One step

Assume a recurrence of the form:

Amplification factor

Let for all (by using Fourier Transform > Sequences) Multiply both sides by and sum both sides over :

Hence we find:

with

The method is stable iff for all .

One side is easy, due to the above calculations. In particular, using Parseval’s Identity:

Suppose now that for some . Find a small nbd of where . Now define to be large on this nbd and 0 otherwise.

Now we calculate:

The integral diverges so the method is unstable.

Suppose we end up with an equation of the form:

Then we would find the solutions to the equation and see if they are both for all .

Tricks like this are used for other multi-step methods.

Suppose now that varies over two spatial coordinates. Then the Fourier Transform is given by:

and it is an isometry from to ie:

Amplification factor also naturally generalizes and we still require for stability.