The flow equations are expressed in finite-difference form for all Ax, reaches between the first and last (Nth) cross section (i = 1,2, ... ,N) along the channel/ floodplain and then solved simultaneously for the unknowns (Q and h) at each cross section. In essence, the solution technique determines the unknown quantities (Q and h at all specified cross sections along the waterway) at various times into the future; the solution is advanced from one time to a future time over a finite time interval (time step) of magnitude At. Thus, applying Eqs. (50) and (51) recursively to each of the (N — 1) rectangular grids in Figure 1 between the upstream and downstream boundaries, a total of (IN — 2) equations with 2N unknowns are formulated. Then, prescribed boundary conditions for subcritical flow [Froude number less than unity, i.e., Fr = Q/(A^/gD) < 1], one at the upstream boundary and one at the downstream boundary, provide the two additional and necessary equations required for the system to be determinate. Since disturbances can propagate only in the downstream direction in supercritical flow (Fr > 1), two upstream boundary conditions and no downstream boundary condition are required for the system to be determinate when the flow is supercritical throughout the routing reach. The boundary conditions are described later. Due to the nonlinearity of Eqs. (50) and (51) with respect to Q and h, an iterative, highly efficient quadratic solution technique such as the Newton-Raphson method is frequently used. Other solution techniques linearize Eqs. (50) and (51) via a Taylor series expansion or other means. Convergence of the iterative technique is attained when the difference between successive solutions for each unknown is less than a relatively small prescribed tolerance. Convergence for each unknown at all cross sections is usually attained within about one to five iterations with the majority of solutions obtained within two iterations. A more complete description of the solution method may be found elsewhere (Fread, 1985).

The solution of 2N x IN simultaneous equations requires an efficient matrix technique for the implicit method to be feasible. One such procedure requiring 38N computational operations (+, —, *, /) is a compact, penta-diagonal Gaussian elimination method (Fread, 1971, 1985) that makes use of the banded structure of the coefficient matrix of the system of equations. This is essentially the same as the double sweep elimination method (Liggett and Cunge, 1975; Cunge et al., 1980).

When flow is everywhere and at all times supercritical, the solution technique previously described can be somewhat simplified. Two boundary conditions are required at the upstream boundary and none at the downstream boundary since flow disturbances cannot propagate upstream in supercritical flow. The unknown h and Q at the most upstream cross section are determined from the two boundary equations. Then, cascading from upstream to downstream, Eqs. (50) and (51) are solved for the two unknowns (hi+x and Qi+,) at each cross section by using Newton-Raphson iteration applied recursively to the two nonlinear equations, with a — 0 in Eq. (51).

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