1D approximation

The two-dimensional shallow-waters system of equations (3.2-3.4) could be rewritten in one dimension as (4.31) for mass conservation and (4.32) for momentum conservation. External forces had been neglected and the over-line notation has been omitted. In practice, low order terms (without derivatives) do not affect the stability of a finite-difference method.

$$\frac{\partial \eta }{\partial t} + (H_u+\eta) \frac{\partial u} {\partial x} + u \frac{\partial \eta} {\partial x} = 0$$ (7.28)

$$\frac{\partial u} {\partial t} + u \frac{\partial u} {\partia... ...{\partial x} = \varepsilon \frac{\partial^2 u} {\partial x^2}$$ (7.29)

The finite-difference scheme for the above system is written as:

$$\begin{split} & \frac{\eta^{n+1}_j - \eta^{n}_j} {\Delta t... ... + D_+ \eta^{n}_j) - \varepsilon D_+D_- u^n_j = 0 \end{split}$$ (7.30)

Freezing coefficients, the one dimensional system of equations could be re-written as:

$$\begin{split} & \frac{\eta^{n+1}_j - \eta^{n}_j} {\Delta t... ... + D_+ \eta^{n}_j) - \varepsilon D_+D_- u^n_j = 0 \end{split}$$ (7.31)

Where the constant coefficients introduced have the meaning: $a=\overline{H_u} + \overline{\eta}$ - $b= \overline{u}$.

The Von Neumann method requires to make the Fourier expansion of the solution. This solution can be expressed by:

$$\begin{split} & \eta_j^n = \tfrac{1}{\sqrt{2 \pi}} \sum\li... ...a (x_j+{\scriptscriptstyle \frac{1}{2}}\Delta x)} \end{split}$$ (7.32)

Replacing $\xi = \omega \Delta x$ and $\lambda = \frac {\Delta t}{\Delta x}$ on the above solution and employing it in 4.34 we get:

$$\begin{split} &\frac{\tilde{\eta}^{n+1}_\omega - \tilde{\e... ...in^2 \tfrac {\xi}{2} \; \tilde{u} ^n_\omega = 0 \end{split}$$ (7.33)

The implicit (4.36) model is rewritten employing matrix notation by $\hat Q_{-1} \mathbf{\tilde w^{n+1}_\omega} = \hat Q_{\sigma} \mathbf{\tilde w^{n}_\omega}$ expanded in (4.37).

 $$\begin{multline} \jot = 10pt \begin{pmatrix}1 & 2 i a \tfrac{\Delta t}{\Delta x... ...}\tilde{\eta}^n_\omega \\ [12pt] \tilde{u}^n_\omega \end{pmatrix}\end{multline}$$

Von Neumann method requires for stability that the spectral radius of the amplification matrix, $\hat Q(\xi) = ({\hat Q_{-1}})^{-1} \hat Q_{\sigma}$, should not be greater than one : $\vert\vert \hat Q(\xi) \vert\vert \leq 1$ for all $\xi$ in $[0,2\pi]$.

The model is stable for practical values of $\Delta t$ and $\Delta x$, without heavy dependence on the maximum depth and velocity, a and b respectively. For a practical value of $\Delta x=1000$ m the feasible time step is shown in figure 4.2. For $\varepsilon leq 10$ it is required that $\Delta t \leq 65$ 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 have chosen time steps smaller than 45 minutes.

  

Figure 4.2: Valid $\Delta t$ values for stable system, 1D, $\Delta x=1000$ m