A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

90 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation
Finally, the vertical velocity w in the water column can be derived from the continuity equation as
n + 1
wi, j, k – 1⁄ 2
=
n + 1
n + 1
n + 1
wi, j, k+ 1⁄ – ⎛U 2 i + 1⁄ 2,
j, k–
Ui – 1⁄ 2,
jk , ⎞ ⁄ Δx – Vi j 1⁄ 2
⎝ ⎠
( k = 234... , , , , km).
n + 1 n + 1
( , + , k – , – ⁄ , k ) ⁄ Δy
V i j 1 2
This equation is solved explicitly, starting from the bottom layer in which w i,j, km + ½ is assigned a zero value.
4.3.1.3 Matrix Solution
n + 1
The matrix { ζi, j } can be solved by either direct (noniterative) or iterative techniques. The primary difference is that direct
techniques (such as Gaussian elimination and its variants) produce a solution after a predetermined number of arithmetic opera-
tions, whereas iterative methods produce a sequence of approximate solutions which converge to the correct value. For large sys-
tems of linear equations where the coefficient matrix is sparsely populated (few nonzero elements), iterative techniques generally
represent the best approach since they require less storage 42 and are less susceptible to round off errors. If a good first estimate for
the unknowns in an iterative solution is available, it can also require less computer time to obtain a solution of comparable accuracy
to a direct solution. The preconditioned conjugate-gradient method (PCGM) (Meijerink and van der Vorst, 1977) is an especially
attractive iterative technique that is ideally suited for matrix solutions that have symmetric and positive definite coefficient matri-
ces; convergence of the iterative process in those solutions is generally rapid. Various forms of the PCGM have been tested and
compare favorably with other iterative methods when applied to ground-water flow modeling problems where the matrix system
of equations is similar in form to that solved here (see for example, Kuiper, 1981, 1987; Meyer and others, 1989; Hill, 1990).
The preference for direct or iterative solution techniques may well depend on the size of the problem being solved. Estuarine
and coastal modeling problems can easily involve large matrix solutions with 100,000 or more unknowns. Because iterative tech-
niques can efficiently solve these large systems with considerably lower storage requirements than direct techniques, they are usu-
ally the best choice. However, there is a trend toward solving larger and larger systems by direct methods, so further research
directly comparing the two approaches for a variety of actual 3-D estuarine modeling problems is still needed. Here the PCGM
was adopted without doing any comparison with a direct technique. For the small problem tested in this report, a direct solution
technique certainly could have been chosen.
n + 1
The matrix solution for { ζi, j } is implemented by using the non-symmetric preconditioned conjugate-gradient (NSPCG)
software package developed at the Center for Numerical Analysis at the University of Texas at Austin (Kincaid and others, 1989).
This package provides many preconditioning and acceleration methods to aid in selecting an iterative technique that is optimal for
a particular matrix structure, computer architecture (vector or scalar), and computer storage limitations. The reader is referred to
Oppe and others (1988) for the preconditioners and acceleration methods that are available. Without testing the 3-D estuarine
model on actual large scale field problems, it is not possible to judge fairly the iterative techniques available in the NSPCG pack-
age. A few comparisons of techniques were made for the simple problem in this report, and the conjugate-gradient accelerator
clearly was more efficient than the next best choice, which was successive over relaxation. The best preconditioner was modified
incomplete Cholesky (see Axelsson and Lindskog, 1986), which was used for all test runs in this report. Also promising, however,
was using a red-black (or odd-even) reduction scheme on the matrix system and then solving the reduced matrix system using the
42 In direct solution techniques, many of the elements of a sparse coefficient matrix that are originally zero will become nonzero in the course of the
computation, and they will require storage space. In computations with iterative techniques, no new nonzero elements are introduced in the coefficient matrix.
(4.37)