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**Finite Difference Method** (**FDM**) refers to a means of devising close
estimates of solutions to questions involving calculus, e.g.,
questions for which relevant differential equations can be devised.
It allows calculation of "near" estimates using a lot of arithmetic,
with the tradeoff of: more calculation yields a closer estimate, and
in some cases, it is possible to decide how close an estimate you
want and find one, e.g., if you are calculating a distance, to
identify an estimate no more than 1 km/sec off.
The general field of solving such problems through a lot
of calculation is termed Numerical Analysis, and the
term **Brute Force** is used to indicate that rather than
first solving the equation, so a single calculation yields the
answer, the (estimated) answer is found using a general method that
requires relatively huge amounts of calculation.

Given equations that describe interrelations of various physical
characteristics as variables, one variable can be calculated from
the others if the equation can be solved for the given variable.
In the case of **Differential Equations** (which include not only
variables themselves but the rates of change of some variables with
respect to others) solving the equation for the given variable is
often difficult or impossible, and FDM provides a means of finding
what the variable is given values for the other variables.

A key to constructing a finite difference method for an equation
and ascertaining the accuracy of the estimate it yields is the use of
the **Taylor Series**, a means of breaking down parts of the
solution into a rough estimate and a string of successively finer
estimates. A tradeoff exists between using a rough estimate
with very small differences versus using some finer
estimates, but that do not need such tiny differences to achieve the
same accuracy. These latter are referred to as **Higher-Order Methods**.
Depending upon the problem, this offers multiple strategies
to calculation, and human analysis as well as experimentation
may be used to identify a method that keeps the amount
of computation tractable. Since for every scientific problem, there
is always a harder problem that would need more calculation,
the number of arithmetic operations that can practically be carried out
is the limitation to many scientific studies: if a
question is of sufficient scientific consequence, months may be
spent on a calculation; year-long and decade-long
calculations are less practical, but as computing capacity
and the means to use it increase, more questions are so-addressed.

The **Courant Condition** (or **Courant-Friedrichs-Lewy Condition** or
**CFL Condition**) is a necessary condition to the solution of
Partial Differential Equations using the FDM.

http://en.wikipedia.org/wiki/Courant-Friedrichs-Lewy_condition

Finite Volume Method (FVM)

MITgcm

Numerical Analysis