Values of water-surface elevation (h) and discharge (Q) for each cross section must be specified initially at time t = 0 to obtain solutions to the Saint-Venant equations. Initial conditions may be obtained from any of the following: (a) observations at gaging stations and using interpolated values between gaging stations for intermediate cross sections in large rivers; (b) computed values from a previous unsteady-flow solution (used in real-time flood forecasting); and (c) computed values from a steady-flow backwater solution. The backwater method is most commonly used in which the steady discharge at each cross section is determined by:
in which Qx is the assumed steady flow at the upstream boundary at time t — 0, and qt is the known average lateral inflow or outflow along each Ax reach at t = 0. The water-surface elevations (/?,) are computed according to the following steady-flow simplification of the momentum equation (30):
in which A and Sj are defined by Eqs. (52) and (53), respectively. The computations proceed in the upstream direction (i = TV — 1,... 3, 2, 1) for subcritical flow (they must proceed in the downstream direction for supercritical flow). The starting water-surface elevation (hN) can be specified or obtained from the appropriate downstream boundary condition for the discharge (QN) obtained via Eq. (58). The Newton-Raphson iterative solution method (Fread and Harbaugh, 1971) for a single equation and/or a simple, less efficient, but more stable bisection iterative technique can be applied to Eq. (59) to obtain ht. Due to friction, small errors in the initial conditions will dampen-out after several computational time steps during the solution of the Saint-Venant equations.
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