,
Full Equations (FEQ) Model for the Solution of the Full, Dynamic Equations
of Motion for One-Dimensional Unsteady Flow in Open Channels and Through
Control Structures
Four points in the x-t plane are used in the weighted four-point scheme to define the rectangle on the x-t plane over which equations 47 and 63 are approximately integrated to produce the system of algebraic equations that must be solved for a branch. These four points are shown in figure 14. The subscripts denoting these points are selected to assist in coding the computer program. The subscript L denotes the station on the left end of the control volume, and the subscript R denotes the station on the right end. The subscript U denotes the time point that is up, and the subscript D denotes the time point that is down relative to the center of the rectangular box on the x-t plane used to define the scheme. These subscripts are applied in the computer-program code by composing variable names for the various elements and values needed. For example, the Fortran name "ALU" specifies the area of flow at the left end of the control volume and at the upper time. The last two letters of the variable name are reserved to refer to the point on the x-t plane. This convention proves to be much simpler than use of the cumbersome indices for subscripts that may lead to many errors. The equations are written for a typical control volume corresponding to the computational element on the branch between adjacent cross sections.
The approximate rule of integration used in the weighted four-point scheme is the weighted-trapezoidal rule wherein
(66)
,
where a and b are the boundaries of the function region integrated and W is the weight on the function at the upper limit of the integral. The weight must satisfy 0 W 1. According to traditional error analysis, the most accurate approximation results when W = 0.5; however, the most accurate value for W is not always the best value. Other considerations enter into the choice of weight when approximating the integrals in the governing equations for unsteady, open-channel flow. For integrals with respect to time the weight is 
. The integrals with respect to distance will have two weights, 
and 
. The use of these weights will be discussed later. Equation 66 is recast so as to be more convenient in computation to
(67)
,
thereby reducing the number of multiplications by one.