Superconvergence Analysis of the Runge–Kutta Discontinuous Galerkin Methods for a Linear Hyperbolic Equation

Abstract

In this paper, we shall establish the superconvergence property of the Runge–Kutta discontinuous Galerkin (RKDG) method for solving a linear constant-coefficient hyperbolic equation. The RKDG method is made of the discontinuous Galerkin (DG) scheme with upwind-biased numerical fluxes coupled with the explicit Runge–Kutta algorithm of arbitrary orders and stages. Superconvergence results for the numerical flux, cell averages as well as the solution and derivative at some special points are shown, which are based on a systematical study of the $\hbox {L}^2$-norm stability for the RKDG method and the incomplete correction techniques for the well-defined reference functions at each time stage. The result demonstrates that the superconvergence property of the semi-discrete DG method is preserved, and the optimal order in time is provided under the smoothness assumption that is independent of the number of stages. As a byproduct of the above superconvergence study, the expected order of the post-processed solution is obtained when a special initial solution is used. Some numerical experiments are also given.

Appendix

In this section, the supplement proofs of three technical results are given.

1.1 Proof of (3.16)

Substituting the offset into the relationship in Lemma 2.2, we have

$$\begin{aligned} e^z + \sum _{i=r+1}^\infty \tilde{\alpha }_i(m) z^i = \Big [e^{\frac{z}{m}} + \sum _{i=r+1}^\infty \tilde{\alpha }_i(1) \Big (\frac{z}{m}\Big )^i\Big ]^m=\Big [e^{\frac{z}{m}} +\frac{z^{r+1}}{m^{r+1}}q(z)\Big ]^m, \end{aligned}$$

(9.1)

where $q(z)=\sum _{i=0}^\infty q_{i} z^i =\sum _{i=0}^\infty \frac{\tilde{\alpha }_{i+r+1}(1)}{m^i} z^i$. Denote $\tilde{\alpha }_{\max }=\max \limits _{\forall \kappa }|\tilde{\alpha }_\kappa (1)|$. By a direct calculation, the coefficient of $[q(z)]^{j}=\sum _{i=0}^{\infty } q_{i,j}z^i$ satisfies

$$\begin{aligned} |q_{i,j}|\le C(\tilde{\alpha }_{\max })^j, \quad 0\le i, j\le 2\zeta -1, \end{aligned}$$

where the bounding constant $C>0$ solely depends on the termination index $\zeta$.

Subtracting $e^z$ from both sides of (9.1) we have

$$\begin{aligned} \sum _{i=r+1}^\infty \tilde{\alpha }_i(m) z^i= & {} \sum _{1\le j\le m}\left( {\begin{array}{c}m\\ j\end{array}}\right) \Big (e^{\frac{z}{m}}\Big )^{m-j}\Big (\frac{z^{r+1}q(z)}{m^{r+1}}\Big )^j \\= & {} \sum _{1\le j\le m}\left( {\begin{array}{c}m\\ j\end{array}}\right) \frac{z^{j(r+1)}}{m^{j(r+1)}} \Big [\sum _{i=0}^\infty \frac{1}{i!}\Big (\frac{m-j}{m}\Big )^i z^i\Big ] \Big [\sum _{i=0}^\infty q_{i,j} z^i\Big ]. \end{aligned}$$

and get

$$\begin{aligned} \tilde{\alpha }_i(m) = \sum _{1\le j\le m}\left[ \left( {\begin{array}{c}m\\ j\end{array}}\right) \frac{1}{m^{j(r+1)}} \sum _{0\le \ell \le \sigma _{ij}}q_{\ell ,j} \Big (\frac{m-j}{m}\Big )^{\sigma _{ij}-\ell }\frac{1}{(\sigma _{ij}-\ell )!} \right] , \end{aligned}$$

where $\sigma _{ij}=i-j(r+1)$. Hence

$$\begin{aligned} |\tilde{\alpha }_i(m)|\le C\sum _{1\le j\le m}\Big [\frac{\tilde{\alpha }_{\max }}{m^{r}}\Big ]^j\le \frac{C\tilde{\alpha }_{\max }}{m^{r}}, \end{aligned}$$

provided $m^r \ge 2\tilde{\alpha }_{\max }$. This completes the proof of this inequality.

1.2 Proof of (4.40)

It is no harm in assuming that $q\ge 1$. Substituting (4.32) into the definition of $\tilde{\xi }^0$ yields

$$\begin{aligned} \tilde{\xi }^0= & {} {\mathcal {H}}_h \Big ({\mathbb {G}}_hU_0 - \sum _{1\le p\le q_\mathrm{init}} {\mathcal {F}}_p (-\partial _x)^p U_0\Big ) -\Big ({\mathbb {G}}_h\Pi _0 -\sum _{1\le p\le q}{\mathcal {F}}_p (-\partial _x)^p \Pi _0\Big ) \nonumber \\= & {} {\mathcal {H}}_h {\mathbb {G}}_hU_0 - \sum _{1\le p\le q_\mathrm{init}} {\mathcal {H}}_h {\mathcal {F}}_p (-\partial _x)^p U_0 +\beta {\mathbb {G}}_h(U_0)_x - \beta \sum _{1\le p\le q}{\mathcal {F}}_p (-\partial _x)^p (U_0)_x,\nonumber \\ \end{aligned}$$

(9.2)

since $\Pi _0 = -\beta (U_0)_x$. Because $U_0\in H^1(I)$ is continuous in I, for any $v\in V_h$ we have

$$\begin{aligned} ({\mathcal {H}}_h {\mathbb {G}}_hU_0 ,v) = {\mathcal {H}}({\mathbb {G}}_hU_0 ,v) = {\mathcal {H}}(U_0 ,v) =-\beta ((U_0)_x ,v) = -\beta ({\mathbb {P}}_h(U_0)_x ,v), \end{aligned}$$

where the definitions of the two projections are used. Similarly, due to Lemma 4.4, each term in the first summation of (9.2) satisfies

$$\begin{aligned} ({\mathcal {H}}_h {\mathcal {F}}_p (-\partial _x)^p U_0,v)= {\mathcal {H}}({\mathcal {F}}_p (-\partial _x)^p U_0 ,v) = \beta ({\mathcal {F}}_{p-1} (-\partial _x)^p U_0 ,v),\quad \forall v\in V_h. \end{aligned}$$

Hence, ${\mathcal {H}}_h {\mathbb {G}}_hU_0=-\beta {\mathbb {P}}_h(U_0)_x$ and ${\mathcal {H}}_h {\mathcal {F}}_p (-\partial _x)^p U_0= \beta {\mathcal {F}}_{p-1} (-\partial _x)^p U_0$. Substituting them into (9.2), we arrive at

$$\begin{aligned} \tilde{\xi }^0=-\beta \sum _{1\le p\le q_\mathrm{nt}-1} {\mathcal {F}}_p (-\partial _x)^{p+1} U_0 +\beta \sum _{1\le p\le q} {\mathcal {F}}_p (-\partial _x)^{p+1} U_0. \end{aligned}$$

(9.3)

Since $q\le q_\mathrm{nt}\le k$, we can get (4.40), along the same line as for (4.34).

A supplement is given for $q=0$. Since the summation is equal to zero if the index set is empty, the formula (9.3) also holds for $q=0$ and $q_\mathrm{nt}\ge 1$. If $q=q_\mathrm{nt}=0$, the two summations in (9.2) vanish such that $\tilde{\xi }^0=-\beta {\mathcal {F}}_0(U_0)_x$. For these special cases, it is easy to see that (4.40) holds.

1.3 Proof of Lemma 5.2

By the definitions of the two projections we have

$$\begin{aligned} ({\mathbb {G}}_hw - {\mathbb {C}}_hw)|_{I_j} = \widetilde{w}_j L_{j,k}, \quad j=1,2,\ldots , J, \end{aligned}$$

and the undetermined constants $\widetilde{w}_j$ satisfy the following system of linear equations

$$\begin{aligned} \theta \widetilde{w}_j+(1-\theta )(-1)^k \widetilde{w}_{j+1} =\{\!\!\{{\mathbb {C}}_h^\perp w\}\!\!\}^{(\theta )}_{j+\frac{1}{2}}, \quad j=1,2,\ldots , J. \end{aligned}$$

(9.4)

It is proved in [8] that this linear system has a unique solution since $\theta \ne 1/2$, and

$$\begin{aligned} \left\| {{\mathbb {G}}_hw - {\mathbb {C}}_hw}\right\| _{L^2(I)}^2 \le Ch\sum _{1\le j\le J} |\widetilde{w}_j|^2 \le Ch\sum _{1\le j\le J}|\{\!\!\{{\mathbb {C}}_h^\perp w\}\!\!\}^{(\theta )}_{j+\frac{1}{2}}|^2. \end{aligned}$$

(9.5)

Hence, it is sufficient to prove this lemma by showing

$$\begin{aligned} |\{\!\!\{{\mathbb {C}}_h^\perp w\}\!\!\}^{(\theta )}_{j+\frac{1}{2}}| \le Ch^{k+\frac{3}{2}}\left\| {w}\right\| _{H^{k+2}(I_j\cup I_{j+1})}, \quad j=1,2,\ldots ,J. \end{aligned}$$

(9.6)

To this end, let us consider the decomposition

$$\begin{aligned} \{\!\!\{{\mathbb {C}}_h^\perp w\}\!\!\}^{(\theta )}_{j+\frac{1}{2}} =\{\!\!\{{\mathbb {C}}_h^\perp ({\mathbb {P}}_h^{k+1})^\perp w\}\!\!\}^{(\theta )}_{j+\frac{1}{2}} +\{\!\!\{{\mathbb {C}}_h^\perp {\mathbb {P}}_h^{k+1}w\}\!\!\}^{(\theta )}_{j+\frac{1}{2}} =b_{j1}+b_{j2}, \end{aligned}$$

where ${\mathbb {P}}_h^{k+1}$ denotes the local $\hbox {L}^2$-projection on $V_h^{k+1}$. By using the approximation property of the projections ${\mathbb {C}}_h$ and ${\mathbb {P}}_h^{k+1}$, we get

$$\begin{aligned} |b_{j1}|\le Ch^{\frac{1}{2}}\left\| {({\mathbb {P}}_h^{k+1})^\perp w}\right\| _{H^{1}(I_j\cup I_{j+1})} \le Ch^{k+\frac{3}{2}}\left\| {w}\right\| _{H^{k+2}(I_j\cup I_{j+1})}. \end{aligned}$$

(9.7)

Using (5.10), we know that ${\mathbb {C}}_h^\perp {\mathbb {P}}_h^{k+1}w(x)=w_{j,k+1}(L_{j,k+1}(x)-\vartheta _j L_{j,k}(x))$ for $x\in I_j$, where

$$\begin{aligned} w_{j,k+1}= \frac{2k+3}{2} \int _{-1}^1 w\Big (x_j+\frac{h_j\hat{x}}{2}\Big ) L_{k+1}(\hat{x})\mathrm {d}\hat{x} = h_j^{k+1} \int _{-1}^1\partial _x^{k+1} w\Big (x_j+\frac{h_j\hat{x}}{2}\Big ) \Phi (\hat{x})\mathrm {d}\hat{x}, \end{aligned}$$

and the kernel function $\Phi (\hat{x})=\frac{(-1)^{k+1}(2k+3)}{2^{2k+3}(k+1)!}(\hat{x}^2-1)^{k+1}$ is independent of j. In the above manipulations the Rodrigue’s formula of the Legendre polynomials and integration by parts are used. Using (5.1), we get

$$\begin{aligned} b_{j2}= & {} \theta (1-\vartheta _j)w_{j,k+1} +(1-\theta )[(-1)^{k+1}-\vartheta _{j+1}(-1)^k]w_{j+1,k+1} \nonumber \\= & {} \theta (1-\vartheta _j)h_j^{k+1} \int _{-1}^{1} \Big [ \partial _x^{k+1}w\Big (x_j+\frac{h_j\hat{x}}{2}\Big ) -\partial _x^{k+1}w \Big (x_{j+1}+\frac{h_{j+1}\hat{x}}{2}\Big ) \Big ]\Phi (\hat{x})\mathrm {d}\hat{x} \nonumber \\\le & {} C h^{k+\frac{3}{2}}\left\| {w}\right\| _{H^{k+2}(I_j\cup I_{j+1})}. \end{aligned}$$

(9.8)

where the Holder’s inequality is used at the last step. We have now proved (9.6) and hence completed the proof of this lemma.