where (j) represents any dependent variable or functional quantity (Q, sc, sm, A, A0, q, h). Spatial derivatives are approximated at point (x', t') by a forward-difference quotient located between two adjacent time lines according to weighting factors of 9 (the ratio At'/At shown in Fig. 1) and 1 - 6, i.e.,
Nonderivative terms are approximated with weighting factors at the same time level [point (x', /)] where the spatial derivatives are evaluated, i.e.,
The weighted four-point implicit scheme is unconditionally, linearly stable for 0 > 0.5 (Fread, 1974); however, the sizes of the At and Ax computational steps are limited by the accuracy of the assumed linear variations of functions between the grid points in the x-t solution domain. Values of 9 greater than 0.5 dampen parasitic oscillations that have wavelengths of about 2Ax that can grow enough to invalidate or destroy the solution. The 9 weighting factor causes some loss of accuracy as it departs from 0.5, a box scheme, and approaches 1.0, a fully implicit scheme. This effect becomes more pronounced as the magnitude of the ratio (Tr/At) decreases where Tr is the time of rise of the hydrograph (time interval from beginning of significant rise to peak of the hydrograph). Usually, a 9 weighting factor of 0.60 is used to minimize the loss of accuracy while avoiding the possibility of weak (pseudo) instability for 9 values of 0.5 when frictional effects are minimal.
Selection of At and Ax Computational Parameters. The computational time step (At) can be either specified or automatically determined to best suit the most rapidly rising hydrograph occurring within a system of rivers that may contain one or more breaching dams or other dynamic internal boundary conditions. The time step is selected according to the following:
where Tr is the minimum time of rise (seconds) of any hydrograph that has been specified at upstream boundaries or in the process of being generated at a breaching dam; M is user specified according to the following guidance (Fread, 1993):
in which n' = 3.97 (3.13 SI units), n is the Manning friction coefficient, q is the peak flow per unit channel width (Q/B), and S0 is the channel bottom slope; M usually varies within the range, 6 < M < 40, with M often assumed to be approximately 20.
The Ax computational distance step can be specified or automatically determined according to the smaller of two criteria (Fread, 1993). The first criterion is
in which c is the bulk wave celerity (the celerity or velocity associated with an essential characteristic of the unsteady flow such as the peak of the hydrograph). In most applications, the wave velocity is well approximated as a kinematic wave, and c is estimated as 3/2F (V is the flow velocity) or c can be obtained by dividing the distance between two points along the channel by the difference in the times of occurrence of the peak of an observed or computed flow hydrograph at each point. Since c can vary along the channel, and depending upon the extent of this variation, Ax may not be constant along the channel.
The second criterion for selecting Ax is the restriction imposed by rapidly varying cross-sectional changes along the x axis of the waterway. Such expansion/ contraction is limited to the following inequality (Samuels, 1985):
This condition results in the following approximation (Fread, 1988) for the maximum computational distance step:
in which L' is the distance between two adjacent (i and i + 1) cross sections differing from one another by approximately 50% or greater, A is the active cross-sectional area, A = Aj+] if At > Aj+] (contracting reach) or A = At if At < Ai+i (expanding reach), and N' is rounded to the nearest integer value.
Significant changes in the bottom slope of the waterway also require small distance steps in the vicinity of the change. This is required particularly when the flow changes from subcritical to supercritical or conversely along the waterway. Such changes can require computational distance steps in the range of 50 to 200 ft (15 to 63 m).
Automatic Interpolation. It is convenient to automatically provide linearly interpolated cross sections at a user-specified spatial resolution to increase the spatial frequency at which solutions to the Saint-Venant equations are obtained. This is often required for purposes of attaining numerical accuracy/stability when (a) routing very sharp peaked hydrographs such as those generated by breached dams, (b) when adjacent cross sections either expand or contract by more than about 50%, and (c) where mixed flow changes from subcritical to supercritical or vice versa.
Algebraic Routing Equations. Using the finite-difference operators of Eqs. (41) to (43) to replace the derivatives and other variables in Eqs. (29) and (30), the
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