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UNIT 6 -- FINITE DIFFERENCE THEORY:

Up to this time we have dealt with very simple ground-water systems. The properties have been homogeneous, geometry and boundary conditions have been simple and constant, and often we have dealt with variation in only one dimension. We can tackle complex problems by using numerical models. So at this point in the course, we begin to talk about the issues that are important to numerical modeling.

* 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.

FINITE DIFFERENCE MAIN PAGE

EXERCISES

If you chose to purchase Applied Ground-water Modeling,

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

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

COMMUNICATION

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

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