In the default numerical solver of the Saint-Venant equations, Eq. 14.1 and 14.2, implicit finite-difference numerical method is used and a weighted four-point implicit scheme is applied to get numerical solutions. The implicit method is preferred over explicit since these methods have the advantage of maintaining good stability for large computational time steps and exhibit robustness in modeling systems that integrate the complex hydraulic interactions encountered in gravity sewer systems. The scheme was adopted since it handles unequal distance steps, its stability-convergence properties can be conveniently modified, and the internal (any hydraulic structures, such as dams, weirs, pumps, manholes etc) and external boundary conditions can be easily applied. The dynamic model is developed using the following four-point finite-difference scheme:

in which θ is a weighting factor and the weighted four-point implicit scheme is unconditionally stable for >0.5. The value of θ of 0.6-0.8 is found to be optimal in maintaining stability and accuracy for large computational time steps. This is one of the calculation options which user can modify (NR weighting coefficient). The computation

x and

t domain and resulting finite differential equations are shown below.

Applying equations 14.3 though 14.5 to the non-linear dynamic equations to each computational

x reach results in a complex matrix of linear system, with all Q and h at every section are unknown. In order to derive all Q and h, the Newton-Raphson iteration method is used to solve the finite-difference equations (the linear matrix system). Convergence of the iterative technique is attained when the difference between successive iterations for each unknown Q and h is less than a specified tolerance which can be modified by user as a calculation option as well (Y-iteration tolerance).

Exceptional computational efficiency is achieved by special algorithm to iterate a banded matrix so that a convergence is mostly obtained within 1-5 iterations for each time step.