Astrophysics (Index)About

finite difference method

(FDM)
(means of estimating calculus and differential equation solutions)

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.


(mathematics,physics,computation)
Further reading:
http://en.wikipedia.org/wiki/Finite_difference_method
http://en.wikipedia.org/wiki/Courant-Friedrichs-Lewy_condition

Referenced by pages:
finite volume method (FVM)
MITgcm
numerical analysis

Index