Astrophysics (index)about

Finite Difference Method

(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/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.


Referenced by:
Finite Volume Method (FVM)
Numerical Analysis