2D approximation

Neglecting external forces in the two dimensional shallow waters system, (3.2-3.4), and freezing coefficients the finite differences scheme is written as (4.38) for the row-wise j sweep.

$$\begin{split} & \frac{\eta^{n+{\scriptscriptstyle \frac{1}... ...{\scriptscriptstyle \frac{1}{2}}}_{ij} = v^{n}_{ij} \end{split}$$ (7.34)

With coefficient approximation: $a=\overline{H_u} + \overline{\eta}$, $b= \overline{u}$, $c=\overline{v}$.

For the column-wise i sweep we get:

      $$\begin{eqnarray*} & \frac{\eta^{n+1}_{ij} - \eta^{n+{\scriptscriptstyle \frac{1... ...tyle \frac{1}{2}}}_{ij}) - \varepsilon \mathcal{D} v^n_{ij} = 0 \end{eqnarray*}$$

We can make the Fourier expansion again using:

$$\eta_{ij}^n = \tfrac{1}{\sqrt{2 \pi}} \sum\limits_{\omega_x, \; \omega_y} \tilde{\eta}^{n}_{\omega_x, \; \omega_y} e^{i(\omega_x x_j + \omega_y y_i)}$$ (7.35)

$$u_{ij}^n = \tfrac{1}{\sqrt{2 \pi}} \sum\limits_{\omega_x, \; \omega_... ...e^{i(\omega_x (x_j + {\scriptscriptstyle \frac{1}{2}}\Delta x) + \omega_y y_i)}$$ (7.36)

$$v_{ij}^n = \tfrac{1}{\sqrt{2 \pi}} \sum\limits_{\omega_x, \; \omega_... ...^{i(\omega_x x_j + \omega_y (y_i + {\scriptscriptstyle \frac{1}{2}}\Delta x ))}$$ (7.37)

$$\xi=\omega_x \Delta x \qquad \chi=\omega_y \Delta x$$ (7.38)

The implicit model is formulated, employing matrix notation, by:
$\hat Q_{-1} \mathbf{\tilde w^{n+{\scriptscriptstyle \frac{1}{2}}}_{\omega_x, \; \omega_y}} = \hat Q_{\sigma} \mathbf{\tilde w^{n}_{\omega_x, \; \omega_y}}$ expanded in (4.43) and
$\hat P_{-1} \mathbf{\tilde w^{n+1}_{\omega_x, \; \omega_y}} = \hat P_{\sigma} \mathbf{\tilde w^{n+{\scriptscriptstyle \frac{1}{2}}}_{\omega_x, \; \omega_y}}$expanded in (4.44).

 $$\begin{multline} \begin{pmatrix}1 & \S i a \tfrac{\Delta t}{\Delta x} \sin \tf... ...a_y}\\ [12pt] \tilde{v}^{n}_{\omega_x, \; \omega_y} \end{pmatrix}\end{multline}$$

 $$\begin{multline} \begin{pmatrix}1 & 0 & \S i a \tfrac{\Delta t}{\Delta x} \sin ... ... {\scriptstyle \frac{1}{2}}}_{\omega_x, \; \omega_y} \end{pmatrix}\end{multline}$$

In this case we study stability by examining the spectral radius of the amplification matrices: $\hat Q(\xi) = ({\hat Q_{-1}})^{-1} \hat Q_{\sigma}$ and $\hat P(\xi)=(\hat P_{-1})^{-1} \hat P_{\sigma}$. They should not be greater than one: $\vert\vert \hat Q(\xi) \vert\vert \leq 1$and $\vert\vert \hat P(\xi) \vert\vert \leq 1$ for all $\xi$ in $[0,2\pi]$.

Again, the system is stable for practical values of $\Delta t$ and $\Delta x$without heavy dependence on the maximum depth a and velocities band c. For a practical value of $\Delta x=1000 m$ the allowed time step is shown in figure 4.3. For $\varepsilon leq 10$it is required that $\Delta t \leq 45 min$. The 12-hour period astronomic wave should be modeled employing a time step with at least 15 grid points per wave length. In every case we must choose time steps smaller than 45 minutes, in agreement with our previous result for the explicit model.

  

Figure 4.3: Valid $\Delta t$ values for stable system, 2D

The Río de la Plata wind storm tide simulation requires a $\varepsilon$ rather big, as high as 1000, due to the large depth and steep variations in the sea bottom. In that case, the time step must be selected below 5 minutes due to the stability requirements.