When the complete Saint-Venant equations [(1) and (2)] are used, the routing model is known as a dynamic routing model. With the advent of high-speed computers, Stoker (1953) and Isaacson et al. (1954, 1956) first attempted to use the complete Saint-Venant equations for routing Ohio River floods. Since then, much effort has been expended on the development of dynamic routing models. Many models have been reported in the literature (Fread, 1985, 1992; Liggett and Cunge, 1975).
Dynamic routing models can be categorized as characteristic or direct methods of solving the Saint-Venant equations. In the characteristic methods, the Saint-Venant equations are first transformed into an equivalent set of four ordinary differential equations that are then approximated with finite differences to obtain solutions. Characteristic methods (Abbott, 1966; Henderson, 1966; Streeter and Wylie, 1967; Baltzer and Lai, 1968) have not proven advantageous over the direct methods for practical flow routing applications.
Direct methods can be classified further as either explicit or implicit. Explicit schemes (Stoker, 1953, 1957; Isaacson et al., 1954; Garrison et al., 1969; Dronkers, 1969; Strelkoff", 1970; Liggett and Cunge, 1975; Veissman et al., 1977; Linsley et al., 1986) transform the differential equations into a set of algebraic equations that are solved sequentially for the unknown flow properties at each cross section at a given time. However, implicit schemes (Preissman, 1961; Amein and Fang, 1970; Strelkoff, 1970; Fread, 1973, 1977, 1978, 1985; Liggett and Cunge, 1975; Cunge et al., 1980; Schafiranek, 1987; Fread and Lewis, 1998; Chow et al., 1988; Barkow, 1990) transform the Saint-Venant equations into a set of algebraic equations that must be solved simultaneously for all Ax computational reaches at a given time; this set of simultaneous equations may be either linear or nonlinear, the latter requiring an iterative solution procedure.
Explicit methods, although simpler in application, are restricted by numerical stability considerations. Stability problems arise when inevitable errors in computational roundoff and those introduced in approximating the partial differential equations via finite differences accumulate to the point that they destroy the usefulness and integrity of the solution, if not the total breakdown of the computations, by creating artificial oscillations of length about 2Ax in the solution. Due to stability requirements, explicit methods require very small computational time steps on the order of a few seconds or minutes depending on the ratio of the computational reach length (Ar) to the minimum dynamic wave celerity («), i.e., At < Ax/u, where u = V + (gA/B)1/2. This is known as the Courant condition, and it restricts the time step to less than that required for an infinitesimal disturbance to travel the Ax distance. Such small time steps cause explicit methods to be inefficient in the use of computer time.
Implicit finite-difference techniques, however, have no restrictions on the size of the time step due to mathematical stability; however, numerical convergence (accuracy) considerations require some limitation in time step size (Fread, 1974; Cunge et al., 1980). Implicit techniques are generally preferred over explicit because of their computational efficiency. Therefore, an implicit scheme will be subsequently described in detail herein. Rather than using finite-difference approximation techniques to solve the Saint-Venant equations, finite-element techniques (Gray et al., 1977; DeLong, 1986, 1989) can be used; however, their greater complexity offsets any apparent advantages when compared to a weighted, four-point implicit finite-difference scheme (described later) for solving the one-dimensional flow equations. However, finite-element techniques are often applied to two- and three-dimensional flow computations.
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