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DRAFT: 4/10/02
calculations. Because the governing equations are hyperbolic, the
solution procedure for the flows and elevations is quite different from the
transport solution; therefore, this part presents the solution procedure for
the continuity and momentum equations.
6.3.2. Numerical Approximations
Three numerical procedures are useful for solving hyperbolic equations:
the finite element method, the method of characteristics, and implicit, finite
difference methods. The method of characteristics is quite accurate but
can be difficult to program by anyone but a specialist. Reviews of this
method are found in Liggett and Cunge (1975) and Abbott (1979).
Implicit, finite difference methods are simpler to program because they are
much more direct numerical techniques to approximate partial derivatives.
These methods also possess favorable stability behavior even in applications with variable space and time-steps. Explicit finite difference methods
are generally unstable and are not considered.
There are many implicit procedures, but the method to be used here is the
four-point implicit method first used by Preissman (1961) with subsequent
applications by, among others, Amein and Fang (1970) and Amein and
Chu (1975). This formulation is the most widely used and accepted
method presently available. The method is weighted implicit at each time
level, is unconditionally stable for 0.5 < θ < 1.0, and permits relatively
unequal space and time-steps. The scheme has second-order accuracy
when θ = 0.5 and first-order accuracy when θ = 1.0 . It is fully nonlinear
but yet is a compact scheme requiring just two points at each time level for
second-order spatial accuracy.
The river system is discretized (Figure 6-1) by a network of time and
space nodes separated by time and space increments ∆xi , ∆tj . If β
denotes the point about which the governing equation is to be discretized,
then the values of the variables at the four points surrounding β are used
to form the appropriate derivatives and weighted averages.
For a general variable ω , then
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EPD-RIV1