Discretisation

It is not possible to solve the Saint-Venant equations analytically - hence the need for numerical solution. As long ago as 1958 Isaacson E. et al (1958) demonstrated the feasibility of using the full Saint-Venant equations to obtain reliable results for practical engineering purposes. Their work was the first computational model implemented in the field of open channel flow and used an explicit finite difference scheme.

The first computational model using an implicit finite difference scheme was implemented by Preissmann A. (1961). In the course of the following years various different numerical schemes were proposed and implemented by other authors, but it appears now that the Preissmann scheme has more or less become the accepted standard.

Flood Modeller also employs the Preissmann implicit scheme - which is popularly referred to as the 4-point Box scheme. The scheme is outlined below.

Let f be the value of depth or discharge or a function of depth or discharge at point (i + ½, j + q ) as shown in the figure below:

The value of f or its continuous derivatives with respect to time or space can be discretised as:

Using the above, both Saint-Venant equations can be transformed into the linear form:

The coefficient matrix, which comprises largely of the a, b, c, d and e values described above must be inverted to solve the set of simultaneous difference equations for Q and H at the following iteration or timestep. Flood Modeller takes advantage of the banded structure of this matrix by employing a powerful sparse matrix solver.