7 Examples

7.1 Semi-discrete Navier-Stokes equations

By scalings and state transforms, we find that the coefficients of the spatially discretized Navier-Stokes equations are equivalent to: \[\begin{align*} (\mathcal E, \mathcal A) &= \left( \begin{bmatrix} M & 0 \\ 0 & 0 \end{bmatrix} , \begin{bmatrix} A & B^H \\ B & 0 \end{bmatrix} \right) \\ & \backsim \left( \begin{bmatrix} M^{-1/2} & 0 \\ 0 & I \end{bmatrix} \begin{bmatrix} M & 0 \\ 0 & 0 \end{bmatrix} , \begin{bmatrix} A & B^H \\ B & 0 \end{bmatrix} \begin{bmatrix} M^{-1/2} & 0 \\ 0 & I \end{bmatrix} \right) \\ & \backsim \left( \begin{bmatrix} Q^H & 0 \\ 0 & I \end{bmatrix} \begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix} , \begin{bmatrix} M^{-1/2}AM^{-1/2} & M^{-1/2}B^H \\ B M^{-1/2} & 0 \end{bmatrix} \begin{bmatrix} Q & 0 \\ 0 & I \end{bmatrix} \right) \\ & \backsim \left( \begin{bmatrix} I & 0 \\ 0 & R^{-H} \end{bmatrix} \begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix} , \begin{bmatrix} M^{-1/2}AM^{-1/2} & \begin{bmatrix} R \\ 0 \end{bmatrix} \\ \begin{bmatrix}R^H & 0\end{bmatrix} & 0 \end{bmatrix} \begin{bmatrix} I & 0 \\ 0 & R^{-1} \end{bmatrix} \right) \\ & \quad = \left( \begin{bmatrix} I_{n_1} & 0 & 0 \\ 0 & I_{n_2} & 0 \\ 0 & 0 & 0\end{bmatrix} , \begin{bmatrix} A_{11} & A_{12} & I_{n_1} \\ A_{21} & A_{22} & 0 \\ I_{n_1} & 0 & 0\end{bmatrix} \right). \end{align*}\] 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.