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Input Initial Conditions

Increment

Time Step

Input

New Boundary

Conditions

Iterate

Ice Stability Check

All

Reaches

Yes

Stable

No

Solve Full System

of Equations

Solve Water

Reset Variables

Equations Only

Write Output

*Figure 33. Block diagram for the fully coupled solution scheme.*

The block diagram for the fully coupled solution scheme is presented in Figure

33. The initial conditions are read and the timer incremented. Boundary conditions

for the new time step are read and local jam stability checked. The coefficient

matrix is then computed and solved, tolerances are checked, and variables are reset

for the next time step. For special cases when the entire ice cover is deemed stable,

the solution reverts to only the water variables, as the ice variables will remain

constant. In this case, an abbreviated coefficient matrix is developed and solved

that contains only two equations per reach and the two water boundary condition

equations (water discharge upstream and water depth downstream). This abbrevi-

ated matrix is a banded symmetrical matrix with two bands above and two below

the main diagonal. The reversion to the abbreviated coefficient matrix reduces com-

putation time.

*Loosely coupled solution*

The loosely coupled mode of solution also uses the full equations of mass and

momentum conservation for the water and ice, including the simplifications for

sections and reaches that are determined to be stable. However, in this mode, the

water variables are solved separately and sequentially from the ice variables. Both

solutions are by the Newton-Raphson iteration scheme presented above. The coef-

ficient matrix for each solution is a banded symmetrical matrix with two bands

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