CFD forward, central and backward difference methods.?

The derivative of a function is the slope of the line tangent at that point (if the function is smooth and continuous at the point). In computer simulations such as computational fluid dynamics (i.e. computer simulations), the derivative is often needed and is usually numerically computed.

The value of the derivative is the limit of the slope of the tangent at that point or δy/δx, which can be approximated by Δy/Δx or (y2-y1)/(x2-x1).

The easiest ways to calculate the value of the derivative uses Δy/Δx, where y = f(x). Since you need two points, and have typically already calculated f(x), if you calculate f(x+h) you have two points, and the Δx = x+h - x = h.

Δy/Δx = ( f(x+h) - f(x) ) / h.

This is called the forward difference. If you reverse it to use f(x) and f(x-h),

Δy/Δx = ( f(x) - f(x-h) ) / h

The problem with these methods is that since one point is f(x), the average of the points is ±1/2 h away from x. Thus, x has to be really small to get an accurate estimate for the value of the slope.

A better method is to do a few extra calcs. Instead of f(x), use f(x+½h) and f(x-½h). These average to f(x), so they give a highly accurate estimate. Thus,

Δy/Δx = ( f(x+½h) - f(x-½h) ) /h

This central difference method normally closes faster, but there are sometimes advantages to using the other methods.

According to Wikipedia, the three will give different results. I'm guessing the differences are similar to the differences between finite integration methods (trapezoid, Simpson's, etc).

What's a CFD ??