What is the finite difference method and how does it involve a grid?

[courtrigrad]

What exactly is the finite difference method and difference equations? I know you work backwards, but could you provide some examples? Also how does the finite difference method involve a grid?

Thanks

 

[arildno]

Effectively, the finite difference method replaces a differential equation with a difference equation.
Since a differential equation must hold for all points on a domain (like the real line, or plane), we require that the difference equation must be satisfied on a grid (which is the discrete version of the domain).

 

[HallsofIvy]

A very simple example is the best I can do here.

Suppose you have the differential equation y"- y= 0 with boundary conditions y(0)= 1, y(1)= 2. Divide the interval from 0 to 1 into 4 equal parts (I SAID "simple"). We can approximate the derivative y' (which is limit (y(x+h)- y(x))/h with h going to 0)by the "finite difference" (y(x+y)- y(x))/h. Obviously, the smaller h is, the more accurate this is- here h= 0.25 which isn't all that accurate. The more partitions, the more accurate.
There are a variety of possible "finite difference" formulas for the second derivative- the best is the "centered difference": y" is approximately ((y(x)- y(x-h))/h - (y(x+y)- y(x))/h)/h= (2y(x)- y(x+h)-y(x-h))/h². Let y_i be y(x) at each of the "partition" points (also called "knots") and plug into the differential equation using the finite difference equation. That will give an equation (linear if the de is linear) for the "unknowns" y_i. Because we are using the centered difference we can only do that at the "internal" knots, not the endpoints. If we have n intervals, there will be n-2 internal knots and so that only gives n-2 equations for the n y_i values. Fortunately, we are given the values at the endpoints so we have n (linear) equations for the n values.

In the example I gave, The equations would be
y₁= 1
-y₁+ 2y₂- y₃= 0
-y₂+ 2y₃- y₄= 0
y₄= 2


Notice, by the way, that, for each i, the equation involves only y_i, y_i-1, and y_i+1: 3 values only out of the possible n values. Written as a matrix, we would have a matrix in which only the central 3 diagonals have non-zero entries. That's a "tri-diagonal" matrix, a special case that is well studied. There are methods for solving tri-diagonal matrix equations using only the three diagonal values. If we divided into, say, 10000 intervals to get really good accuracy, that would give us 10000 equations in 10000 unknowns- but the three central diagons have 10000+ 9991+ 9991= 29998 entries- still large but only 0.03% of 10000x10000 possible entries- a "sparce" matrix.

 

[courtrigrad]

Also, could you graph finite difference methods on as a curve? Are there any sources that are good fot this topic?

Thanks