Local Error Estimates for Runge–Kutta Discontinuous Galerkin Methods with Upwind-Biased Numerical Fluxes for a Linear Hyperbolic Equation in One-Dimension with Discontinuous Initial Data

Abstract

In this paper we present the local error estimate of Runge–Kutta discontinuous Galerkin (RKDG) methods with upwind-biased numerical fluxes for solving linear hyperbolic equations in one dimension with discontinuous initial data. Under the temporal–spatial condition to ensure the $\hbox {L}^2$-norm stability, the convergence orders in both space and time are optimal outside the pollution region due to the discontinuous initial data, whose width is nearly optimal to be one-half order of the mesh size, no matter in the upwind or the downwind direction. The kernel analysis is achieving the weighted $\hbox {L}^2$-norm stability result for the RKDG methods, based on the matrix transferring process and the local $\hbox {L}^2$-projection. Furthermore, we need to set up the weighted approximation property for the generalized Gauss–Radau projection. Finally some numerical experiments are also given to support the theoretical results.

Appendix

In this section we present the detailed proof of (4.9), where Cauchy–Schwarz inequality and Young inequality are the elemental tools.

The former five terms in (4.7) are easily bounded along the same line as in [19, 20]. Limited by the length of this paper, we only give a quick description below. In details,

$$\begin{aligned} \Pi ^n_1 \le \Big (a_{\zeta \zeta }+\frac{1}{2}\vert a_{\zeta \zeta }\vert \Big ) \Vert \mathbb {D}_\zeta u_h^n\Vert _{\psi ^n,I_h}^2 +C\mathcal {G}^n_1. \end{aligned}$$

(6.1)

Lemma 3.2 implies

$$\begin{aligned} \begin{aligned} \Pi ^n_2 \le&\; C(1+hK) \sum _{0\le i,j\le ms-1} \Vert \mathbb {D}_{j}u_h^n\Vert _{\psi ^n,I_h} \Vert \mathbb {D}_{i}f^n\Vert _{\psi ^n,I_h}\tau \\ \le&\; CT^{-1}(1+hK)^2\sum _{0\le j\le ms-1} \Vert \mathbb {D}_{j}u_h^n\Vert _{\psi ^n,I_h}^2\tau + CT\sum _{0\le i\le ms-1} \Vert \mathbb {D}_{i}f^n\Vert _{\psi ^n,I_h}^2\tau \\ \le&\; C(1+h^2K^2)\mathcal {G}^n_2+C\mathcal {G}^n_3. \end{aligned} \end{aligned}$$

(6.2)

Lemma 3.3 and the second inverse inequality in (3.2) yield

$$\begin{aligned} \begin{aligned} \Pi ^n_3 \le&\; C\tau \beta Kh^{\frac{1}{2}}\sum _{0\le i,j\le ms-1} \Vert \![\![\mathbb {D}_{i}u_h^n]\!]\!\Vert _{\psi ^n,\varGamma _{\!h}} \Vert \mathbb {D}_{j}u_h^n\Vert _{\psi ^n,I_h} \\ \le&\; \frac{1}{4}\varepsilon \Big (\theta -\frac{1}{2}\Big ) m\tau \beta \sum _{0\le i\le ms-1}\Vert \![\![\mathbb {D}_iu_h^n]\!]\!\Vert _{\psi ^n,\varGamma _{\!h}}^2 + C\tau \beta hK^2 \sum _{0\le j\le ms-1} \Vert \mathbb {D}_{j}u_h^n\Vert _{\psi ^n,I_h}^2 \\ \le&\; \frac{1}{2}\mathcal {Y}^n + C\lambda \Vert \mathbb {D}_\zeta u_h^n\Vert _{\psi ^n,I_h}^2 + C\mathcal {G}^n_1 + CT\beta hK^2\mathcal {G}^n_2. \end{aligned} \end{aligned}$$

(6.3)

Since the matrix $\{b_{ij}\}_{0\le i,j\le \zeta -1}$ is symmetric positive definite with the smallest eigenvalue $\varepsilon >0$, we can additionally use the second inverse inequality in (3.2) to yield

$$\begin{aligned} \Pi ^n_4 \le -\mathcal {Y}^n + C\tau \beta \sum _{\zeta \le i\le ms-1} \Vert \![\![\mathbb {D}_iu_h^n]\!]\!\Vert _{\psi ^n,\varGamma _{\!h}}^2 \le -\mathcal {Y}^n + C\lambda \Vert \mathbb {D}_\zeta u_h^n\Vert _{\psi ^n,I_h}^2 + C\mathcal {G}^n_1. \end{aligned}$$

(6.4)

Since $\psi (x,t)=\psi _{\star }(x-\beta t)$ and $\lambda \le 1$, we use the Taylor expansion together with assumptions W1 and W2 to get

$$\begin{aligned} \vert \varDelta \psi ^n(x)\vert \le m\tau \beta \int ^{x}_{x-m\tau \beta } \vert \partial _x^2\psi ^n(x')\vert \,\mathrm {d}x' \le C\tau \beta hK^2\vert \psi ^n(x)\vert , \end{aligned}$$

which implies

$$\begin{aligned} \Pi ^n_5 \le C\tau \beta hK^2\sum _{0\le i\le ms} \Vert \mathbb {D}_iu_h^n\Vert _{\psi ^n,I_h}^2 \le CT\beta hK^2\mathcal {G}^n_2. \end{aligned}$$

(6.5)

However, the estimate to the last term $\Pi ^n_6$ is a little difficult. Without losing generality, we assume that $\psi ^n$ is monotone increasing and take $\phi =\partial _x\psi ^n$ in (3.21). Based on the second strategy of the matrix transferring process (see Lemma 3.9), we have

$$\begin{aligned} \Pi ^n_6= \Pi ^n_{61}+\Pi ^n_{62}+\Pi ^n_{63}+\Pi ^n_{64}+\Pi ^n_{65}, \end{aligned}$$

(6.6)

where

$$\begin{aligned} \Pi ^n_{61} =&\; m\tau \beta \sum _{i,j\in \{0,ms\}, i+j\ge r} p_{ij}\Big (\mathbb {D}_iu_h^n,\partial _x\psi ^n\mathbb {D}_ju_h^n\Big )_{I_h}, \end{aligned}$$

(6.7a)

$$\begin{aligned} \Pi ^n_{62} =&\; -(m\tau \beta )^2 \Big (\theta -\frac{1}{2}\Big ) \sum _{0\le i,j\le ms-1} q_{ij} \Big \langle [\![\mathbb {D}_iu_h^n]\!],\partial _x\psi ^n[\![\mathbb {D}_ju_h^n]\!]\Big \rangle _{\varGamma _{\!h}}, \end{aligned}$$

(6.7b)

$$\begin{aligned} \Pi ^n_{63} =&\; (m\tau )^2\beta \sum _{0\le i,j\le ms-1} q_{ij} \Big (\mathbb {D}_if^n,\mathbb {P}_h(\partial _x\psi ^n\mathbb {D}_ju_h^n)\Big )_{I_h}, \end{aligned}$$

(6.7c)

$$\begin{aligned} \Pi ^n_{64} =&\; -(m\tau )^2\beta \sum _{0\le i,j\le ms-1} q_{ij} \mathcal {H}(\mathbb {D}_iu_h^n,\mathbb {P}_h^\perp (\partial _x\psi ^n\mathbb {D}_ju_h^n)), \end{aligned}$$

(6.7d)

$$\begin{aligned} \Pi ^n_{65} =&\; \frac{1}{2}(m\tau \beta )^2 \sum _{0\le i,j\le ms-1} q_{ij} \Big (\mathbb {D}_iu_h^n,\partial _x^2\psi ^n\mathbb {D}_ju_h^n\Big )_{I_h}. \end{aligned}$$

(6.7e)

Each term in (6.7) can be similarly bounded by using Cauchy-Schwarz inequality and Young inequality, with the additional help of

$$\begin{aligned} \Vert v_h\Vert _{\vert \partial _x\phi \vert ,I_h} \le K^{\frac{1}{2}}\Vert v_h\Vert _{\phi ,I_h}, \quad \Vert \![\![v_h]\!]\!\Vert _{\vert \partial _x\phi \vert ,\varGamma _{\!h}} \le K^{\frac{1}{2}}\Vert \![\![v_h]\!]\!\Vert _{\phi ,\varGamma _{\!h}}, \end{aligned}$$

(6.8)

for $\phi =\psi ^n$ or $\phi =\partial _x\psi ^n$. Note that (6.8) follows from assumption W2.

The estimate to the first term $\Pi ^n_{61}$ is kernel. Noticing the shallowing structure of the matrix $\{p_{ij}\}$, we have

$$\begin{aligned} \begin{aligned} \Pi ^n_{61} \le&\; C\tau \beta K\Bigg ( \sum _{r\le i\le ms} \Vert \mathbb {D}_iu_h^n\Vert _{\psi ^n,I_h}\Vert u_h^n\Vert _{\psi ^n,I_h} + \sum _{0\le j\le ms} \Vert \mathbb {D}_{ms}u_h^n\Vert _{\psi ^n,I_h}\Vert \mathbb {D}_ju_h^n\Vert _{\psi ^n,I_h}\Bigg ) \\ \le&\; C\tau \beta K\Bigg (\sum _{r\le i\le ms} \Vert \mathbb {D}_iu_h^n\Vert _{\psi ^n,I_h}^2\Bigg )^{\frac{1}{2}} \Bigg (\sum _{0\le j\le ms} \Vert \mathbb {D}_ju_h^n\Vert _{\psi ^n,I_h}^2\Bigg )^{\frac{1}{2}} \\ \le&\; C\tau \beta h^{-1} \sum _{r\le i\le ms} \Vert \mathbb {D}_iu_h^n\Vert _{\psi ^n,I_h}^2 +C\tau \beta hK^2\sum _{0\le j\le ms} \Vert \mathbb {D}_ju_h^n\Vert _{\psi ^n,I_h}^2 \\ \le&\; C\lambda \Vert \mathbb {D}_\zeta u_h^n\Vert _{\psi ^n,I_h}^2 +C\mathcal {G}^n_1 +CT\beta hK^2\mathcal {G}^n_2, \end{aligned} \end{aligned}$$

where $\zeta \le r$ and $\lambda \le 1$ are used at the last step. An application of (6.8) yields

$$\begin{aligned} \Pi ^n_{62} \le C\tau ^2\beta ^2K \sum _{0\le i\le ms-1} \Vert \![\![\mathbb {D}_iu_h^n]\!]\!\Vert _{\psi ^n,\varGamma _{\!h}}^2. \end{aligned}$$

Since $\tau ^2\beta ^2=\tau \beta \lambda h=\lambda ^2h^2$, using the inverse inequality (3.2) to deal with the high order temporal information, we have

$$\begin{aligned} \begin{aligned} \Pi ^n_{62} \le&\; C\beta \lambda hK\sum _{0\le i\le \zeta -1}\Vert \![\![\mathbb {D}_iu_h^n]\!]\!\Vert _{\psi ^n,\varGamma _{\!h}}^2\tau +C\lambda ^2 hK \sum _{\zeta \le i\le ms-1} \Vert \mathbb {D}_iu_h^n\Vert _{\psi ^n,I_h}^2 \\ \le&\; C\lambda hK\mathcal {Y}^n +C\lambda hK\Vert \mathbb {D}_\zeta u_h^n\Vert _{\psi ^n,I_h}^2 +ChK\mathcal {G}^n_1. \end{aligned} \end{aligned}$$

Using Lemma 3.2 with $\phi =\partial _x\psi ^n$, we have

$$\begin{aligned} \begin{aligned} \Pi ^n_{63} \le&\; C\tau ^2\beta \sum _{0\le i,j\le ms-1} \Vert \mathbb {P}_h(\partial _x\psi ^n\mathbb {D}_ju_h^n)\Vert _{(\partial _x\psi ^n)^{-1},I_h} \Vert \mathbb {D}_{i}f^n\Vert _{\partial _x\psi ^n,I_h} \\ \le&\; C\tau ^2\beta K(1+hK) \sum _{0\le i,j\le ms-1} \Vert \mathbb {D}_{j}u_h^n\Vert _{\psi ^n,I_h} \Vert \mathbb {D}_{i}f^n\Vert _{\psi ^n,I_h} \\ \le&\; CT^{-1}(1+hK)^4 \sum _{0\le j\le ms-1} \Vert \mathbb {D}_{j}u_h^n\Vert _{\psi ^n,I_h}^2\tau + CT \lambda ^2 \sum _{0\le i\le ms-1} \Vert \mathbb {D}_{i}f^n\Vert _{\psi ^n,I_h}^2\tau \\ \le&\; C(1+h^4K^4)\mathcal {G}^n_2+C\mathcal {G}^n_3. \end{aligned} \end{aligned}$$

Using Lemma 3.3 with $\phi =\partial _x\psi ^n$ and (6.8), we have

$$\begin{aligned} \begin{aligned} \Pi ^n_{64} \le&\; C\tau ^2\beta ^2h^{\frac{1}{2}}K^2 \sum _{0\le i,j\le ms-1} \Vert \mathbb {D}_{i}u_h^n\Vert _{\psi ^n,I_h} \Vert \![\![\mathbb {D}_{j}u_h^n]\!]\!\Vert _{\psi ^n,\varGamma _{\!h}} \\ \le&\; C\tau ^2\beta ^2hK^3 \sum _{0\le i\le ms-1} \Vert \mathbb {D}_{i}u_h^n\Vert _{\psi ^n,I_h}^2 + C\tau ^2\beta ^2K \sum _{0\le j\le ms-1} \Vert \![\![\mathbb {D}_{j}u_h^n]\!]\!\Vert _{\psi ^n,\varGamma _{\!h}}^2 \\ \le&\; CT\beta h^2K^3\lambda \mathcal {G}^n_2+C\lambda hK\mathcal {Y}^n +C\lambda hK\Vert \mathbb {D}_\zeta u_h^n\Vert _{\psi ^n,I_h}^2 +ChK\mathcal {G}^n_1, \end{aligned} \end{aligned}$$

where the treatment for $\Pi ^n_{62}$ have been also used here. Similarly, an application of (6.8) twice yields

$$\begin{aligned} \Pi ^n_{65} \le C\tau ^2\beta ^2 K^2 \sum _{0\le i\le ms-1} \Vert \mathbb {D}_iu_h^n\Vert _{\psi ^n,I_h}^2 \le CT\beta hK^2\lambda \mathcal {G}^n_2 \le CT\beta hK^2\mathcal {G}^n_2. \end{aligned}$$

Collecting the above inequalities into (6.6), we have the estimate

$$\begin{aligned} \begin{aligned} \Pi ^n_6 \le&\; C\lambda (1+hK)\Vert \mathbb {D}_\zeta u_h^n\Vert _{\psi ^n,I_h}^2 +C\lambda hK\mathcal {Y}^n \\ {}&\; +C(1+hK)\mathcal {G}^n_1 +C(1+T\beta hK^2)(1+h^4K^4)\mathcal {G}^n_2 +C\mathcal {G}^n_3. \end{aligned} \end{aligned}$$

(6.9)

Now we are able to get the expected inequality (4.9) by summing up all estimates for the terms $\Pi ^n_1$ through $\Pi ^n_6$.