Implicit Four Point Finite Difference Approximations

The extended Saint-Venant equations [(29) and (30)] constitute a system of partial differential equations with two independent variables, x and t, and two dependent variables, h and Q, the remaining terms are either functions of x, t, h, and/or Q, or they are constants. The partial differential equations can be solved numerically by approximating each with a finite-difference algebraic equation; then the system of algebraic equations are solved in conformance with prescribed initial and boundary conditions.

Of various implicit, finite-difference solution schemes that have been developed, a four-point scheme first suggested by Issacson et al. (1954, 1956) and first used by Preissmann (1961) and later by Amein and Fang (1970) and then a weighted version by others (Fread, 1974, 1977, 1985, 1988; Cunge et al., 1980) is most advantageous. It is readily used with unequal distance steps, its stability-convergence properties are conveniently modified, and boundary conditions are easily applied.

Space—Time Plane. In the weighted four-point implicit scheme, the continuous x-t region in which solutions of h and Q are sought is represented by a rectangular grid of discrete points as shown in Figure 1. An x-t plane (solution domain) is a convenient means for expressing relationships among the variables. The grid points are determined by the intersection of lines drawn parallel to the x and t axes. Those

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