline{\varvec{A}}^\star (m)^\top =\{b_{ij}^{\star }(m)\}_{0\le i,j\le ms-1}. \end{aligned}$$

(7.4)

The entry at the lower triangular zone is defined as

$$\begin{aligned} b_{ij}^{\star }(m) = {\left\{ \begin{array}{ll} 2a_{i+1,j}^{(j)}(m),&{}0\le j\le \zeta -1 \text{ and } j\le i\le ms-1, \\ 0,&{}\text{ otherwise }, \end{array}\right. } \end{aligned}$$

(7.5)

as the same as in the ultimate spatial matrix in [26] for the RKDG methods with a fixed numerical flux parameter.

To investigate the property of the second term in (7.3), we just need to study the perturbation matrix

$$\begin{aligned} \underline{\varvec{Z}}(m;\vartheta )=\underline{\varvec{B}}^{\star }(m)\underline{\varvec{Q}}(m;\vartheta )= \{z_{ij}(m;\vartheta )\}_{0\le i,j\le ms-1}. \end{aligned}$$

(7.6)

Taking into account on definition of the contribution index, we only pay attention on those left-top entries in (7.6). In what follows we try to deduce a convenient and unified formula for

$$\begin{aligned} z_{ij}(m;\vartheta ) = \sum _{0\le \ell \le ms-1} b_{i\ell }^{\star }(m)q_{\ell ,j}(m;\vartheta ), \quad 0\le i,j\le \zeta -1. \end{aligned}$$

(7.7)

The formula of every $b_{i\ell }^{\star }(m)$ has been given in [26], but is variant according to the size relationship of i and $\ell$. In this paper we have to rebuild an equivalent and unified formula, as stated in the next lemma.

Lemma 7.1

For $0\le i\le \zeta -1$, there holds

$$\begin{aligned} b_{i\ell }^{\star }(m)= 2\sum _{0\le \kappa \le i}(-1)^\kappa \alpha _{i-\kappa }(m)\alpha _{\ell +1+\kappa }(m), \quad 0\le \ell \le ms-1. \end{aligned}$$

(7.8)

Here and below we define $\alpha _{i'}(m)=0$ if $i'>ms$ for simplicity.

We put aside the proof of this lemma in Sect. 7.1.3. Substituting (7.8) into (7.7) deduces for any $0\le i,j\le \zeta -1$ that

$$\begin{aligned} z_{ij}(m;\vartheta ) = \sum _{0\le \kappa \le i} 2(-1)^{\kappa }\alpha _{i-\kappa }(m) \underbrace{\sum _{0\le \ell \le ms-1} \alpha _{\ell +1+\kappa }(m)q_{\ell ,j}(m;\vartheta )}_{\pi _{\kappa ,j}(m;\vartheta )}. \end{aligned}$$

(7.9)

In what follows we would like to set up a useful formula of $\pi _{\kappa ,j}(m;\vartheta )$ by those data to define the ESTDG method.

1.1.2 Formula of $\pi _{\kappa ,j}(m;\vartheta )$

Due to (3.13) and (3.8), we can respectively obtain

$$\begin{aligned} \underline{\varvec{Q}}(m;\vartheta )\underline{\varvec{\varSigma }}(m)=\underline{\varvec{\varPhi }}(m)\underline{\varvec{W}}(m;\vartheta ), \quad \underline{\varvec{\varPhi }}(m)\underline{\varvec{D}}(m)=\underline{\varvec{\varSigma }}(m). \end{aligned}$$

(7.10)

This implies $\underline{\varvec{Q}}(m;\vartheta ) =\underline{\varvec{\varSigma }}(m)\underline{\varvec{D}}(m)^{-1}\underline{\varvec{W}}(m;\vartheta )\underline{\varvec{\varSigma }}(m)^{-1}$. With the short notation

$$\begin{aligned} \varvec{y}^{\top }(m)= \sum _{0\le \ell \le ms-1} \alpha _{\ell +1+\kappa }(m)\varvec{e}_{\ell }^{\top }(m) \underline{\varvec{\varSigma }}(m), \end{aligned}$$

(7.11)

it follows from (7.9) and $q_{\ell ,j}(m;\vartheta )=\varvec{e}_{\ell }^{\top }(m)\underline{\varvec{Q}}(m;\vartheta )\varvec{e}_j(m)$ that

$$\begin{aligned} \pi _{\kappa ,j}(m;\vartheta ) = \varvec{y}^{\top }(m) \cdot \Big [\underline{\varvec{D}}^{-1}(m)\underline{\varvec{W}}(m;\vartheta )\Big ]\cdot \Big [\underline{\varvec{\varSigma }}(m)^{-1}\varvec{e}_j(m)\Big ]. \end{aligned}$$

(7.12)

Below we are going to express three terms in (7.12). To that end, we start this work from the calculation of $\underline{\varvec{\varSigma }}(m)^{-1}$.

By denoting (here and below we omit (m) for the matrix entry)

$$\begin{aligned} \underline{\varvec{S}}(m)=\underline{\varvec{I}}(m)-\underline{\varvec{C}}(m)\underline{\varvec{E}}(m) = \begin{pmatrix} 1\\ -c_{11}&{}1\\ -c_{21}&{}-c_{22}&{}1\\ \vdots &{}\vdots &{}&{}\ddots \\ -c_{ms-1,1}&{}-c_{ms-1,2}&{}\cdots &{}-c_{ms-1,ms-1}&{}1 \end{pmatrix}, \end{aligned}$$

the definition procedure of the temporal differences of stage solutions can be written into the matrix form

Recalling the definition of the evolution identity, the matrix inversion on both sides of the above identity yields

where we have used (7.10) to get $\underline{\varvec{\varPhi }}(m)^{-1}=\underline{\varvec{D}}(m)\underline{\varvec{\varSigma }}(m)^{-1}$. Comparing with the matrices entries on both sides, we can achieve the following equalities for every column in the matrix $\underline{\varvec{\varSigma }}(m)^{-1}$,

$$\begin{aligned} \underline{\varvec{\varSigma }}(m)^{-1}\varvec{e}_0(m) =&\; [\underline{\varvec{I}}(m)+\underline{\varvec{E}}(m)\underline{\varvec{S}}(m)^{-1}\underline{\varvec{C}}(m)]\varvec{e}_0(m) \overset{{\tiny \text{ def }}}{=}\varvec{q}(m), \end{aligned}$$

(7.13a)

$$\begin{aligned} \underline{\varvec{\varSigma }}(m)^{-1}\varvec{e}_j(m) =&\; \underbrace{\underline{\varvec{E}}(m)\underline{\varvec{S}}(m)^{-1}\underline{\varvec{D}}(m)}_{\underline{\varvec{K}}(m)} \underline{\varvec{\varSigma }}(m)^{-1}\varvec{e}_{j-1}(m), \quad j\ge 1, \end{aligned}$$

(7.13b)

and for every evolution coefficient in (3.11),

$$\begin{aligned} \alpha _0(m) =&\; \varvec{e}_{ms-1}(m)^{\top }\underline{\varvec{S}}(m)^{-1}\underline{\varvec{C}}(m)\varvec{e}_0(m), \end{aligned}$$

(7.14a)

$$\begin{aligned} \alpha _j(m) =&\; \underbrace{\varvec{e}_{ms-1}(m)^{\top }\underline{\varvec{S}}(m)^{-1}\underline{\varvec{D}}(m)}_{\varvec{p}^{\top }(m)} \underline{\varvec{\varSigma }}(m)^{-1}\varvec{e}_{j-1}(m), \quad j\ge 1. \end{aligned}$$

(7.14b)

Then, an induction process for (7.13) yields that

$$\begin{aligned} \underline{\varvec{\varSigma }}(m)^{-1}\varvec{e}_j(m)=\underline{\varvec{K}}(m)^j\varvec{q}(m), \quad j\ge 0, \end{aligned}$$

(7.15)

and the matrix identity

$$\begin{aligned} \underline{\varvec{\varSigma }}(m)^{-1}\underline{\varvec{E}}(m) = \underline{\varvec{K}}(m)\underline{\varvec{\varSigma }}(m)^{-1}. \end{aligned}$$

(7.16)

For any $\kappa \ge 0$, substituting (7.14b) into (7.11) yields

$$\begin{aligned} \begin{aligned} \varvec{y}^{\top }(m) =&\; \varvec{p}(m)^{\top }\underline{\varvec{\varSigma }}(m)^{-1} \left[ \sum _{0\le \ell \le ms-1} \varvec{e}_{\ell +\kappa }(m)\varvec{e}_{\ell }(m)^{\top }\right] \underline{\varvec{\varSigma }}(m)\\ =&\; \varvec{p}(m)^{\top }\underline{\varvec{\varSigma }}(m)^{-1}\underline{\varvec{E}}(m)^{\kappa }\underline{\varvec{\varSigma }}(m)\\ =&\; \varvec{p}(m)^{\top }[\underline{\varvec{\varSigma }}(m)^{-1}\underline{\varvec{E}}(m)\underline{\varvec{\varSigma }}(m)]^{\kappa } = \varvec{p}(m)^{\top }\underline{\varvec{K}}(m)^{\kappa }, \end{aligned} \end{aligned}$$

(7.17)

where (7.16) is used at the last step. Substituting (7.17) and (7.15) into (7.12), we finally have

$$\begin{aligned} \pi _{\kappa ,j}(m;\vartheta ) = \varvec{p}(m)^{\top }\underline{\varvec{K}}(m)^{\kappa } \underline{\varvec{D}}^{-1}(m)\underline{\varvec{W}}(m;\vartheta ) \underline{\varvec{K}}(m)^j\varvec{q}(m). \end{aligned}$$

(7.18)

In order to investigate the relationship between this quantity and the multistep number, we would like to make some (right) Kronecker product of matrices [25] to simplify each term in (7.18). For example, we will use

$$\begin{aligned} \varvec{e}_0(m)=\hat{\varvec{e}}_0\otimes \varvec{e}_0, \quad \varvec{e}_{ms-1}(m)^{\top } = \hat{\varvec{e}}_{m-1}^\top \otimes \varvec{e}_{s-1}^{\top }, \quad \underline{\varvec{I}}(m) = \hat{\underline{\varvec{I}}}\otimes \underline{\varvec{I}}, \end{aligned}$$

(7.19)

which implies

$$\begin{aligned} \underline{\varvec{E}}(m) = \hat{\underline{\varvec{I}}}\otimes \underline{\varvec{E}}+\hat{\underline{\varvec{E}}}\otimes \varvec{e}_0\varvec{e}_{s-1}^{\top }. \end{aligned}$$

(7.20)

Due to the definition (3.3), we derive

$$\begin{aligned} \underline{\varvec{C}}(m) =\hat{\underline{\varvec{I}}}\otimes \underline{\varvec{C}}, \quad \underline{\varvec{D}}(m) =\frac{1}{m}\hat{\underline{\varvec{I}}}\otimes \underline{\varvec{D}}, \quad \underline{\varvec{W}}(m;\vartheta ) =\frac{1}{m}\hat{\underline{\varvec{I}}}\otimes \underline{\varvec{W}}(\vartheta ), \end{aligned}$$

(7.21)

where $\underline{\varvec{W}}(\vartheta )=\underline{\varvec{W}}(1;\vartheta )$. Based on these identities, by some lengthy and tedious matrices manipulations, we can get the following important conclusions

$$\begin{aligned} \underline{\varvec{S}}(m)^{-1} =&\; \hat{\underline{\varvec{L}}}\otimes \underline{\varvec{S}}^{-1}\underline{\varvec{C}}\varvec{e}_0\varvec{e}_{s-1}^{\top } \underline{\varvec{S}}^{-1}+ \hat{\underline{\varvec{I}}}\otimes \underline{\varvec{S}}^{-1}, \end{aligned}$$

(7.22a)

$$\begin{aligned} \underline{\varvec{K}}(m) =&\; \frac{1}{m}\Big [\hat{\underline{\varvec{L}}}\otimes \varvec{q}\varvec{p}^{\top }+ \hat{\underline{\varvec{I}}}\otimes \underline{\varvec{E}}\,\underline{\varvec{S}}^{-1}\underline{\varvec{D}}\Big ], \end{aligned}$$

(7.22b)

$$\begin{aligned} \varvec{p}(m)^{\top } =&\; \frac{1}{m}\hat{\varvec{1}}^\top \otimes \varvec{p}^{\top }, \end{aligned}$$

(7.22c)

$$\begin{aligned} \varvec{q}(m) =&\; \hat{\varvec{1}}\otimes \varvec{q}. \end{aligned}$$

(7.22d)

In this process, we have used the following simple conclusions

$$\begin{aligned} \hat{\underline{\varvec{E}}}+\hat{\underline{\varvec{E}}}\,\hat{\underline{\varvec{L}}}=\hat{\underline{\varvec{L}}}, \quad \hat{\varvec{e}}_{m-1}^{\top }+\hat{\varvec{e}}_{m-1}^{\top }\hat{\underline{\varvec{L}}}= \hat{\varvec{1}}^{\top }, \quad \hat{\varvec{e}}_0+\hat{\underline{\varvec{L}}}\hat{\varvec{e}}_0=\hat{\varvec{1}}, \end{aligned}$$

(7.23)

and an important identity as a corollary of (7.14a) and $\alpha _0(m)=1$,

$$\begin{aligned} \varvec{e}_{ms-1}^{\top }(m)\underline{\varvec{S}}(m)^{-1}\underline{\varvec{C}}(m)\varvec{e}_0(m)=1. \end{aligned}$$

(7.24)

For ease of reading, we present the verifications of (7.22) in Sect. 7.1.4.

With the help of (7.21), substituting (7.22c) and (7.22d) into (7.18) yields the final simplification expression

$$\begin{aligned} \pi _{\kappa ,j}(m;\vartheta ) =\frac{1}{m} \Big (\hat{\varvec{1}}^{\top }\otimes \varvec{p}^{\top }\Big ) \underline{\varvec{K}}(m)^\kappa \Big (\hat{\underline{\varvec{I}}}\otimes \underline{\varvec{D}}^{-1}\underline{\varvec{W}}(\vartheta )\Big ) \underline{\varvec{K}}(m)^j \Big (\varvec{1}\otimes \varvec{q}\Big ). \end{aligned}$$

(7.25)

If needed, we can use (7.22b) to further deal with $\underline{\varvec{K}}(m)$.

1.1.3 Proof of Lemma 7.1

To end this subsection, we need to prove the skipped Lemma 7.1. Since all related manipulation does not depend on the spatial discretization, the results given in [26, Lemma 3.1] still hold. Hence, for $0\le j'\le \zeta$ and $j'< i'\le ms$ we have

$$\begin{aligned} a_{i'j'}^{(j')}(m)= \sum _{0\le \kappa \le j'} (-1)^\kappa \alpha _{i'+\kappa }(m) \alpha _{j'-\kappa }(m), \end{aligned}$$

(7.26a)

and for $1\le i'\le \zeta$ we have

$$\begin{aligned} a_{i'i'}^{(i')}(m)= \sum _{-i'\le \kappa \le i'} (-1)^\kappa \alpha _{i'+\kappa }(m)\alpha _{i'-\kappa }(m). \end{aligned}$$

(7.26b)

Based on the formulas in (7.26), we can prove this lemma by simple discussions for different cases of $\ell$.

If $\ell >i$, since $\mathbb {B}^{\star }(m)$ is symmetric, it follows from (7.5) that

$$\begin{aligned} b_{i\ell }^{\star }(m) =b_{\ell i}^{\star }(m) =2a_{\ell +1,i}^{(i)}(m). \end{aligned}$$

This proves (7.8) by using (7.26a) with $i'=\ell +1$ and $j'=i$.

Otherwise, if $\ell \le i$, we similarly have from (7.26a) that

$$\begin{aligned} b_{i\ell }^{\star }(m)=2a_{i+1,\ell }^{(\ell )}(m) = 2\sum _{0\le \kappa \le \ell } (-1)^\kappa \alpha _{i+1+\kappa }(m)\alpha _{\ell -\kappa }(m). \end{aligned}$$

To show it can be written in (7.8), we just need to show $\varUpsilon =0$, with

$$\begin{aligned} \begin{aligned} \varUpsilon \overset{{\tiny \text{ def }}}{=}&\; \sum _{0\le \kappa \le \ell } (-1)^\kappa \alpha _{i+1+\kappa }(m)\alpha _{\ell -\kappa }(m) - \sum _{0\le \kappa \le i} (-1)^\kappa \alpha _{i-\kappa }(m)\alpha _{\ell +1+\kappa }(m)\\ =&\; \sum _{0\le \kappa \le \ell +i+1} (-1)^{\ell -\kappa } \alpha _{\kappa }(m)\alpha _{\ell +i+1-\kappa }(m). \end{aligned} \end{aligned}$$

Here we have respectively used the replacements of index $\kappa '=\ell -\kappa$ and $\kappa '=\ell +1+\kappa$ in two summations of the first equality. The verification is easy as follows.

• If $\ell +i+1$ is odd, the replacement $\kappa '=i+\ell +1-\kappa$ implies $\varUpsilon =(-1)^{i+\ell +1}\varUpsilon$ and hence $\varUpsilon =0$.

• Otherwise, if $\ell +i+1$ is even, denoted by 2L, a simple replacement of summation index again reduces

  $$\begin{aligned} (-1)^{\ell -L}\varUpsilon = \sum _{-L\le \kappa \le L} (-1)^\kappa \alpha _{L+\kappa }(m)\alpha _{L-\kappa }(m) =a_{L,L}^{(L)}(m), \end{aligned}$$

  where the last step uses (7.26b). Since $L<\zeta$, it follows $a_{L,L}^{(L)}(m)=0$ from the definition of $\zeta$. This implies $\varUpsilon =0$ also.

Till now we sum up the above conclusions and complete the proof of this lemma.

1.1.4 Verifications of (7.22)

To verify the first identity (7.22a), we start from the definition of $\underline{\varvec{S}}(m)$. Substituting the identities (7.19), (7.21) and (7.20), we have

$$\begin{aligned} \underline{\varvec{S}}(m) = \underline{\varvec{I}}(m) - \underline{\varvec{C}}(m)\underline{\varvec{E}}(m) = \hat{\underline{\varvec{I}}}\otimes \underline{\varvec{I}}- (\hat{\underline{\varvec{I}}}\otimes \underline{\varvec{C}}) (\hat{\underline{\varvec{I}}}\otimes \underline{\varvec{E}}+\hat{\underline{\varvec{E}}}\otimes \varvec{e}_0\varvec{e}_{s-1}^\top ). \end{aligned}$$

Expanding the right-hand side and using the definition of $\underline{\varvec{S}}$, after some manipulations we have

$$\begin{aligned} \begin{aligned} \underline{\varvec{S}}(m) =&\; \hat{\underline{\varvec{I}}}\otimes (\underline{\varvec{I}}-\underline{\varvec{C}}\,\underline{\varvec{E}}) - \hat{\underline{\varvec{E}}}\otimes \underline{\varvec{C}}\varvec{e}_0\varvec{e}_{s-1}^\top \\ =&\; \hat{\underline{\varvec{I}}}\otimes \underline{\varvec{S}}- \hat{\underline{\varvec{E}}}\otimes \underline{\varvec{C}}\varvec{e}_0\varvec{e}_{s-1}^\top = ( \hat{\underline{\varvec{I}}}\otimes \underline{\varvec{I}}- \hat{\underline{\varvec{E}}}\otimes \underline{\varvec{C}}\varvec{e}_0\varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1} ) (\hat{\underline{\varvec{I}}}\otimes \underline{\varvec{S}}). \end{aligned} \end{aligned}$$

Since $(\hat{\underline{\varvec{E}}})^m$ is a zero matrix, the inverse of the first matrix is expressed by

$$\begin{aligned} \hat{\underline{\varvec{I}}}\otimes \underline{\varvec{I}}+ \sum _{1\le i\le m-1} (\hat{\underline{\varvec{E}}})^i \otimes (\underline{\varvec{C}}\varvec{e}_0\varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1})^i. \end{aligned}$$

Using (7.24), we have for any $i\ge 1$ that

$$\begin{aligned} (\underline{\varvec{C}}\varvec{e}_0\varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1})^i = \underline{\varvec{C}}\varvec{e}_0 ( \varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1} \underline{\varvec{C}}\varvec{e}_0 )^{i-1} \varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1} =\underline{\varvec{C}}\varvec{e}_0\varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1}. \end{aligned}$$

Summing up the above identities, we have

$$\begin{aligned} \begin{aligned} \underline{\varvec{S}}(m)^{-1} =&\; (\hat{\underline{\varvec{I}}}\otimes \underline{\varvec{S}}^{-1}) \Big [ \hat{\underline{\varvec{I}}}\otimes \underline{\varvec{I}}+ \sum _{1\le i\le m-1} (\hat{\underline{\varvec{E}}})^i \otimes \underline{\varvec{C}}\varvec{e}_0\varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1} \Big ]\\ =&\; (\hat{\underline{\varvec{I}}}\otimes \underline{\varvec{S}}^{-1}) \Big ( \hat{\underline{\varvec{I}}}\otimes \underline{\varvec{I}}+\hat{\underline{\varvec{L}}} \otimes \underline{\varvec{C}}\varvec{e}_0\varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1} \Big )\\ =&\; \hat{\underline{\varvec{I}}}\otimes \underline{\varvec{S}}^{-1} +\hat{\underline{\varvec{L}}} \otimes \underline{\varvec{S}}^{-1}\underline{\varvec{C}}\varvec{e}_0\varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1}, \end{aligned} \end{aligned}$$

where we have used the definition (7.1) of $\hat{\underline{\varvec{L}}}$ at the second step. This completes the verification of (7.22a).

We start the verification of (7.22b) from the definition (7.13b) of $\underline{\varvec{K}}(m)$. Substituting the identities (7.20), (7.22a) and (7.21), we have

$$\begin{aligned} \begin{aligned}&\; m\underline{\varvec{K}}(m) = m\underline{\varvec{E}}(m)\underline{\varvec{S}}(m)^{-1}\underline{\varvec{D}}(m)\\&\quad =\;(\hat{\underline{\varvec{I}}}\otimes \underline{\varvec{E}}+\hat{\underline{\varvec{E}}}\otimes \varvec{e}_0\varvec{e}_{s-1}^\top ) ( \hat{\underline{\varvec{I}}}\otimes \underline{\varvec{S}}^{-1} + \hat{\underline{\varvec{L}}}\otimes \underline{\varvec{S}}^{-1}\underline{\varvec{C}}\varvec{e}_0\varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1} ) (\hat{\underline{\varvec{I}}}\otimes \underline{\varvec{D}}). \end{aligned} \end{aligned}$$

Expanding the right-hand side, using (7.24) and the first identity in (7.23), we achieve

$$\begin{aligned} \begin{aligned} m\underline{\varvec{K}}(m) =&\; \hat{\underline{\varvec{I}}}\otimes \underline{\varvec{E}}\,\underline{\varvec{S}}^{-1}\underline{\varvec{D}}+ \hat{\underline{\varvec{L}}}\otimes \underline{\varvec{E}}\,\underline{\varvec{S}}^{-1}\underline{\varvec{C}}\varvec{e}_0\varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1}\underline{\varvec{D}}+ \hat{\underline{\varvec{L}}}\otimes \varvec{e}_0\varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1}\underline{\varvec{D}}\\ =&\; \hat{\underline{\varvec{I}}}\otimes \underline{\varvec{E}}\,\underline{\varvec{S}}^{-1}\underline{\varvec{D}}+ \hat{\underline{\varvec{L}}}\otimes (\underline{\varvec{I}}+\underline{\varvec{E}}\,\underline{\varvec{S}}^{-1}\underline{\varvec{C}}) \varvec{e}_0\varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1}\underline{\varvec{D}}\\ =&\; \hat{\underline{\varvec{I}}}\otimes \underline{\varvec{E}}\,\underline{\varvec{S}}^{-1}\underline{\varvec{D}}+ \hat{\underline{\varvec{L}}}\otimes \varvec{q}\varvec{p}^\top , \end{aligned} \end{aligned}$$

where at the last step we have used the definitions of $\varvec{q}$ and $\varvec{p}^\top$ in (7.13a) and (7.14b). This completes the verification of (7.22b).

The third identity (7.22c) is verified along the same line. Starting from the definition of $\varvec{p}(m)^{\top }$ in (7.14b), and substituting the identities (7.19), (7.22a) and (7.21), we have

$$\begin{aligned} \begin{aligned} m\varvec{p}(m)^{\top } =&\; m\varvec{e}_{ms-1}(m)^{\top }\underline{\varvec{S}}(m)^{-1}\underline{\varvec{D}}(m)\\ =&\; (\hat{\varvec{e}}_{m-1}^\top \otimes \varvec{e}_{s-1}^\top ) ( \hat{\underline{\varvec{I}}}\otimes \underline{\varvec{S}}^{-1} + \hat{\underline{\varvec{L}}}\otimes \underline{\varvec{S}}^{-1}\underline{\varvec{C}}\varvec{e}_0\varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1} ) (\hat{\underline{\varvec{I}}}\otimes \underline{\varvec{D}}). \end{aligned} \end{aligned}$$

Expanding the above expression and using (7.24), we have

$$\begin{aligned} \begin{aligned} m\varvec{p}(m)^{\top } =&\; \hat{\varvec{e}}_{m-1}^\top \otimes \varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1}\underline{\varvec{D}}+ \hat{\varvec{e}}_{m-1}^\top \hat{\underline{\varvec{L}}} \otimes \varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1}\underline{\varvec{C}}\varvec{e}_0 \varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1}\underline{\varvec{D}}\\ =&\; \hat{\varvec{e}}_{m-1}^\top \otimes \varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1}\underline{\varvec{D}}+ \hat{\varvec{e}}_{m-1}^\top \hat{\underline{\varvec{L}}} \otimes \varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1}\underline{\varvec{D}}\\ =&\; (\hat{\varvec{e}}_{m-1}^\top +\hat{\varvec{e}}_{m-1}^\top \hat{\underline{\varvec{L}}}) \otimes \varvec{e}_{s-1}^\top \underline{\varvec{S}}^{-1}\underline{\varvec{D}}= \hat{\varvec{1}}^\top \otimes \varvec{p}^\top , \end{aligned} \end{aligned}$$

where at the last step we have used the second identity in (7.23) and the definition of $\varvec{p}^\top$ in (7.14b). This proves (7.22c).

The fourth identity (7.22d) is verified similarly. To save the length of this paper, we omit the detailed procedure.

1.2 Some Proofs

In this subsection we would like to prove Lemmas 3.6 and 3.7, as well as Propositions 3.1 and 3.2.

1.2.1 Proof of Lemma 3.6

Recalling the definition of $\pi _{\kappa ,j}(m;\vartheta )$, given in (7.9), it follows from (3.16) and (3.27) that $\varTheta (m)=\vartheta +\pi _{00}(m;\vartheta )$. Substituting (7.25) implies that

$$\begin{aligned} \varTheta (m) = \vartheta + \frac{1}{m} \Big (\hat{\varvec{1}}^\top \otimes \varvec{p}^{\top }\Big ) \Big (\hat{\underline{\varvec{I}}}\otimes \underline{\varvec{D}}^{-1}\underline{\varvec{W}}(\vartheta )\Big ) \Big (\hat{\varvec{1}}\otimes \varvec{q}\Big ) = \vartheta + \varvec{p}^{\top }\underline{\varvec{D}}^{-1}\underline{\varvec{W}}(\vartheta )\varvec{q}, \end{aligned}$$

(7.27)

where the simple fact $\hat{\varvec{1}}^\top \hat{\underline{\varvec{I}}}\hat{\varvec{1}}=m$ is used. This completes the proof of Lemma 3.6.

Remark 7.1

Taking $m=1$ and $\vartheta =\varTheta$ in (7.27), we use Lemma 3.6 to get

$$\begin{aligned} \varvec{p}^{\top }\underline{\varvec{D}}^{-1}\underline{\varvec{W}}(\varTheta )\varvec{q}=0. \end{aligned}$$

(7.28)

This is just the conclusion in Lemma 3.5 with $m=1$. As an essence property of the averaged numerical flux parameter, it plays an important role in the proof of Lemma 3.7.

1.2.2 Proof of Lemma 3.7

For convenience of notations, in what follows we use a generic notation C to denote a positive constant independent of m. Recalling the proof of [26, Proposition 3.3], we have for $0\le i,j\le \zeta -1$ that

$$\begin{aligned} \left| b_{ij}^{\star }(m)-\frac{2}{i!j!(i+j+1)} \right| \le \frac{C}{m}, \end{aligned}$$

(7.29)

where $\{\frac{2}{i!j!(i+j+1)}\}_{0\le i,j\le \zeta -1}$ forms a symmetric positive definite matrix congruent to an Hilbert matrix. Since $\varTheta >1/2$, it follows from (7.3) and (7.6) with $\vartheta =\varTheta$ that we can prove this lemma by showing that $z_{ij}(m;\varTheta )$ for $0\le i,j\le \zeta -1$ all tends to zero as m goes to infinity. By (7.9), it is sufficient to prove

$$\begin{aligned} \vert \pi _{\kappa ,j}(m;\varTheta )\vert \le \frac{C}{m}, \quad 0\le \kappa , j\le \zeta -1, \end{aligned}$$

(7.30)

since [26, inequality (3.16)] shows that $\alpha _{i-\kappa }(m)$ is bounded independent of m.

Denote $\pi _{\kappa ,j}=\pi _{\kappa ,j}(m;\varTheta )$ and $\underline{\varvec{W}}=\underline{\varvec{W}}(\varTheta )$ for simplicity. Below we are going to prove (7.30) for different cases of $\kappa$ and j, where (7.28) plays an important role to well control the accumulation and growth as m goes to infinity.

• If $\kappa =j=0$, we have $\pi _{0,0}= (\hat{\varvec{1}}^{\top }\hat{\underline{\varvec{I}}}\hat{\varvec{1}})\otimes (\varvec{p}^{\top }\underline{\varvec{D}}^{-1}\underline{\varvec{W}}\varvec{q})=0$, due to (7.28).

• If $\kappa >0$ and $j>0$, we have

  $$\begin{aligned} \pi _{\kappa ,j} = \frac{1}{m} \Big (\hat{\varvec{1}}^{\top }\otimes \varvec{p}^{\top }\Big ) [\underline{\varvec{K}}(m)]^{\kappa -1} \varvec{\varPi }_{\kappa ,j}(m) [\underline{\varvec{K}}(m)]^{j-1} \Big (\hat{\varvec{1}}\otimes \varvec{q}\Big ), \end{aligned}$$

  (7.31)

  where $\varvec{\varPi }_{\kappa ,j}(m) = \underline{\varvec{K}}(m) \Big (\hat{\underline{\varvec{I}}}\otimes \underline{\varvec{D}}^{-1}\underline{\varvec{W}}\Big ) \underline{\varvec{K}}(m)$. Substituting (7.22b) into this formula and then using (7.28) to eliminate the term involving $\hat{\underline{\varvec{L}}}^2$. After some manipulations we yield

  $$\begin{aligned} \begin{aligned} \varvec{\varPi }_{\kappa ,j}(m) =&\; \frac{1}{m^2}\hat{\underline{\varvec{L}}}\otimes [ \varvec{q}\varvec{p}^{\top }\underline{\varvec{D}}^{-1}\underline{\varvec{W}}\underline{\varvec{E}}\underline{\varvec{S}}^{-1}\underline{\varvec{D}}+ \underline{\varvec{E}}\underline{\varvec{S}}^{-1}\underline{\varvec{W}}\varvec{q}\varvec{p}^{\top } ]\\&\; +\frac{1}{m^2}\hat{\underline{\varvec{I}}}\otimes \underline{\varvec{E}}\underline{\varvec{S}}^{-1}\underline{\varvec{W}}\underline{\varvec{E}}\underline{\varvec{S}}^{-1}\underline{\varvec{D}}. \end{aligned} \end{aligned}$$

  The row norms for all matrices (including the row vectors and column vectors) do not depend on m, except that $\Vert \hat{\underline{\varvec{L}}}\Vert _\infty =m-1$. Hence we have

  $$\begin{aligned} \Vert \varvec{\varPi }_{\kappa ,j}(m)\Vert _{\infty }\le \frac{C}{m}. \end{aligned}$$

  Noticing $\Vert \frac{1}{m} (\hat{\varvec{1}}^{\top }\otimes \varvec{p}^{\top })\Vert _{\infty }\le C$ and $\Vert \underline{\varvec{K}}(m)\Vert _{\infty }\le C$, we get from (7.31) what we want to prove.

• If $\kappa =0$ and $j>0$, we have $\pi _{0,j}=\frac{1}{m} \varvec{\varPi }_{0,j}(m)[\underline{\varvec{K}}(m)]^{j-1}(\hat{\varvec{1}}\otimes \varvec{q})$ with

  $$\begin{aligned} \varvec{\varPi }_{0,j}(m)= \Big (\hat{\varvec{1}}^{\top }\otimes \varvec{p}^{\top }\Big ) \Big (\hat{\underline{\varvec{I}}}\otimes \underline{\varvec{D}}^{-1}\underline{\varvec{W}}\Big )\underline{\varvec{K}}(m) = \frac{1}{m}\hat{\varvec{1}}^{\top }\otimes \varvec{p}^{\top } \underline{\varvec{D}}^{-1}\underline{\varvec{W}}\underline{\varvec{E}}\underline{\varvec{S}}^{-1}\underline{\varvec{D}}, \end{aligned}$$

  by some manipulations with the help of (7.22b) and (7.28). The remaining proof follows the same line as above, hence is omitted.

• If $\kappa >0$ and $j=0$, we have $\pi _{\kappa ,0}= \frac{1}{m}(\hat{\varvec{1}}^{\top }\otimes \varvec{p}^{\top }) [\underline{\varvec{K}}(m)]^{\kappa -1}\varvec{\varPi }_{\kappa ,0}(m)$, where

  $$\begin{aligned} \varvec{\varPi }_{\kappa ,0}(m) = \underline{\varvec{K}}(m) (\hat{\underline{\varvec{I}}}\otimes \underline{\varvec{D}}^{-1}\underline{\varvec{W}}) (\hat{\varvec{1}}\otimes \varvec{q}) = \hat{\varvec{1}}\otimes \underline{\varvec{E}}\underline{\varvec{S}}^{-1}\underline{\varvec{W}}\varvec{q}, \end{aligned}$$

  with the help of (7.22b) and (7.28). Then we can prove (7.31) as above.

Summing up the above conclusions, we verify (7.30) and then prove this lemma.

1.2.3 Proof of Propositions 3.1 and 3.2

Taking $\vartheta =0$ in (7.27) and substituting the definition of $\varvec{p}^{\top }$ and $\varvec{q}$, we have

$$\begin{aligned} \varTheta = \varvec{e}_{s-1}^{\top }\underline{\varvec{S}}^{-1}\underline{\varvec{W}}(0) (\underline{\varvec{I}}+\underline{\varvec{E}}\underline{\varvec{S}}^{-1}\underline{\varvec{C}})\varvec{e}_0. \end{aligned}$$

(7.32)

This identity will be used to prove these propositions.

Since we have assumed $c_{\ell \kappa }\ge 0$ for any $\ell$ and $\kappa$ in this paper, all entries of $\underline{\varvec{S}}^{-1}$ are non-negative due to the simple fact

$$\begin{aligned} \underline{\varvec{S}}^{-1}=(\underline{\varvec{I}}-\underline{\varvec{E}}\underline{\varvec{C}})^{-1} =\underline{\varvec{I}}+\sum _{1\le i\le s-1}(\underline{\varvec{E}}\underline{\varvec{C}})^i. \end{aligned}$$

Hence we can conclude from (7.32) that $\varTheta$ is a non-negative linear combination of the entries of $\underline{\varvec{W}}(0)=\{d_{\ell \kappa }\theta _{\ell \kappa }\}_{0\le \ell ,\kappa \le s-1}$. As a trivial conclusion for special case that all numerical flux parameters are the same, it is easy to conclude that $\varTheta$ is a weighted average of $\theta _{\ell \kappa }$. This proves Proposition 3.1.

Remark 7.2

This is the only place that the condition $c_{\ell \kappa }\ge 0$ is used in this paper.

For the LWDG method with the time marching coefficients (2.10), we have $\underline{\varvec{S}}=\underline{\varvec{I}}$ and then get from (7.32) that

$$\begin{aligned} \varTheta =\varvec{e}_{s-1}^{\top }\underline{\varvec{W}}(0)\varvec{e}_0 =d_{s-1,0}\theta _{s-1,0}=\theta _{s-1,0}, \end{aligned}$$

since $\underline{\varvec{I}}+\underline{\varvec{E}}\underline{\varvec{S}}^{-1}\underline{\varvec{C}}=\underline{\varvec{I}}+\underline{\varvec{E}}\underline{\varvec{C}}=\underline{\varvec{I}}$. This completes the proof of Proposition 3.2.