Analysis of Discontinuous Galerkin Methods with Upwind-Biased Fluxes for One Dimensional Linear Hyperbolic Equations with Degenerate Variable Coefficients

Abstract

In this paper, we analyze the discontinuous Galerkin method with upwind-biased numerical fluxes for one dimensional linear hyperbolic equations with degenerate variable coefficients. The $L^2$-stability is obtained by the choice of upwind-biased fluxes which could provide more flexible numerical viscosity. Furthermore, we construct some new piecewise global projections and present proofs of unique existence and optimal approximation properties. Then the optimal error estimates are derived by the benefits of the specially designed projections, essentially following the energy analysis. Numerical experiments are given which confirm the sharpness of the theoretical results.

Appendices

Appendix A Explanation of (3.11)

Under the modulo $N$ operation, we have $\overline{(N)}_{\scriptscriptstyle N}=\overline{(0)}_{\scriptscriptstyle N}=0$ and $\overline{(N+1)}_{\scriptscriptstyle N}=\overline{(1)}_{\scriptscriptstyle N}=1$. Noting that $\lambda _{-\frac{1}{2}}=\lambda _{N-\frac{1}{2}}$ and $\lambda _{\frac{1}{2}}=\lambda _{N+\frac{1}{2}}$, we can rewrite $\varTheta _{\mathbb {b}\!^{+}}$, $\mathbb {A}_{\mathbb {b}\!^{+}}$ as $\varTheta _{\mathbb {b}\!^{+}}=\text {diag}(\tilde{\theta }_{\beta +1{\scriptscriptstyle -\frac{1}{2}}}, \ldots ,\tilde{\theta }_{\beta +|\mathbb {b}\!^{+}|{\scriptscriptstyle -\frac{1}{2}}})$ and

$$\begin{aligned} \mathbb {A}_{\mathbb {b}\!^{+}}=\left( \begin{array}{ccccc} \theta _{\beta +1{\scriptscriptstyle -\frac{1}{2}}}&{}\tilde{\theta }_{\beta +1{\scriptscriptstyle -\frac{1}{2}}}(-1)^k\\ &{}\theta _{\beta \!+\!2\!-\!\frac{1}{2}} &{}\tilde{\theta }_{\beta \!+\!2\!-\!\frac{1}{2}}(-1)^k\\ &{}&{}\ddots &{}\ddots \\ &{}&{}&{}\theta _{\beta +|\mathbb {b}\!^{+}|\!-\!\frac{3}{2}} &{}\tilde{\theta }_{\beta +|\mathbb {b}\!^{+}|\!-\!\frac{3}{2}}(-1)^k\\ &{}&{}&{}&{}\theta _{\beta +|\mathbb {b}\!^{+}|{\scriptscriptstyle -\frac{1}{2}}} \end{array}\right) . \end{aligned}$$

Assume $\mathbb {F}_+$ is the cofactor matrix of $\mathbb {A}_{\mathbb {b}\!^{+}}$. Owing to the special form of $\mathbb {A}_{\mathbb {b}\!^{+}}$, the entries of $\mathbb {F}_+$ can be written as

$$\begin{aligned} (\mathbb {F}_+)_{ij}=\left\{ \begin{array}{lll} &{}\displaystyle (-\!1)^{i\!+\!j}\!\! \left( \prod _{r=\beta +1}^{\beta +j-1} \!\!\theta _{\overline{(r)}_{\scriptscriptstyle N}\!-\!\frac{1}{2}}\right) \left( \prod _{r=\beta +j}^{\beta +i-1} \!\!(-1)^k\tilde{\theta }_{\overline{(r)}_{\scriptscriptstyle N}\!-\!\frac{1}{2}}\right) \left( \prod _{r=\beta +i+1}^{\beta +|\mathbb {b}\!^{+}|} \!\!\theta _{\overline{(r)}_{\scriptscriptstyle N}\!-\!\frac{1}{2}}\right) , &{}\quad j\!\le \! i,\\ &{}\displaystyle 0,&{}\quad j\!>\!i, \end{array}\right. \end{aligned}$$

where $1\le i,j\le |\mathbb {b}\!^{+}|$. The transposition of $\mathbb {F}_+$ can be obtained by the interchange of indexes $i$ and $j$. Along with $|\mathbb {A}_{\mathbb {b}\!^{+}}| =\prod _{r=\beta +1}^{\beta +|\mathbb {b}\!^{+}|}\theta _{\overline{(r)}_{\scriptscriptstyle N}\!-\!\frac{1}{2}}$, we can derive

$$\begin{aligned} (\mathbb {A}^{-1}_{\mathbb {b}\!^{+}})_{ij}=\left\{ \begin{array}{lll} &{}\displaystyle (-1)^{i+j}\left( \prod _{r=\beta +i}^{\beta +j-1} \lambda _{\overline{(r)}_{\scriptscriptstyle N}\!-\!\frac{1}{2}}\right) \frac{1}{\theta _{\overline{(\beta \!+\!j)}_{\scriptscriptstyle N}\!-\!\frac{1}{2}}}, &{}\quad j\ge i,\\ &{}\displaystyle 0,&{}\quad j<i, \end{array}\right. \end{aligned}$$

where $1\le i,j\le |\mathbb {b}\!^{+}|$. Then, with considering that $\mathbb {M}_+=\mathbb {A}^{-1}_{\mathbb {b}\!^{+}}\varTheta _{\mathbb {b}\!^{+}}$, the conclusion of (3.11) can be finally obtained.

Now we consider the estimates of $\Vert \mathbb {M}_+\Vert _1$ and $\Vert \mathbb {M}_+\Vert _\infty$. For any $1\le i\le |\mathbb {b}\!^{+}|$ and $1\le j\le |\mathbb {b}\!^{+}|$, there holds the following two estimates:

$$\begin{aligned} \sum _{i=1}^{|\mathbb {b}\!^{+}|}|(\mathbb {M}_+)_{ij}|= & {} |\lambda _{\overline{(\beta +j)}_{\scriptscriptstyle N}\!-\!\frac{1}{2}}| +|\lambda _{\overline{(\beta +j-1)}_{\scriptscriptstyle N}\!-\!\frac{1}{2}} \cdot \lambda _{\overline{(\beta +j)}_{\scriptscriptstyle N}\!-\!\frac{1}{2}}|+\cdots + |\prod _{r=\beta +1}^{\beta +j}\lambda _{\overline{(r)}_{\scriptscriptstyle N}\!-\!\frac{1}{2}}|\\\le & {} \lambda _{\scriptscriptstyle +}+\lambda _{\scriptscriptstyle +}^2+\cdots +\lambda _{\scriptscriptstyle +}^{j},\\ \sum _{j=1}^{|\mathbb {b}\!^{+}|}|(\mathbb {M}_+)_{ij}|= & {} |\lambda _{\overline{(\beta +i)}_{\scriptscriptstyle N}\!-\!\frac{1}{2}}|+|\lambda _{\overline{(\beta +i)}_{\scriptscriptstyle N}\!-\!\frac{1}{2}} \cdot \lambda _{\overline{(\beta +i+1)}_{\scriptscriptstyle N}\!-\!\frac{1}{2}}|+\cdots + |\prod _{r=\beta +i}^{\beta +|\mathbb {b}\!^{+}|}\lambda _{\overline{(r)}_{\scriptscriptstyle N}\!-\!\frac{1}{2}}|\\\le & {} \lambda _{\scriptscriptstyle +}+\lambda _{\scriptscriptstyle +}^2+\cdots +\lambda _{\scriptscriptstyle +}^{|\mathbb {b}\!^{+}|-i+1}. \end{aligned}$$

Hence the estimates of $\Vert \mathbb {M}_+\Vert _1$ and $\Vert \mathbb {M}_+\Vert _\infty$ can be obtained.

For readers’ benefits, we give a simple example about the inverse matrix that holds the form like $\mathbb {A}_{\mathbb {b}\!^{+}}$. We set a matrix $\mathcal {A}_{5\times 5}$ as

$$\begin{aligned} \mathcal {A}=\left( \begin{array}{ccccc} a&{}\quad a'\\ {} &{}\quad b&{}\quad b'\\ &{}&{}\quad c&{}\quad c'\\ &{}&{}&{}\quad d&{}\quad d'\\ &{}&{}&{}&{}\quad e \end{array}\right) . \end{aligned}$$

Assume $\mathcal {F}$ is the cofactor matrix of $\mathcal {A}$ and $|\mathcal {A}|=abcde\ne 0$, then we can derive

$$\begin{aligned} \mathcal {A}^{-1}=\frac{\mathcal {F}^\top }{|\mathcal {A}|} =\frac{1}{|\mathcal {A}|} \left( \begin{array}{ccccc} bcde&{}\quad -a'cde&{}\quad a'b'de&{}\quad -a'b'c'e&{}\quad a'b'c'd'\\ &{}\quad acde&{}\quad -ab'de&{}\quad ab'c'e&{}\quad -ab'c'd'\\ &{}&{}\quad abde&{}\quad -abc'e&{}\quad abc'd'\\ &{}&{}&{}\quad abce&{}\quad -abcd'\\ 0&{}&{}&{}&{}\quad abcd \end{array}\right) . \end{aligned}$$

With observing example of $\mathcal {A}^{-1}$, we can easily testify above formula of $\mathbb {A}_{\mathbb {b}\!^{+}}$.

Similarly, we can also write $\mathbb {M}_-$ and show estimates for $\Vert \mathbb {M}_-\Vert _1$ and $\Vert \mathbb {M}_-\Vert _\infty$, which are almost the same with the situation of $\mathbb {M}_-$ only with minor difference. Here we do not present the details to save space.

Appendix B Explanation of Remark 3.3

When $a(x)$ keeps its sign on $I$, we can use projection $P_h^\star u$ presented in [15] in error estimates. Without loss of generality, we assume $a(x)>0,\ x\in I$. The definition of $P_h^\star u$ is as follows. For $u\in H^1(\mathcal {I}_h)$, $P_h^\star u$ is defined as the element of $V_h^k$ that satisfies

$$\begin{aligned} \int _{I_j}(P_h^\star u)\varphi \text {d}x= & {} \int _{I_j}u\varphi \text {d}x, \quad \forall \varphi \in P^{k-1}(I_j),\\ \widehat{(P_h^\star u)}_{j{\scriptscriptstyle +\frac{1}{2}}}= & {} \hat{u}_{j{\scriptscriptstyle +\frac{1}{2}}},\qquad \text {at}\ x_{j{\scriptscriptstyle +\frac{1}{2}}}, \end{aligned}$$

where $\hat{u}_{j{\scriptscriptstyle +\frac{1}{2}}}=\theta _{j{\scriptscriptstyle +\frac{1}{2}}}u^-_{j{\scriptscriptstyle +\frac{1}{2}}}+\tilde{\theta }_{j{\scriptscriptstyle +\frac{1}{2}}}u^+_{j{\scriptscriptstyle +\frac{1}{2}}}$ and $\widehat{(P_h^\star u)}_{j{\scriptscriptstyle +\frac{1}{2}}}=\theta _{j{\scriptscriptstyle +\frac{1}{2}}}(P_h^\star u)^-_{j{\scriptscriptstyle +\frac{1}{2}}}+\tilde{\theta }_{j{\scriptscriptstyle +\frac{1}{2}}}(P_h^\star u)^+_{j{\scriptscriptstyle +\frac{1}{2}}}$ for $\forall j\in \mathbb {Z}^+_{\scriptscriptstyle N}$.

Different from the uniform flux parameter $\theta$ in the study of [6, 15], here $\theta _{j{\scriptscriptstyle +\frac{1}{2}}}>\frac{1}{2},\ j\in \mathbb {Z}^+_{\scriptscriptstyle N}$ can be different values at cell interfaces. However, the conclusion of Lemma 3.2 in [6] still stands. For simplicity, we only give a brief explanation about the difference in the proof of current case.

If we follow the proof line of Lemma 3.2 in [6] and denote $\psi =P_h^-u-u$, we will arrive at the following linear system problem:

$$\begin{aligned} \mathbb {A}_N\alpha _N=\varTheta _N \psi _N, \end{aligned}$$

where $\alpha _N=(\alpha _{1,k},\ldots ,\alpha _{N,k})^\top$, $\psi _N=(\psi ^+_{\frac{1}{2}},\ldots ,\psi ^+_{N+\frac{1}{2}})^\top$, $\varTheta _N=\text {diag}(\tilde{\theta }_{\frac{1}{2}}, \ldots ,\tilde{\theta }_{N+\frac{1}{2}})$ and

$$\begin{aligned} \mathbb {A}_{N}=\left( \begin{array}{ccccc} \theta _{\frac{3}{2}}&{}\tilde{\theta }_{\frac{3}{2}}(-1)^k\\ &{}\theta _{\frac{5}{2}}&{}\tilde{\theta }_{\frac{5}{2}}(-1)^k\\ &{}&{}\ddots &{}\ddots \\ &{}&{}&{}\theta _{N{\scriptscriptstyle -\frac{1}{2}}}&{}\tilde{\theta }_{N{\scriptscriptstyle -\frac{1}{2}}}(-1)^k\\ \tilde{\theta }_{N{\scriptscriptstyle +\frac{1}{2}}}(-1)^k&{}&{}&{}&{}\theta _{N{\scriptscriptstyle +\frac{1}{2}}} \end{array}\right) . \end{aligned}$$

We can easily testify that

$$\begin{aligned} |\mathbb {A}_N|= & {} \prod _{j=1}^{N}\theta _{j{\scriptscriptstyle +\frac{1}{2}}}+(-1)^{(N+1)}\prod _{j=1}^{N}\left( \tilde{\theta }_{j{\scriptscriptstyle +\frac{1}{2}}}(-1)^k\right) \\= & {} \left( \prod _{j=1}^{N}\theta _{j{\scriptscriptstyle +\frac{1}{2}}}\right) \left( 1+(-1)^{(N+1)}\prod _{j=1}^{N}\lambda _{j{\scriptscriptstyle +\frac{1}{2}}}\right) , \end{aligned}$$

where $\lambda _{j{\scriptscriptstyle +\frac{1}{2}}}=\tilde{\theta }_{j{\scriptscriptstyle +\frac{1}{2}}}(-1)^k/\theta _{j{\scriptscriptstyle +\frac{1}{2}}}$. Since $\theta _{j{\scriptscriptstyle +\frac{1}{2}}}>\frac{1}{2},\ j\in \mathbb {Z}^+_{\scriptscriptstyle N}$, we have $0\le |\prod ^N_{j=1}\lambda _{j{\scriptscriptstyle +\frac{1}{2}}}|<1$, hence there holds $|\mathbb {A}_N|\ne 0$, which means $\mathbb {A}_N$ is invertible.

Denoting by $\mathbb {F}_N$ the cofactor matrix of $\mathbb {A}_N$, the transposition of $\mathbb {F}_N$ can be expressed as

$$\begin{aligned} (\mathbb {F}_N^\top )_{i,\overline{(i+m)}_{\scriptscriptstyle N}}=(-1)^m \left( \prod _{r=i+m+1}^{i+N-1}\theta _{\overline{(r)}_{\scriptscriptstyle N}{\scriptscriptstyle +\frac{1}{2}}}\right) \left( \prod _{r=i+N}^{i+N+m-1}(-1)^k \tilde{\theta }_{\overline{(r)}_{\scriptscriptstyle N}{\scriptscriptstyle +\frac{1}{2}}}\right) , \end{aligned}$$

where $i=1,2,\ldots ,N$ and $m=0,1,\ldots ,N-1$. Then we can derive

$$\begin{aligned} (\mathbb {A}_N^{-1})_{i,\overline{(i+m)}_{\scriptscriptstyle N}}= \frac{\varPsi \cdot (-1)^m}{\theta _{\overline{(i+m)}_{\scriptscriptstyle N}{\scriptscriptstyle +\frac{1}{2}}}} \left( \prod _{r=i+N}^{i+N+m-1}\lambda _{\overline{(r)}_{\scriptscriptstyle N}{\scriptscriptstyle +\frac{1}{2}}}\right) , \end{aligned}$$

where $\varPsi =\Big (1+(-1)^{(N+1)}\prod _{j=1}^{N} \lambda _{j{\scriptscriptstyle +\frac{1}{2}}}\Big )^{-1}$. Finally, we can obtain

$$\begin{aligned} (\mathbb {A}_N^{-1}\varTheta _N)_{i,\overline{(i+m)}_{\scriptscriptstyle N}}= {\varPsi \cdot (-1)^{m+k}} \left( \prod _{r=i+N}^{i+N+m}\lambda _{\overline{(r)}_{\scriptscriptstyle N}{\scriptscriptstyle +\frac{1}{2}}}\right) . \end{aligned}$$

Notice that $\varPsi$ is a bounded constant independent of mesh size $h$, then the optimal approximation property of projection can be finally obtained by following the proof line of Lemma 3.2 in [6].

Here we also give an example matrix. We set a matrix $\mathcal {A}$ as

$$\begin{aligned} \mathcal {A}=\left( \begin{array}{ccccc} a&{}\quad a'\\ {} &{}\quad b&{}\quad b'\\ &{}&{}\quad c&{}\quad c'\\ &{}&{}&{}\quad d&{}\quad d'\\ e'&{}&{}&{}&{}\quad e \end{array}\right) . \end{aligned}$$

Assume $|\mathcal {A}|\ne 0$, then we can derive

$$\begin{aligned} \mathcal {A}^{-1}=\frac{1}{|\mathcal {A}|} \left( \begin{array}{ccccc} bcde&{}\quad -cdea'&{}\quad dea'b'&{}\quad -ea'b'c'&{}\quad a'b'c'd'\\ b'c'd'e'&{}\quad cdea&{}\quad -deab'&{}\quad eab'c'&{}\quad -ab'c'd'\\ -bc'd'e'&{}\quad c'd'e'a'&{}\quad deab&{}\quad -eabc'&{}\quad abc'd'\\ bcd'e'&{}\quad -cd'e'a'&{}\quad d'e'a'b'&{}\quad eabc&{}\quad -abcd'\\ -bcde'&{}\quad cde'a'&{}\quad -de'a'b'&{}\quad e'a'b'c'&{}\quad abcd \end{array}\right) . \end{aligned}$$

With observing example of $\mathcal {A}^{-1}$, we can easily testify above formula of $\mathbb {A}_N^{-1}$.