The explicit solution is different from the SOR solution in the sense that there is no iterative matrix solution. In other words, the exchange flows between every cell and to the river are simply calculated based on the individual cell heads. However, the explicit solution is much more restrictive in terms of time step. For the explicit solution to be stable, the flow must be slow relative to the time step. For example, a flood wave cannot cross a cell in one time step. So, this leads to the following 3-step calculation process for the explicit solution of the overland flow:
1. Calculate all flow rates and discharges between cells and between the overland cells and river links based on the current water levels
2. Loop over all the cells and calculate the maximum allowed time step length for the current time step, based on the following criteria
– Courant criteria (see next section)
– Cell volume criteria - the volume in the cell divided by the flow rate
– River link volume criteria - the volume in the river link divided by the flow rate
– River bank criteria - the exchangeable volume in the river link based on the river bank elevation divided by the flow rate
In most cases, the Courant criteria is the critical criteria for the maximum time step, with the Cell volume criteria sometimes being critical. The River link and River bank criteria are less commonly critical, but may become critical when the river is very shallow.
3. Calculate the actual flows between the all the cells and to/from the river links using the maximum allowed time step and update all the cell water depths.