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 \(\ell =1\):
\[\begin{equation} \left ( \mathcal M_1 , \mathcal N_1 \right ) = \left ( \begin{bmatrix} M & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ A & B^H & M & 0\\ B & 0 & 0 & 0 \end{bmatrix} , \begin{bmatrix} A & B^H& 0 & 0\\ B & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \right ) , \quad g_1 = \begin{bmatrix} f_1 \\ f_2 \\ \dot f_1 \\ \dot f_2 \end{bmatrix}. \end{equation}\]