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88 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation Because the matrix A is an M-matrix, its matrix inverse A −1 has all non-negative elements (for proof see Lancaster and Tismenetsky, 1985). Therefore, the matrix products ΞA −1 R and ΞA −1 G in equation 4.35 are each a single non-negative number. Rearranging equation 4.35, the following form can be obtained: where n + 1 n + 1 n + 1 – , ζij , – 1 ri, jζi, j sxi – 1⁄ 2, j ζ – i – 1, j syi j – 1⁄ 2 sxi ± 1⁄ 2 , j syi, j± 1⁄ 2 n + 1 + – syi , j + 1⁄ ζ 2 i, j + 1 sx 1 i ⁄ 2 g Δt ⎛----- ⎞ ⎝Δx⎠ 2 n ρ 1 i ± ⁄ 2 , j, 1 ΞA 1 – = R i ± 1⁄ 2 , j g Δt ⎛----- ⎞ ⎝Δx⎠ 2 n = ri, j = 1 sxi 1⁄ 2, j qi, j ρi, j± 1 2 ⁄ , 1 ΞA 1 – R i, j± 1⁄ 2 + + + sxi – 1⁄ 2, j + syi, j + 1⁄ + sy 2 i j – 1⁄ 2 n – 1 Δt ζ i, j – ----- Δx ΞA 1 – G i + 1⁄ 2, j ΞA 1 – = ⎛ – ⎞ ⎜ G i – 1⁄ 2, j⎟ ⎝ ⎠ , , n + 1 – + , j ζi + 1, j = qi, j (4.36) , , Δt – ----- Δy ΞA 1 – G i, j + 1⁄ ΞA 2 1 – n – 1 ⎛ – ⎞ – d G ij , , ⎜ i, j– 1⁄ ⎟ 2 ⎝ ⎠ n – 1 and di, j is as defined for equation 4.34. Equation 4.36 can be written at each of the interior nodal points of the rectangular grid (excluding the fictitious row and column along each boundary) to form N = (imax − 2) × (jmax − 2) simultaneous linear equations in the unknowns { ζi, j}. If the set of equations is written in matrix form, the coefficient matrix is five-diagonal with a tridiagonal band along the main diagonal and two additional diagonals displaced an equal amount above and below the main diagonal (fig.4.7); the amount the outer diagonals are displaced is referred to as the matrix bandwidth and is dependent on the dimensions of the grid and the ordering of the unknowns. Here the most common “natural” ordering is used in which the numbering is done along the smallest dimension of the finite-difference grid. For example, if the smallest dimension is the y (north-south) direction, the natural ordering scheme numbers from bottom to top (south to north) on a column starting with the first (westernmost) column that is not fictitious (fig. 4.8). Other orderings such as the “ordering along the diagonals” and the “red-black” ordering also are mathematically consistent (see Young, 1971, p. 159). For problems involving irregularly shaped regions, the form of the coefficient matrix does not change. Dry point equations are represented with a value of 1.0 on the main diagonal and zeros for the other elements of the n + 1 n equation row; the continuity equation for a dry point therefore is reduced to ζi, j = ζi, jwhere the assigned water surface elevation is artificial. Because all the dry point equations can be made identical, only one must be stored. The coefficient matrix for the system of water surface elevation equations 4.36 is both symmetric and positive definite, 41 a fortuitous circumstance. Therefore, the equations can be solved efficiently by iteration using the preconditioned conjugate-gradient n + 1 n + 1 method discussed in the next section. Once the ζi, j are determined, equation 4.32 and the corresponding equation for Vi, j+ 1⁄ 2 can be solved explicitly for the new layer volumetric transports. 41 For a formal discussion of positive definite equation systems, see Golub and Van Loan (1989, p. 139ff). In general terms, if A is symmetric with non- negative diagonal elements and diagonally dominant, A is positive definite.
Fictitious row y Bandwidth { 0 0 Figure 4.7. Representation of five-diagonal coefficient matrix. Fictitious column 4 3 2 1 x 8 7 6 5 12 11 10 9 16 15 14 13 20 19 18 17 0 24 23 22 21 imax = 10, jmax = 6 0 4. Finite-difference Formulation 89 Finite-difference grid 28 27 26 25 32 31 30 29 Fictitious row Fictitious column Figure 4.8. Illustration of the "natural" ordering scheme used in forming the matrix system of equations for the free- surface elevation. In the above example, the bandwidth would be five. imax is the total number of columns in the finite-difference grid (including fictitious columns). jmax is the total number of rows in the finite-difference grid (including fictitious rows).
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In cooperation with the Interagency
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vi 4.2 Semi-Implicit One-Dimensiona
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viii Figure 5.6. Graph showing solu
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x Tables Table 2.1. Dimensionless n
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