# 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 $$\ell =1$$:

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