Explicit-Implicit Implementation

In each block of the block DD the ADI method is employed (Leca & Mane, 1992). The process working on a block builds the tri-diagonal system arising from the discrete shallow water equations like in the serial model (see chapter 4 and section 4.2). To allow the system to be defined the missing level value on the inter block boundary must be calculated explicitly. In that cell, equation  (4.2) is replaced by  (5.1) if working by rows. When working by columns  (4.6) is replaced by  (5.2).

$$\frac{\eta^{n+{\scriptscriptstyle \frac{1}{2}}}_{i\;j}-\eta^{... ..._{i\;j} \; u^n_{i\;j}) + D_{-y} (H^n_{i\;j} \; v^n_{i\;j}) = 0$$ (8.1)

$$\frac{\eta^{n+1}_{i\;j}-\eta^{n+{\scriptscriptstyle \frac{1}{... ...{n+{\scriptscriptstyle \frac{1}{2}}}_{i\;j} \; v^n_{i\;j}) = 0$$ (8.2)

The level value at those inter block boundaries is calculated by  (5.3) in the row wise step, and by  (5.4) in the column wise step, at the following half time-step.

$$\eta^{n+{\scriptscriptstyle \frac{1}{2}}}_{i\;j} = \eta^{n}_{... ...H^n_{i\;j} \; u^n_{i\;j}) + D_{-y} (H^n_{i\;j} \; v^n_{i\;j}))$$ (8.3)

$$\eta^{n+1}_{i\;j} = \eta^{n+{\scriptscriptstyle \frac{1}{2}}}... ... (H^{n+{\scriptscriptstyle \frac{1}{2}}}_{i\;j} \; v^n_{i\;j})$$ (8.4)

The explicit calculation described above puts a CFL^8.1 limitation on the time step (Arakawa, 1988; Hirsch, 1991).

   $$\Delta t < \frac{\Delta x}{\sqrt{g\overline{H}}}$$

  

Figure 5.1: Inter-block boundary condition and overlapped area

In figure 5.1 the scheme of the overlapped region between blocks is shown. The steps performed to advance one-time step are:

1.

A first approximation of the value in the overlapped area between blocks in the same row, (denoted by *) is carried out using the explicit computation  (5.3) described above.

2.

Using this estimate as boundary condition the by-row sweep of the ADI method is performed in each block.

3.

Overlapped region values are interchanged between adjacent blocks, updating the explicit approximation.

4.

An approximation of the values in the overlapped area between blocks in the same column is carried out using the explicit computation  (5.4).

5.

The by column sweep of the ADI method is performed.

6.

Overlapped region data is interchanged between adjacent blocks, updating the explicit approximation.

Repetitive use of this approach leads up to an explicit-implicit hybrid method (Cekirge et al., 1994; Dawson & Dupont, 1994).

Variables marked $\circ$ are sent by block $\Omega_{1,1}$, see figure 5.2, and received by block $\Omega_{1,2}$ replacing variables marked *, making a single step iteration. Alternating send-receive in block $\Omega_{1,1}$ and receive-send in block $\Omega_{1,2}$ allows a non-blocking communication with an improved model performance.

  

Figure 5.2: Example of a domain decomposition with the overlapped areas.