7 Examples

7.1 Semi-discrete Navier-Stokes equations

7.1.1 Transformation to a more handy form

By scalings and state transforms, we find that the coefficients of the spatially discretized Navier-Stokes equations $$\begin{align*} \{\lambda \mathcal E - \mathcal A\} &= \left\{\lambda \begin{bmatrix} M & 0 \\ 0 & 0 \end{bmatrix} - \begin{bmatrix} A & B^H \\ B & 0 \end{bmatrix} \right\} \\ & \backsim \begin{bmatrix} M^{-1/2} & 0 \\ 0 & I \end{bmatrix} \left\{\lambda \begin{bmatrix} M & 0 \\ 0 & 0 \end{bmatrix} - \begin{bmatrix} A & B^H \\ B & 0 \end{bmatrix} \right\} \begin{bmatrix} M^{-1/2} & 0 \\ 0 & I \end{bmatrix} \\ & \backsim \begin{bmatrix} Q^H & 0 \\ 0 & I \end{bmatrix} \left\{\lambda \begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix} - \begin{bmatrix} M^{-1/2}AM^{-1/2} & M^{-1/2}B^H \\ B M^{-1/2} & 0 \end{bmatrix} \right\} \begin{bmatrix} Q & 0 \\ 0 & I \end{bmatrix} \\ & \backsim \begin{bmatrix} I & 0 \\ 0 & R^{-H} \end{bmatrix} \left\{ \lambda \begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix} - \begin{bmatrix} M^{-1/2}AM^{-1/2} & \begin{bmatrix} R \\ 0 \end{bmatrix} \\ \begin{bmatrix}R^H & 0\end{bmatrix} & 0 \end{bmatrix} \right \} \begin{bmatrix} I & 0 \\ 0 & R^{-1} \end{bmatrix} \\ & \quad = \left\{\lambda \begin{bmatrix} I_{n_1} & 0 & 0 \\ 0 & I_{n_2} & 0 \\ 0 & 0 & 0\end{bmatrix} - \begin{bmatrix} A_{11} & A_{12} & I_{n_1} \\ A_{21} & A_{22} & 0 \\ I_{n_1} & 0 & 0\end{bmatrix} \right\}. \end{align*}$$ where we have used a QR-decomposition: $$M^{-1/2}B^H=Q\begin{bmatrix}R \\ 0\end{bmatrix}$$ with unitary $Q$ and invertible $R$ in the third step.

7.1.2 Local Characteristic Values

Next we derive the local characteristic as in Theorem 4.1.

We compute the subspaces as defined in (4.5):

Matrix	as the basis of/computed as
$T=\begin{bmatrix} 0 \\ 0 \\I_{n_1} \end{bmatrix}$	$\operatorname{kernel}\begin{bmatrix} I_{n_1} & 0 & 0 \\ 0 & I_{n_2} & 0 \\ 0 & 0 & 0\end{bmatrix}$
$Z=\begin{bmatrix} 0 \\ 0 \\I_{n_1} \end{bmatrix}$	$\operatorname{corange}\begin{bmatrix} I_{n_1} & 0 & 0 \\ 0 & I_{n_2} & 0 \\ 0 & 0 & 0\end{bmatrix}$
$T'=\begin{bmatrix} I_{n_1} & 0 \\ 0 & I_{n_2} \\ 0 & 0 \end{bmatrix}$	$\operatorname{cokernel}\begin{bmatrix} I_{n_1} & 0 & 0 \\ 0 & I_{n_2} & 0 \\ 0 & 0 & 0\end{bmatrix}$
$Z^HAT=0_{n_1}$	$\begin{bmatrix} 0 \\ 0 \\I_{n_1} \end{bmatrix}^H\begin{bmatrix} A_{11} & A_{12} & I_{n_1} \\ A_{21} & A_{22} & 0 \\ I_{n_1} & 0 & 0\end{bmatrix}\begin{bmatrix} 0 \\ 0 \\I_{n_1} \end{bmatrix}$
$V=I_{n_1}$	$\operatorname{corange}(Z^HAT) = \operatorname{kernel}0_{n_1}^H\phantom{\begin{bmatrix} 0 \\ I_1 \end{bmatrix}}$
$Z^HAT'=\begin{bmatrix} I_{n_1} & 0_{n_1\times n_2}\end{bmatrix}$	$\begin{bmatrix} 0 \\ 0 \\I_{n_1} \end{bmatrix}^H\begin{bmatrix} A_{11} & A_{12} & I_{n_1} \\ A_{21} & A_{22} & 0 \\ I_{n_1} & 0 & 0\end{bmatrix}\begin{bmatrix} I_{n_1} & 0 \\ 0 & I_{n_2} \\ 0 & 0 \end{bmatrix}$

and derive the quantities as defined in (4.6):

Name	Value	Derived from
rank	$r=n_1+n_2$	$\operatorname{rank}E = \operatorname{rank}\begin{bmatrix} I_{n_1} & 0 & 0 \\ 0 & I_{n_2} & 0 \\ 0 & 0 & 0\end{bmatrix}$
algebraic part	$a=0$	$\operatorname{rank}Z^HAT = \operatorname{rank}0_{n_1}$
strangeness	$s=n_1$	$\operatorname{rank}V^HZ^HAT' = \operatorname{rank}\begin{bmatrix} I_{n_1} & 0_{n_1\times n_2}\end{bmatrix}$
differential part	$d=n_2$	$d=r-s=(n_1 + n_2) - n_1$
undetermined variables	$u=n_1$	$u=n-r-a=(n_1+n_2+n_1)-(n_1+n_2)-0$
vanishing equations	$v=0$	$v=m-r-a-s=(n_1+n_2+n_1)-(n_1+n_2)-n_1$

7.1.3 Derivative Array and the Condensed Form

Since the differentiation index of (7.1) is $\nu=1$, we anticipate for the strangeness index that $\mu=1$ and consider the derivative array of order $\ell =1$:

$$\begin{equation} \left ( \mathcal M_1 , \mathcal N_1 \right ) = \left ( \begin{bmatrix} M & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ A & B^H & M & 0\\ B & 0 & 0 & 0 \end{bmatrix} , \begin{bmatrix} A & B^H& 0 & 0\\ B & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \right ) , \quad g_1 = \begin{bmatrix} f_1 \\ f_2 \\ \dot f_1 \\ \dot f_2 \end{bmatrix}. \end{equation}$$