Local Discontinuous Galerkin Methods with Generalized Alternating Numerical Fluxes for Two-dimensional Linear Sobolev Equation

Abstract

In this paper we present an efficient and high-order numerical method to solve two-dimensional linear Sobolev equations, which is based on the local discontinuous Galerkin (LDG) method with the upwind-biased fluxes and generalized alternating fluxes. A weak stability is given for both schemes, and a strong stability is established if the initial solutions exactly satisfy the elemental discontinuous Galerkin discretization. Moreover, the sharp error estimate in $L^2$-norm is established, by an elaborate application of the generalized Gauss–Radau projection. A fully-discrete LDG scheme is also considered, where the third-order explicit TVD Runge–Kutta algorithm is adopted. Finally some numerical experiments are given.

Appendix

In this section some detailed proofs for main conclusions are presented.

1.1 Proof of Lemma 5.2

By some certain linear combinations with three equations from (5.1a) to (5.1c), we can yield that

$$\begin{aligned} \Big \langle \mathbb {D}_{\ell +1}{\varvec{z}}_{uq}^n, {\varvec{v}}_{uq} \Big \rangle =\frac{\tau }{\ell +1} \Big \langle \mathbb {D}_{\ell }{\varvec{z}}_{wp}^n+\mathbb {D}_{\ell }{\varvec{\varPsi }}_{uq}^n, {\varvec{v}}_{uq} \Big \rangle , \quad \forall \,{\varvec{v}}_{uq}\in (V_h)^3. \end{aligned}$$

(8.1)

This relationship among the difference of stage solutions will be used many times

Below we take $\ell =2$ as an example to prove this lemma. By taking ${\varvec{v}}_{uq}=\mathbb {D}_3{\varvec{z}}_{uq}^n$ in (8.1), we have

$$\begin{aligned} \begin{aligned} 3\Vert \mathbb {D}_3{\varvec{z}}_{uq}^n\Vert ^2_{\mu } =&\; \tau \Big \langle \mathbb {D}_2{\varvec{z}}_{wp}^n+\mathbb {D}_2{\varvec{\varPsi }}_{uq}^n, \mathbb {D}_3{\varvec{z}}_{uq}^n \Big \rangle \le \Vert \mathbb {D}_3{\varvec{z}}_{uq}^n\Vert ^2_{\mu } +\frac{\tau ^2}{2}\Vert \mathbb {D}_2{\varvec{z}}_{wp}^n\Vert _{\mu }^2 +\frac{\tau ^2}{2}\Vert \mathbb {D}_2{\varvec{\varPsi }}_{uq}^n\Vert _{\mu }^2, \end{aligned} \end{aligned}$$

which implies that

$$\begin{aligned} \Vert \mathbb {D}_3{\varvec{z}}_{uq}^n\Vert ^2_{\mu } \le \frac{\tau ^2}{4}\Vert \mathbb {D}_2{\varvec{z}}_{wp}^n\Vert _{\mu }^2 +\frac{\tau ^2}{4}\Vert \mathbb {D}_2{\varvec{\varPsi }}_{uq}^n\Vert _{\mu }^2. \end{aligned}$$

(8.2)

Take ${\varvec{v}}_{wp}=\mathbb {D}_2{\varvec{z}}_{wp}^n$ in (5.1d) for $\ell =0,1,2$. A linear combination gives the identity

$$\begin{aligned}\Vert \mathbb {D}_2{\varvec{z}}_{wp}^n\Vert _{\mu }^2= \Lambda _1+\Lambda _2+\Lambda _3+\Lambda _4, \end{aligned}$$

where each term is given and estimated as follows:

$$\begin{aligned} \Lambda _1=&\; \sum _{\kappa =1}^2 c_{\kappa }\mathcal {H}_{\kappa }^{\theta _{\kappa }}(\mathbb {D}_2u^n,\mathbb {D}_2w^n) \le 2M ^2|c|^2h^{-2}\Vert \mathbb {D}_2u^n\Vert ^2 + \frac{1}{4}\Vert \mathbb {D}_2{\varvec{z}}_{wp}^n\Vert _{\mu }^2, \end{aligned}$$

(8.3a)

$$\begin{aligned} \Lambda _2=&\; -\varepsilon \sum _{\kappa =1}^2 \mathcal {H}_{\kappa }^{1-\theta _{\kappa }}(\mathbb {D}_2q_{\kappa }^n,\mathbb {D}_2w^n) \le 2M ^2\varepsilon ^2 h^{-2}\sum _{\kappa =1}^2\Vert \mathbb {D}_2q_{\kappa }^n\Vert ^2 +\frac{1}{4}\Vert \mathbb {D}_2{\varvec{z}}_{wp}^n\Vert ^2_{\mu }, \end{aligned}$$

(8.3b)

$$\begin{aligned} \Lambda _3=&\; -\mu \sum _{\kappa =1}^2 \Big [ \mathcal {H}_{\kappa }^{1-\theta _{\kappa }}(\mathbb {D}_2p_{\kappa }^n,\mathbb {D}_2w^n) + \mathcal {H}_{\kappa }^{\theta _{\kappa }}(\mathbb {D}_2w^n,\mathbb {D}_2p_{\kappa }^n) \Big ] =0,\end{aligned}$$

(8.3c)

$$\begin{aligned} \Lambda _4=&\; \Big \langle \mathbb {D}_2{\varvec{\varPsi }}_{wp}^n, \mathbb {D}_2{\varvec{z}}_{wp}^n \Big \rangle \le \Vert \mathbb {D}_2{\varvec{\varPsi }}_{wp}^n\Vert _{\mu }^2+\frac{1}{4}\Vert \mathbb {D}_2{\varvec{z}}_{wp}^n\Vert _{\mu }^2. \end{aligned}$$

(8.3d)

In the above process, the boundedness property (Lemma 3.3) and the skew-symmetrical property (Lemma 3.1) of DG discretization are used. Finally, collecting up the above conclusion completes the proof of this lemma.

1.2 Proof of (5.9)

Firstly in (5.4) we take $v=-\varepsilon q_\kappa ^{n,\ell }$ and $v=-\mu p_\kappa ^{n,\ell }$. Then in (5.1d) we take ${\varvec{v}}_{wp}=(u^{n,\ell },0,0)$. Summing up the resulting equations, we have

$$\begin{aligned} \mathcal {R}_1^n= \sum _{\ell =0}^2d_{\ell } \Big [ \mathcal {R}_{11}^{n,\ell }+\mathcal {R}_{12}^{n,\ell }+\mathcal {R}_{13}^{n,\ell }+ \mathcal {R}_{14}^{n,\ell }+\mathcal {R}_{15}^{n,\ell } \Big ]\tau , \end{aligned}$$

(8.4)

Each term in (8.4) is given and bounded as follows:

$$\begin{aligned} \mathcal {R}_{11}^{n,\ell }=&\; \sum _{\kappa =1}^2 c_{\kappa }\mathcal {H}_{\kappa }^{\gamma _{\kappa }}(u^{n,\ell },u^{n,\ell }) = -\frac{1}{2}\sum _{\kappa =1}^2 c_{\kappa }(2\gamma _{\kappa }-1) \Vert [\![u^{n,\ell }]\!]\Vert ^2_{\varGamma _{h}^{\kappa }},\end{aligned}$$

(8.5a)

$$\begin{aligned} \mathcal {R}_{12}^{n,\ell }=&\; -\varepsilon \sum _{\kappa =1}^2 \Big [ \mathcal {H}_{\kappa }^{1-\theta _{\kappa }}(q_{\kappa }^{n,\ell }, u^{n,\ell }) + \mathcal {H}_{\kappa }^{\theta _{\kappa }}(u^{n,\ell }, q_{\kappa }^{n,\ell }) \Big ] =0,\end{aligned}$$

(8.5b)

$$\begin{aligned} \mathcal {R}_{13}^{n,\ell }=&\; -\mu \sum _{\kappa =1}^2 \Big [ \mathcal {H}_{\kappa }^{1-\theta _{\kappa }}(p_{\kappa }^{n,\ell }, u^{n,\ell }) + \mathcal {H}_{\kappa }^{\theta _{\kappa }}(u^{n,\ell }, p_{\kappa }^{n,\ell }) \Big ] =0,\end{aligned}$$

(8.5c)

$$\begin{aligned} \mathcal {R}_{14}^{n,\ell }=&\; -\varepsilon \sum _{\kappa =1}^2 \Big ( q_{\kappa }^{n,\ell }, q_{\kappa }^{n,\ell } \Big ),\end{aligned}$$

(8.5d)

$$\begin{aligned} \mathcal {R}_{15}^{n,\ell }=&\; \sum _{\kappa =1}^2 \Big ( \Phi _{\kappa }^{n,\ell }, \varepsilon q_\kappa ^{n,\ell }+\mu p_{\kappa }^{n,\ell } \Big ) +\Big ( \varPsi _w^{n,\ell }, u^{n,\ell } \Big )+\Big \langle {\varvec{\varPsi }}_{uq}^{n,\ell }, {\varvec{z}}_{uq}^{n,\ell } \Big \rangle . \end{aligned}$$

(8.5e)

The first one is resulted from Lemma 3.2, and the next two are resulted from Lemma 3.1. Till now we have obtained (5.9).

1.3 Proof of (5.10)

In (8.1) we take ${\varvec{v}}_{uq}=(\mathbb {D}_2 u^n,0,0)$ for $\ell =1$ and ${\varvec{v}}_{uq}=(\mathbb {D}_1 u^n,0,0)$ for $\ell =2$. Noticing (5.1d) with the same test functions, we obtain the identity

$$\begin{aligned} \mathcal {R}_2^n= \mathcal {R}_{21}^n\tau +\mathcal {R}_{22}^n\tau +\mathcal {R}_{23}^n\tau +\mathcal {R}_{24}^n\tau +\Theta _{21}^n\tau , \end{aligned}$$

(8.6)

where

$$\begin{aligned} \mathcal {R}_{21}^n=&\; \sum _{\kappa =1}^2c_{\kappa } \Big [\mathcal {H}_{\kappa }^{\gamma _{\kappa }}(\mathbb {D}_1u^n,\mathbb {D}_2u^n) + \mathcal {H}_{\kappa }^{\gamma _{\kappa }}(\mathbb {D}_2u^n,\mathbb {D}_1u^n) \Big ], \end{aligned}$$

(8.7a)

$$\begin{aligned} \mathcal {R}_{22}^n=&\; -\varepsilon \sum _{\kappa =1}^2 \Big [ \mathcal {H}_{\kappa }^{1-\theta _{\kappa }} (\mathbb {D}_1q_{\kappa }^n,\mathbb {D}_2u^n) +\mathcal {H}_{\kappa }^{1-\theta _{\kappa }} (\mathbb {D}_2q_{\kappa }^n,\mathbb {D}_1u^n) \Big ],\end{aligned}$$

(8.7b)

$$\begin{aligned} \mathcal {R}_{23}^n=&\; -\mu \sum _{\kappa =1}^2 \Big [ \mathcal {H}_{\kappa }^{1-\theta _{\kappa }} (\mathbb {D}_1p_{\kappa }^n,\mathbb {D}_2u^n) +\mathcal {H}_{\kappa }^{1-\theta _{\kappa }} (\mathbb {D}_2p_{\kappa }^n,\mathbb {D}_1u^n) \Big ],\end{aligned}$$

(8.7c)

$$\begin{aligned} \mathcal {R}_{24}^n=&\; \mu \sum _{\kappa =1}^2 \Big [ \Big ( \mathbb {D}_1p_{\kappa }^n, \mathbb {D}_2q_{\kappa }^n \Big ) +\Big ( \mathbb {D}_2p_{\kappa }^n, \mathbb {D}_1q_{\kappa }^n \Big ) \Big ]. \end{aligned}$$

(8.7d)

Each term can be estimated one by one.

Combining Lemma 3.2, Young’s inequality and the inverse property, we yield that

$$\begin{aligned} \begin{aligned} \mathcal {R}_{21}^n\le&\; \sum _{\kappa =1}^2 c_{\kappa }(2\gamma _{\kappa }-1) \Big [ 2\Vert [\![\mathbb {D}_2u^n]\!]\Vert ^2_{\varGamma _{h}^{\kappa }} + \frac{1}{8}\Vert [\![\mathbb {D}_1u^n]\!]\Vert ^2_{\varGamma _{h}^{\kappa }} \Big ]\\ \le&\; 4\nu ^2 h^{-1}\sum _{\kappa =1}^2 c_{\kappa }(2\gamma _{\kappa }-1) \Vert \mathbb {D}_2{\varvec{z}}_{uq}^n\Vert ^2_{\mu } + \sum _{\ell =0}^1\sum _{\kappa =1}^2 \frac{1}{4} d_{\ell }c_{\kappa }(2\gamma _{\kappa }-1) \Vert [\![u^{n,\ell }]\!]\Vert ^2_{\varGamma _{h}^{\kappa }}. \end{aligned} \end{aligned}$$

(8.8)

Taking $v=-\varepsilon \mathbb {D}_1q_{\kappa }^n$ and $v=-\varepsilon \mathbb {D}_2q_{\kappa }^n$ in (5.4) with $\ell =0,1,2$, and adding the resulted identities into $\mathcal {R}_{22}^n$ with suitable weight, we can get

$$\begin{aligned} \mathcal {R}_{22}^n=\mathcal {R}_{221}^n+\mathcal {R}_{222}^n+\mathcal {R}_{223}^n+\Theta _{22}^n, \end{aligned}$$

(8.9)

where each term is given and/or estimated as follows:

$$\begin{aligned} \mathcal {R}_{221}^n=&\; -\varepsilon \sum _{\kappa =1}^2 \Big [ \mathcal {H}_{\kappa }^{1-\theta _{\kappa }}(\mathbb {D}_1q_{\kappa }^n,\mathbb {D}_2u^n) + \mathcal {H}_{\kappa }^{\theta _{\kappa }}(\mathbb {D}_2u^n,\mathbb {D}_1q_{\kappa }^n) \Big ] =0, \end{aligned}$$

(8.10a)

$$\begin{aligned} \mathcal {R}_{222}^n=&\; -\varepsilon \sum _{\kappa =1}^2 \Big [ \mathcal {H}_{\kappa }^{1-\theta _{\kappa }}(\mathbb {D}_2q_{\kappa }^n,\mathbb {D}_1u^n) + \mathcal {H}_{\kappa }^{\theta _{\kappa }}(\mathbb {D}_1u^n,\mathbb {D}_2q_{\kappa }^n) \Big ]=0,\end{aligned}$$

(8.10b)

$$\begin{aligned} \mathcal {R}_{223}^n=&\; -2\varepsilon \sum _{\kappa =1}^2 \Big [ \Big ( \mathbb {D}_2q_{\kappa }^n, \mathbb {D}_1q_{\kappa }^n \Big ) - \Big ( \mathbb {D}_2\Phi _\kappa ^n, \mathbb {D}_1q_{\kappa }^n \Big ) \Big ]. \end{aligned}$$

(8.10c)

The first two conclusions are directly resulted from Lemma 3.1. Taking $v=-2\varepsilon \mathbb {D}_1q_\kappa$ in (5.4) with $\ell =0,1,2$, we can get that

$$\begin{aligned} \begin{aligned} \mathcal {R}_{223}^n=&\; 2\varepsilon \sum _{\kappa =1}^2 \mathcal {H}_{\kappa }^{\theta _{\kappa }} (\mathbb {D}_2u^n,\mathbb {D}_1q_{\kappa }^n) \le 2M \varepsilon h^{-1}\sum _{\kappa =1}^2 \Vert \mathbb {D}_2u^n\Vert \Vert \mathbb {D}_1q_{\kappa }^n\Vert \\ \le&\; \frac{1}{4\tau }\Vert \mathbb {D}_2u^n\Vert ^2 +8M ^2\varepsilon ^2h^{-2}\tau \Vert \mathbb {D}_1{\varvec{q}}^n\Vert ^2 \le \frac{1}{4\tau }\Vert \mathbb {D}_2u^n\Vert ^2+ 16M^2\varepsilon ^2 h^{-2}\sum _{\ell =0}^1 d_{\ell }\Vert {\varvec{q}}^{n,\ell }\Vert ^2\tau . \end{aligned} \end{aligned}$$

(8.11)

Collecting up the above analysis yields the estimate to $\mathcal {R}_{22}^n$.

Along the same line as before, it is easy to get that

$$\begin{aligned} \mathcal {R}_{23}^n+\mathcal {R}_{24}^n=\Theta _{23}^n. \end{aligned}$$

(8.12)

Finally, summing up the above conclusions into (8.6), and noticing the definitions of $\lambda _\mathrm{c},\lambda _\mathrm{d},\mathcal {S}_\mathrm{c}$ and $\mathcal {S}_\mathrm{d}$, we can obtain (5.10).