Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation

Abstract

In this paper, we shall establish the superconvergence properties of the Runge-Kutta discontinuous Galerkin method for solving two-dimensional linear constant hyperbolic equation, where the upwind-biased numerical flux is used. By suitably defining the correction function and deeply understanding the mechanisms when the spatial derivatives and the correction manipulations are carried out along the same or different directions, we obtain the superconvergence results on the node averages, the numerical fluxes, the cell averages, the solution and the spatial derivatives. The superconvergence properties in space are preserved as the semi-discrete method, and time discretization solely produces an optimal order error in time. Some numerical experiments also are given.

Appendix A

In this section, we supplement some technical proofs.

1.1 Proofs of (37)

As an example, below we present the detailed proof with respect to $\Vert [\![\mathbb {F}_{1,p}w]\!]\Vert _{L^2(\varGamma _{\!h}^2)}$. It depends on the one-dimensional correction technique.

Fix $y\in [0,1]$ and assume $v(x,y^{\pm })\in H^1(I_h)$. For $0\leqslant p\leqslant k$, the correction function along the x-direction is defined in the form [30]

$$\begin{aligned} \mathbb {F}_{1,p}^\mathrm{1d}v(x,y^{\pm })=(-\mathbb {X}_{\theta _1}\partial _{x}^{-1})^p (\mathbb {X}_{\frac{1}{2}}-\mathbb {X}_{\theta _1})v(x,y^{\pm }) \in \mathcal {P}^k(I_h). \end{aligned}$$

(A1)

Consider the separation function $v(x,y)=v_1(x)v_2(y)$, where either $v_1(x)$ or $v_2(y)$ is the piecewise polynomial of degree at most k. A direct application of (20) yields

$$\begin{aligned} \big (\mathbb {F}_{1,p}v\big )^\pm _{x,j\mp \frac{1}{2}} -\mathbb {F}_{1,p}^\mathrm{1d}\big (v^\pm _{x,j\mp \frac{1}{2}}\big ) = (\mathbb {F}_{1,p}^\mathrm{1d}v_1) \Big [(\mathbb {Y}_{\frac{1}{2}}v_2)^\pm _{j\mp \frac{1}{2}}-(v_2)^\pm _{j\mp \frac{1}{2}}\Big ]=0. \end{aligned}$$

As we have mentioned in Remark 2, we have

$$\begin{aligned} \big (\mathbb {F}_{1,p}v\big )^\pm _{x,j\mp \frac{1}{2}} -\mathbb {F}_{1,p}^\mathrm{1d}\big (v^\pm _{x,j\mp \frac{1}{2}}\big )=0, \quad \forall v\in \mathcal {P}^{2k+1}(\varOmega _h). \end{aligned}$$

(A2)

Note that $\mathbb {F}_{1,p}^\mathrm{1d}(w^+_{x,j+1/2})=\mathbb {F}_{1,p}^\mathrm{1d}(w^-_{x,j+1/2})$, since $w\in H^R(\varOmega )\subset H^2(\varOmega )$ is continuous. Then using (A2) and the triangle inequality, we get $\Vert [\![\mathbb {F}_{1,p}w]\!]\Vert _{L^2(\varGamma _{\!h}^2)}\leqslant ({\text{I}})+({\text{II}})$, where

$$\begin{aligned} \begin{aligned} {\rm{(I)}}&=\Vert (\mathbb {F}_{1,p}(w-v))^{+}\Vert _{L^2(\varGamma _{\!h}^2)}+\Vert (\mathbb {F}_{1,p}(w-v))^{-}\Vert _{L^2(\varGamma _{\!h}^2)}, \\ {\rm{(II)}}&=\Vert \mathbb {F}_{1,p}^\mathrm{1d}(w-v)^{+}\Vert _{L^2(\varGamma _{\!h}^2)}+\Vert \mathbb {F}_{1,p}^\mathrm{1d}(w-v)^{-}\Vert _{L^2(\varGamma _{\!h}^2)}. \end{aligned} \end{aligned}$$

Each term in (I) is bounded in the form

$$\begin{aligned} \Vert (\mathbb {F}_{1,p}(w-v))^{\pm }\Vert _{L^2(\varGamma _{\!h}^2)} \leqslant Ch^{-\frac{1}{2}}\Vert \mathbb {F}_{1,p}(w-v)\Vert _{L^2(\varOmega _h)} \leqslant Ch^{-\frac{1}{2}}h^{p+1}\Vert w-v\Vert _{H^{1}(\varOmega )}, \end{aligned}$$

(A3)

where the inverse inequality (9b) and Lemma 3 with $R=1$ are used. Each term in (II) is bounded in the form

$$\begin{aligned} \Vert \mathbb {F}_{1,p}^\mathrm{1d}(w-v)^{\pm }\Vert _{L^2(\varGamma _{\!h}^2)} \leqslant Ch^{p+1}\Vert (w-v)^{\pm }\Vert _{H^1(\varGamma _h^2)}, \end{aligned}$$

(A4)

where the result in [30, Lemma 4.3] is used for any horizontal edges. Finally using (21), we can get the boundedness of $\Vert [\![\mathbb {F}_{1,p}w]\!]\Vert _{L^2(\varGamma _{\!h}^2)}$ as stated in (37).

Similarly, we can get the boundedness of the second term in (37), by showing

$$\begin{aligned} \partial _y(\mathbb {F}_{1,p} v)-\mathbb {F}_{1,p}(\partial _y v)=0, \quad \forall v\in \mathcal {P}^{2k+1}(\varOmega _h). \end{aligned}$$

(A5)

The detailed process is omitted here.

1.2 Proof of (59)

It is easy to prove the right inequality by taking $v=\mathbb {H}_1 w$ in (39) and using Lemma 1. To prove the left inequality, we start from the formulation for any function $w(x,y)\in V_h$, namely

$$\begin{aligned} w(x,y)=\sum _{i}\sum _{j}\sum _{\kappa }\sum _{\ell } w_{ij}^{\kappa ,\ell } L^x_{i,\kappa }(x)L^y_{j,\ell }(y) = \sum _{j}\sum _{\ell }w_{j,\ell }(x)L^y_{j,\ell }(y). \end{aligned}$$

For simplification of notations, the ranges in the summations are omitted. Due to the $L^2$-orthogonality of $L^y_{j,\ell }(y)$, we can easily get that

$$\begin{aligned} \begin{aligned} \Vert \partial _x w(x,y)\Vert _{L^2(\varOmega _h)}^2&= \sum _{j}\sum _{\ell } \Vert \partial _xw_{j,\ell }(x)\Vert _{L^2(0,1)}^2 \Vert L^y_{j,\ell }(y)\Vert _{L^2(J_j)}^2, \\ \Vert \mathbb {H}_1w(x,y)\Vert _{L^2(\varOmega _h)}^2&= \sum _{j}\sum _{\ell } \Vert \mathbb {H}_1w_{j,\ell }(x)\Vert _{L^2(0,1)}^2 \Vert L^y_{j,\ell }(y)\Vert _{L^2(J_j)}^2, \end{aligned} \end{aligned}$$

where (55) has been used in the second conclusion. Then we can prove the left inequality using the inequality [29] for the single-variable function

$$\begin{aligned} \Vert \partial _x w_{j,\ell }(x)\Vert _{L^2(0,1)} \leqslant C\Vert \mathbb {H}_1 w_{j,\ell }(x)\Vert _{L^2(0,1)}, \quad j=1,2,\cdots , N_x, \quad \ell =0,1,\cdots ,k. \end{aligned}$$

1.3 Proof of (64)

Since $w\in H^3(\varOmega )$ is continuous everywhere, we have

$$\begin{aligned} \{\!\!\{\mathbb {G}_{\theta _1,\theta _2}^\perp w\}\!\!\}^{\theta _1,y}_{i+\frac{1}{2},y}=\mathbb {Y}_{\theta _2}^\perp w(x_{i+\frac{1}{2}},y), \quad i=1,2,\cdots , N_x, \quad y\in [0,1]. \end{aligned}$$

It has been proved in [30, Lemma 5.3] for every i that

$$\begin{aligned} \mathrm{RMS}\{\{\!\!\{\mathbb {Y}_{\theta _2}^\perp w(x_{i+\frac{1}{2}},y)\}\!\!\}^{\theta _1,y}_{i+\frac{1}{2},y} :y\in S_h^{\mathrm{R},y}\}&\leqslant Ch^{k+2}\Vert w(x_{i+\frac{1}{2}},\cdot )\Vert _{H^{k+2}(0,1)}, \\ \mathrm{RMS}\{\{\!\!\{\partial _y \mathbb {Y}_{\theta _2}^\perp w(x_{i+\frac{1}{2}},y)\}\!\!\}^{\theta _1,y}_{i+\frac{1}{2},y} :y\in S_h^{\mathrm{L},y}\}&\leqslant Ch^{k+1}\Vert w(x_{i+\frac{1}{2}},\cdot )\Vert _{H^{k+2}(0,1)}, \end{aligned}$$

which together with the standard trace inequity [12] yield (64a).

The proof of (64b) is almost the same, so omitted here. In what follows we devote to proving (64c).

The proof depends on the local projection related to (62), the parameter-dependent Radau polynomials. For any given function $w\in L^2(\varOmega _h)$, the projection $\mathbb {C}w$ is defined element by element, namely,

$$\begin{aligned} \mathbb {C}w|_{K_{ij}} = \mathbb {R}w|_{K_{ij}} -w_{i,j}^{k+1,0}R^x_{i,k+1}(x) -w_{i,j}^{0,k+1}R^y_{j,k+1}(y), \end{aligned}$$

(A6)

which belongs to $\mathcal {P}^{k+1}(K_{ij})\cap \mathcal {Q}^k(K_{ij})$. Here

$$\begin{aligned} \mathbb {R}w|_{K_{ij}} =\sum _{0\leqslant \ell _1+\ell _2\leqslant k+1} w_{i,j}^{\ell _1,\ell _2}L^x_{i,\ell _1}(x)L^y_{j,\ell _2}(y) \end{aligned}$$

(A7)

is the $L^2$ projection of w onto $\mathcal {P}^{k+1}(K_{ij})$. Note that the definition of this local projection is a little different to that in [2, 30], and we do not need to discuss whether $\vartheta ^x_i$ and/or $\vartheta ^y_j$ are equal to 0.

Let $\mathbb {C}^\perp w=w-\mathbb {C}w$ be the projection error. By standard scaling argument, we have the approximation property

$$\begin{aligned} \Vert \mathbb {C}^\perp w\Vert _{L^2(\varOmega _h)} +h\Vert \mathbb {C}^\perp w\Vert _{H^{1}(\varOmega _h)} +h^2\Vert \mathbb {C}^\perp w\Vert _{H^{2}(\varOmega _h)} \leqslant Ch^{\min (R,k+1)}\Vert w\Vert _{H^{R}(\varOmega _h)}. \end{aligned}$$

(A8)

Furthermore, we can easily obtain the following lemmas.

Lemma A1

There exists a constant $C>0$ independent of h and w, such that

$$\begin{aligned} \vert \!\vert \!\vert {\mathbb {C}^\perp w}\vert \!\vert \!\vert _{L^2(S_h^\mathrm{R,R})} +h\vert \!\vert \!\vert {\partial _x(\mathbb {C}^\perp w)}\vert \!\vert \!\vert _{L_y^2(S_h^{\mathrm{L},x})} +h\vert \!\vert \!\vert {\partial _y(\mathbb {C}^\perp w)}\vert \!\vert \!\vert _{L_x^2(S_h^{\mathrm{L},y})} \leqslant C h^{k+2}\Vert w\Vert _{H^{k+2}(\varOmega )}. \end{aligned}$$

Proof

As the applications of the Bramble-Hilbert lemma and the scaling argument, it is sufficient to prove for any $w\in \mathcal {P}^{k+1}(K_{ij})$ that

$$\begin{aligned} \mathbb {C}^\perp w(x,y)&=0, \quad x\in S_h^{\mathrm{R},x},\quad y\in S_h^{\mathrm{R},y}, \\ \partial _x(\mathbb {C}^\perp w) (x,y)&=0, \quad x\in S_h^{\mathrm{R},x},\quad y\in J_j, \\ \quad \partial _y(\mathbb {C}^\perp w) (x,y)&=0, \quad x\in I_i,\quad y\in S_h^{\mathrm{R},y}, \end{aligned}$$

which is implied by the definition of projection $\mathbb {C}$, say (A6).

Lemma A2

There exists a constant $C>0$ independent of h and w, such that

$$\begin{aligned} \Vert \mathbb {G}_{\theta _1,\theta _2}w - \mathbb {C}w\Vert _{L^2(\varOmega _h)} \leqslant Ch^{k+2}\Vert w\Vert _{H^{k+2}(\varOmega )}. \end{aligned}$$

Proof

By the definitions of $\mathbb {G}_{\theta _1,\theta _2}$ and $\mathbb {C}$, we have

$$\begin{aligned} (\mathbb {G}_{\theta _1,\theta _2}- \mathbb {C})w = (\mathbb {G}_{\theta _1,\theta _2}- \mathbb {C})\mathbb {R}^\perp w +(\mathbb {G}_{\theta _1,\theta _2}- \mathbb {C})w_1 +(\mathbb {G}_{\theta _1,\theta _2}- \mathbb {C})w_2, \end{aligned}$$

(A9)

where $\mathbb {R}w$ is the $L^2$ projection of w onto $\mathcal {P}^k(\varOmega _h)$, and $\mathbb {R}^\perp w=w-\mathbb {R}w$ is the projection error. Here $w_1$ and $w_2$ are defined element-by-element by

$$\begin{aligned} w_1|_{K_{ij}}=w_{i,j}^{k+1,0}L^x_{i,k+1}(x), \quad w_2|_{K_{ij}}=w_{i,j}^{0,k+1}L^y_{j,k+1}(y). \end{aligned}$$

(A10)

Next we will estimate each term on the right-hand side of (A9).

Using the approximation properties of projections (16) and (A8), as well as the approximation property of projection $\mathbb {R}$ (refer to (21)), we get

$$\begin{aligned} \Vert (\mathbb {G}_{\theta _1,\theta _2}- \mathbb {C})\mathbb {R}^\perp w\Vert _{L^2(\varOmega _h)} \leqslant Ch^2\Vert \mathbb {R}^\perp w\Vert _{H^{2}(\varOmega )} \leqslant Ch^{k+2}\Vert w\Vert _{H^{k+2}(\varOmega )}. \end{aligned}$$

After some technical and direct manipulations, almost the same as that in [30], we also have

$$\begin{aligned} \Vert (\mathbb {G}_{\theta _1,\theta _2}- \mathbb {C})w_1\Vert _{L^2(\varOmega _h)} +\Vert (\mathbb {G}_{\theta _1,\theta _2}- \mathbb {C})w_2\Vert _{L^2(\varOmega _h)} \leqslant Ch^{k+2}\Vert w\Vert _{H^{k+2}(\varOmega )}, \end{aligned}$$

(A11)

where definition (61), with respect to $\{\vartheta ^x_{i}\}_{i=1}^{N_x}$ and $\{\vartheta ^y_{j}\}_{j=1}^{N_y}$, plays an important role. For more details, please see [30].

Finally, summing up the above conclusions yields this lemma.

Using the inverse inequity $\Vert v\Vert _{L^\infty (\varOmega _h)}\leqslant C h^{-1}\Vert v\Vert _{L^2(\varOmega _h)}$ for $v\in V_h$, Lemma A2 implies

$$\begin{aligned} \vert \!\vert \!\vert {(\mathbb {G}_{\theta _1,\theta _2}-\mathbb {C}) w}\vert \!\vert \!\vert _{L^2(S_h^\mathrm{R,R})} \leqslant C\Vert (\mathbb {G}_{\theta _1,\theta _2}-\mathbb {C})w\Vert _{L^2(\varOmega _h)} \leqslant Ch^{k+2}\Vert w\Vert _{H^{k+2}(\varOmega )}. \end{aligned}$$

Hence it follows from Lemma A1 that

$$\begin{aligned} \vert \!\vert \!\vert {\mathbb {G}_{\theta _1,\theta _2}^\perp w}\vert \!\vert \!\vert _{L^2(S_h^\mathrm{R,R})} \leqslant \vert \!\vert \!\vert {\mathbb {C}^\perp w}\vert \!\vert \!\vert _{L^2(S_h^\mathrm{R,R})} +\vert \!\vert \!\vert {(\mathbb {G}_{\theta _1,\theta _2}-\mathbb {C}) w}\vert \!\vert \!\vert _{L^2(S_h^\mathrm{R,R})} \leqslant Ch^{k+2}\Vert w\Vert _{H^{k+2}(\varOmega )}. \end{aligned}$$

The others can be estimated similarly. Now we complete the proof of (54).