The ADI method

A finite differences scheme is being employed over a staggered mesh (figure 4.1), implicit with directional splitting. This Arakawa class C grid (Arakawa, 1988) has good conservative properties and is also well suited to the DD method with overlapped regions as will be described later (Arakawa & Lamb, 1980). A uniform grid has been employed: $\Delta x = \Delta y$.

  

Figure 4.1: Distribution of the dependent variables on a 2D grid.

An ADI^7.1 (Kim & Lee, 1994; Leendertse & Liu, 1975) (implicit, alternating directions in the horizontal and vertical directions successively, with a splitting in the time-step) method is employed (Leca & Mane, 1992).

Advective processes, corresponding to the terms (4.1) are dominant in atmospheric and oceanic circulation systems governed by shallow waters equations, while diffusive effects are important only in boundary layer regions (Neta, 1992). Any numerical model working on those equations should treat advective effects accurately. In this model, such terms are treated in implicit form.

$$\overline{u}\frac{\partial \overline{u}} {\partial x} + \over... ...x} + \overline{v}\frac{\partial \overline{v} } {\partial y}$$ (7.1)