Using first-derivatives across elements as an indication of error, and the target
Peclet number convergence criteria, the simulation proceeds as described in the
following sections.
3.4.1 Model Inputs
To start the simulation, inputs are required for the physical characteristics of the groundwater contaminant transport system and for the simulation parameters (see Section 4.2 for tables of the input variables used in a simulation).
3.4.2 Initial Mesh I
At each level of the solution process, a uniform mesh is set up consisting of rectangular elements aligned with the x-y coordinate system. To determine the
initial mesh. Ax and Ay are chosen such that a node will be present where the
peak of the plume is located at the start of the simulation. This will ensure that
a non-zero value will be picked-up initially, regardless of how sharp the front is.
These Ax and Ay will also be chosen such that the maximum initial Pe limitation
is satisfied. Thus the boundaries of the domain are determined by the initial peaJc
3.4 Simulation Procedure
These variables represent the maximum x-coordinate of the left side of the domain
(vertical line), the minimum x-coordinate of the right side of the domain (vertical
line), the maximum y-coordinate of the bottom of the domain (horizontal line), and
the minimum y-coordinate of the top of the domain (horizontal line) respectively.
Once this initial discretization is established, the next step is to determine the
integer by which the elements will be divided from one level to the next. This is done by using the initial Pe, the maximum final Pe, and the total number of levels
available according to the following formula:
,. . . / initial Pe \l/nlevela-l
division > I---——;---1 (6)
\max final Pe/
The smallest integer satisfying the above expression is then used as the divider. Next, using the initial grid and the divider, a check is made to see if the Ax and Ay at any level will be less than or equal to the distance traveled by the plume
in one overall time step. If the Ax or Ay at any level will be less than the distance
traveled by the plume in one overall time step, an adjustment is made to ensure equality. If an adjustment is made, then the calculation of the divider is repeated. Adjustment of the Ax and Ay and recalculation of the divider are repeated until no
further adjustments are necessary. Once Ax and Ay are established, the boundaries
may have to be adjusted outward so that uniformity of the grid is maintained.Thus an initial uniform mesh (level 0) is now in place, consisting of rectangles aligned with the x-y coordinate system. An example of such a mesh is shown in Figure 1.
Inputs are now required concerning the magnitude and number of time steps
to be carried out. Using the given overall At (length of time intervals for desired
output), the discretization parameters Ax and Ay, and the given velocities in the
X and y directions, the Crj; and Cry are then calculated and checked to ensure that
(0.5)0—m—o
m—© (5,5)
^1
(0,0)
(5,0)
dx = dy = 1
> X
Figure 1. Sample Initial Mesh
they are below the maximum allowable Cr. If they are not below the maximum
allowable, the time step is reduced (by dividing by successively greater integers)
and the number of time steps increased (by multiplying by the respective integers)
^^fl^^*^w^l^''
3.4 Simulation Procedure
3.4.3 Multi-Level Solution Procedure
The FEM is then applied to this initial mesh using given initial values and
boundary conditions to yield an approximate solution at the initial nodes. Deriva¬
tives are then calculated in both the x and y directions between adjoining nodes.
The boundaries of the next level's domain are determined by comparing these
derivatives with the maximum allowable de;rivative. Those areas where the deriva¬
tives exceed this limitation are included in the subsequent domain. Steps are taken
in this process to ensure that the domains at every level will be rectangular. Work¬
ing with rectangular domains greatly simplifies the code, however it does sacrifice
some of the efficiency of the model in using more nodes than is absolutely necessary.
The new domain is then refined by dividing each element by the previously
calculated divider in both the x and y directions. Examples of the refinement of a
grid using a divider of 2 and a divider of 3 are shown in Figure 2. The FEM is then
carried out on this refinement (level 1) of the original mesh, again using the given
initial values but now the boundary conditions to be used are natural (second-kind,
no-flux) boundary conditions wherever the sub-domain boundary does not coincide
with a global domain boundary. Also, for this new mesh the time step is halved
and the number of time steps doubled to ensure that the Courant number remains
less than the minimum allowable — in fact it stays constant.This refinement method will ensure that for each of the nodes in the original
mesh there will be a node in the refined mesh that assumes the same spatial location.
Because of this, it is possible to (if storage allows) compare the values obtained for
the independent variable (concentration in our case) on the original mesh to those
obtained on the refined mesh. Moreover, the convergence of the approximation on
a given original element can be evaluated by calculating the differences in the nodal
Level I (# time steps = s) Level i+1 (# time steps = 2s)
Divider = 2
#
Level i (# time steps = s) Level i+1 (# time steps = 3s)Divider = 3
A - elements flagged for adaption
- regular nodes Q " 'tegular nodes
3.4 Simulation Procedure
approximations from one mesh to the other. Using the maximum absolute value of
these differences:
WCi'-Ci^W^ . (7)
where r— refined mesh, o= original mesh, and i = 1,..., number of nodes in refined
mesh, as a measure of convergence, it can then be determined whether the solution
has converged or not. Due to storage limitations, this approach is not taken in this
model. Instead, the target Peclet number criteria, as described in Section 3.3, is
used as aji indication of convergence.
This procedtire (identification of sub-domain boundaries, refinement in space
and time, and application of FEM) is continued within the same "overall" time step
until the maximtmi nvimber of levels is exhausted or until uniform convergence is
achieved. Thus, at each time step the domain is adaptively refined until satisfactory
approximation is achieved. The results of each of the levels within this "overall"
time step are then stored to be used as initial conditions for the next "overall" time
step. The procedure is then repeated for as majiy "overall" time steps as is called
for in the original input to the simulation.