Firstly the entire procedure is split into two main fractional-steps, which constitute an alternating direction implicit (ADI) form algorithm. This idea has been used in a number of numerical models (e.g. DIVAST, originally developed by Falconer, 1980b, 1984). In the following parts each of these two main fractional-steps is referred as one half-time-step. The ADI algorithm provides a three-dimensional implicit scheme, but considering one vertical plane implicitly for each half-time-step which requires the solution of a two-dimensional block tri-diagonal matrix for each direction for each half-time-step. The planes in alternate directions, which are swept in one complete time-step, form the columns and rows of the projection of the whole domain on the xoy reference plane as has been demonstrated in figure (4.4). In other words the three-dimensional procedure is accomplished by sweeping all the vertical planes in, for instance, x-direction for the first half-time-step and sweeping all the vertical planes in, therefore, y-direction for the second half-time-step. The numerical procedure for each of the two half-time-steps has been described in the following paragraphs.
Figure (4.8) - Temporal levels of calculation of the unknowns t
x t = n + 1
t = n + 1/2
t = n
t = n - 1/2
t = n - 1 Scalar quantity
location u-velocity location v-velocity location
w-velocity y
In the first half-time-step, for instance, the u- and w-momentum equations are solved together with continuity equation. In the second half-time-step, therefore, the v-momentum equation and again w-momentum and continuity equations are solved.
In each half-time-step the alternate horizontal velocity component has an explicit contribution to the solution. w and pressure are calculated in both half-time-steps.
This has been illustrated in figure (4.8) in which the z-direction spatial coordinate has been replaced by time to demonstrate the temporal procedure of the solution and therefore the z-component of the velocity (w), originally located at points half-way between the pressure points (or in general; scalar quantity locations) in the z-direction, has been demonstrated at the same location as the pressure. Recalling figures (4.4) and (4.5), the vertical planes do not necessarily have same dimensions in either direction. In general the vertical planes can be characterised by:
The xoz planes with an arbitrary length and an independent longitudinal location with respect to the neighbouring planes in x-direction.
The yoz planes with an arbitrary width and an independent transverse location with respect to the neighbouring planes in y-direction.
This provides better and more accurate boundary-fitting in the horizontal plane, especially in irregular geometries, and facilitates consideration of islands, or permanently dry zones in the domain without any discontinuity. Therefore any vertical plane in either direction contains only wet cells.
Secondly the time advancement of each half-time-step is decomposed into two fractional-time-steps and instead of simultaneously satisfying two of the momentum equations and the continuity equation, the method proceeds, for example for the first half-time-step, as follows:
o The first fractional-step, which includes advective and diffusive terms, consists of finding , providing that Vn is known, an intermediate or provisional velocity, V*:
velocity which were last updated at (n-1/2) and n time levels, have contributed to
the V velocity. So for the solution of the first half-time-step in the x-direction, for example, u which was last updated at time level (n-1/2) and v and w which were last updated at time level n are used. D is the implicit weighting factor for the diffusion, where θ = 1 implies a fully implicit diffusion solution and θ = 0, denotes a fully explicit diffusion solution. The first step of the method can be thought of as a Burgers' equation.
o The second fractional-step makes use of the Hodge decomposition theorem, which states that any vector function can be decomposed into a divergence-free part plus the gradient of a scalar potential (Brown, 2001). So the second step proceeds by solving the Poisson equation as follows:
pn t
Thirdly, the first fractional-step of each half-time-step is further split into two sub-fractional-steps, so that allowing for separately computing of advective and diffusive terms. This approach allows the use of suitable approximation for each term. Therefore equation (4.41) is split into two equations which are computed sequentially as follows:
second equation, the same definition applies.Fourthly the advective contribution of the transport term in equation (4.44) is further split into three sub-sub-fractional-steps. For the first half-time-step (i.e.
considering u- and w-momentum equations to be solved) and for the x-component of velocity (u) this sub-fractional-time-step comprises of:
Ax, denotes the time level of completion of the advection in both x and z directions, and A denotes the time level of completion of the whole advection process including the grid velocity. The treatment for the z-velocity component (w), however, is somehow different. w is computed in both half-time-steps, therefore its time advancement should be halved. This also applies for the advection of all scalar quantities. There exist two strategies:
o The advection of w in all directions is only advanced one half-time-step. This scheme involves the newly updated u-velocity for the first half-time-step but employs the v-velocity from the previous half-time-step and vice versa for the second half-time-step.
o The advection of w in the x- and y-directions, terms wu x and wv y respectively, is advanced for a full-time-step at their own share of each half-time-step and only its advection in the z-direction, term w2 z, is advanced a half-time-step. This scheme involves the newly updated u and v velocities and therefore uses the more implicit values than the previous scheme.
The diffusive contribution is split into two sub-sub-fractional steps. For the first half-time-step, considering u- and w-momentum equations to be solved, and for the x-component of velocity (u) the diffusion sub-fractional-step comprises of equation (4.46) in which "D" stands for diffusion. D denotes the time level of completion of x diffusion process in x-direction and D denotes the time level of the completion of the whole diffusion process (i.e. D and x D ). For the treatment of z-velocity component z same discussion which was provided for the advection of w applies.
On the free surface a special treatment is applied. The pressure within the top layer has been defined with a hydrostatic pressure assumption as follows:
where P is the atmospheric pressure acting on the free surface. By defining (Namin a et al., 2001):
r
P p
(4.48)
then the pressure within the top layer becomes:
2
The free surface equation is obtained by integrating the continuity equation over the depth with the kinematic conditions at the bed and free surface as follows:
0 equation (4.50) is equal to zero.
Equations (4.51) and (4.52) show the integrated temporal procedure of the solution in the domain and on the free surface for the first half-time-step respectively. θ scheme is used for the pressure in the domain (P) and for the