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**
UNIT 6 -- FINITE DIFFERENCE THEORY:
**

*** The OBJECTIVE
of UNIT 6 is for you to:**

*** UNDERSTAND
THE BASIS FOR FINITE DIFFERENCES **

*** APPRECIATE
THE COMPUTATIONS THAT A NUMERICAL CODE ACCOMPLISHES FOR YOU **

*** REALIZE
HOW A DISCRETE CALCULATION SIMPLIFIES PROCESSES IN THE PHYSICAL SYSTEM SO YOU
CAN UNDERSTAND THE RESULTS OF NUMERICAL MODELS **

*** UNDERSTAND
THE CONCEPT OF INTERATIVELY SOLVING A MATRIX REPRESENTING A SET OF GROUND-WATER
FLOW EQUATIONS TO CONVERGENCE AND KNOW WHAT TO DO IF THE SOLUTION DOES NOT CONVERGE
**

*** UNDERSTAND
MASS BALANCE AND KNOW WHAT TO DO IF YOU HAVE A POOR MASS BALANCE **

** DISCUSSION**

** We will
use finite differences to solve the ground-water flow equations. Using the finite
difference approach, we divide our system into many pieces
(sometimes called grid blocks, cells or elements). This
is called discretizing or discretization. Each cell can be represented by only
one value of either an independent or dependent variable (that is, only one
value each of hydraulic conductivity, storage coefficient, head, and flow rate
applies to each cell).
The ground-water flow equation is written for each cell.
All but one unknown is defined for each cell, that is,
we must know either the head or the flow rate for each cell. This is easier
than it seems because we know that the net flow for all internal cells must
be zero, because the inflow must balance with the outflow and the change in
storage.
Next we solve the equations simultaneously to determine
the unknown at each cell. Often the models are so large
that we cannot solve all the equations at once in the available memory of our
computer, consequently there are a variety of matrix solution schemes that may
be called upon to approximate the solution. In this unit,
we will discuss finite differences and the items that you will need to be aware
of in order to achieve an acceptable approximate solution. **

** EXERCISES**

**
If you chose to purchase Applied Ground-water Modeling,
read section 2.2
**

** If you
chose to purchase Introduction to Ground-water Modeling, **

read Chapters 2 and 3

** If you
haven't undertaken the calculations at the bottom of the pages illustrating
the explicit and implicit approaches for the simple finite difference problem,
then undertake the following exercises now: **

**
EXPLICIT FINITE DIFFERENCE EXERCISE **

**
IMPLICIT FINITE DIFFERENCE EXERCISE **

** COMMUNICATION**

**Please
bring up any concerns you may have about finite differencing, convergence, or
mass balance. epoeter@mines.edu**