A sewer network can be as simple as a system of a few dendritic branches or as complicated as a system of hundreds of branches with many loops and various junctions containing different hydraulic structures and facilities, such as weirs and pumps. In many situations, the mutual flow interaction must be accounted for to achieve realistic results, particularly for unsteady flows since those confluence junctions can have significant effects to the traveling dynamic wave in a sewer system.

In order to simulate any complicated sewer networks using implicit numerical schemes, special algorithms have to be developed for any efficient and accurate solutions. An extended relaxation technique is used in the implicit dynamic sewer model to decompose the network of a sewer system into many single branches and solve each individual branch by the four-point implicit scheme described above. In doing so, it treats the influences of other branches at a junction as either lateral flows (when other branch joins the junction) or as a stage boundary condition (when the branch downstream end joins a junction). During the numerical solution process, when each branch is solved by the Newton-Raphson iteration, an assumed lateral inflow or outflow is added at each junction reach to replace the confluence branch.

The branches are automatically numbered such that the dendritic branches are always treated before the loop branches and higher order branches (more downstream) are always solved before the lower order branches. This numbering scheme enables a stage boundary condition at the downstream of a branch to be determined using the average computed stages at the two confluence cross-sections at the junction which the branch joins. In this way, each branch is independently solved one by one using the estimated lateral flows at each of the branch junctions. If the system has a total of J junctions, the relaxation is to iterate these J junction-related lateral flows.

The relaxation equation for the lateral flows is:

Values of α between 0.8 and 0.9 provide the most efficient convergence for the relaxation iteration. Extensive tests have shown that the relaxation iteration convergence is achieved within one to three iterations for almost all situations using α = 0.6 (a default value in the calculation option).

A looped branch or split branch is the one linked to a junction node at both upstream and downstream boundaries. When solving a looped branch for a time step, the implicit solver does the normal relaxation for the downstream boundary and uses an estimated flow as the upstream boundary condition, the iterative upstream flow is based on the diversion junction flow distribution factors which are derived by estimating the exiting link dynamic flow capacities.

The relaxation technique retains the super efficiency of solving the St. Venant equations for each individual branch and applying iterative proper boundary conditions so that it converges very fast for both boundary relaxation and branch solving.