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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 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.

One key is the use of a **Taylor series** which can provide formulas
for constructing finite difference methods and also for
and ascertaining the accuracy of the estimate that calculation
has yielded.
A Taylor series breaks a tough equation into
solvable equations for a rough estimate and for a string of successively
finer estimates. A tradeoff exists between using very small distances
(increments)
with a simple formula versus using larger differences with a more
complicated formula and such a Taylor series offers many choices of
"estimate formulas" of varying complexity. Use of the
more complex formulas are referred to as **higher-order methods**.
Depending upon the problem, this offers a choice of strategies
to calculation, and human analysis as well as experimentation
can be used to seek a method that provides a sufficient
solution with a tractable amount of computation.

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/Finite_difference_method

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

finite volume method (FVM)

MITgcm

numerical analysis