s ah, ‘ - r a t

h2 + h, + hq + h, - 4h - AX’ - . (1.4)

This leaves the time derivative on the right side to evaluate. Using an expression similar t o that in equation (1 .3 ) gives

where n is the new time level and A t is the time step. Substitution for the time derivative in equation

(1.4) and using general notation, a finite-difference equation for node (i , j) may be written as

n n n n hi,j-1 + hi+l,j + hi-l,j + hi,j+I

n S Ax2 - 4 h . . = - - (h!j - hrJ1) . (1.6) T A t

An equation similar t o (1.6) is obtained for each node in our grid; however, boundary nodes may require special consideration. In summary, applica- tion of the FDM to a ground-water flow problem involves: (1) divide aquifer into grid blocks with nodes; (2) node spacing is determined by Ax and Ay; ( 3 ) take time steps, A t ; (4) obtain an algebraic equation for each node, and ( 5 ) solve matrix equation.

APPENDIX 2. NUMERICAL MODEL TERMINOLOGY

Numerical models are often referred to on the basis of minor characteristics. For example, a finite- difference model that uses the AD1 method for matrix solution may be called the “AD1 model.” Because of the large number of minor variations, the terminology is confusing. A partial list of terminology (including those terms most commonly used) is presented in outline form. Many of the

Potentiometric _ - I h2/----- Surface

node It- A y * 2 1

Derivative Midway Between Node 2

Fig. 1.3. Approximation of hydraulic head derivative.

terms are defined in the text; others are self- explanatory. For those terms that are not defined, the interested reader is referred to standard texts on numerical methods, such as those listed in the references.

I. Finite-difference methods may be classified by: A. Type of Grid

1. Block-centered 2. Mesh-centered 3. Irregular shape (integrated finite-difference)

1. Explicit in time 2. Implicit in time 3 . Central difference in space 4. Upstream weighting in space

C. Type of matrix equation solution 1. Direct methods

B. Type of approximation

a. Gauss elimination b. Fholesky method c. bparse matrix methods d. Special ordering techniques (e.g. D4)

a. Alternating direction implicit (ADI) b. Strongly implicit procedure (SIP) c. Line successive over-relaxation (LSOR) d. Point successive over-relaxation (PSOR)

2. Itecative methods

11. Finite-element methods may be classified by: A. Type of element

1. Tripngles, 2-D 2. Quadrilateral, 2-D 3 . Tetrahedron, 3-D 4. Prism, 3-D

1. Linear 2. Quadratic 3. Cubic 4. Hermite

equation 1. Variational 2. Weighted residual

a. Calerkin b. Collocation

B. Type of approximating (basis) function

C. Type of method used to obtain integral

D. Type of time approximation E. Type of matrix equarion solution

111. Special’numerical techniques include : A. Method of characteristics B. Nonlinear techniques

1, Newton-Raphson method (fully implicit) 2. Semi-implicit method 3. Quasi-linearization 4. Picard iteration 5 . Extrapolation

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