Finite differencing on non-uniform grids

[hunt_mat]

Hi,

Recently I had to find a derivative on a uniform grid. Being naive I tried the following scheme:
$$f'(x_{n})=Af(x_{n+2})+Bf(x_{n+1})+Cf(x_{n})+Df(x_{n-1})+Ef(x_{n-2})$$

Then write the $f(x_{n\pm i})$ in terms of $f^{(n)}(x_{n})$ by use of Taylor's theorem. This lead to a system of linear equations for the A,B,C,D,E which required inverting a Vandermonde matrix.

I tried it out a couple of times and it worked okay for the first derivative but when I applied it to higher derivatives it became unstable. Does anyone know what is going wrong?

I also tried a seven point stencil in the same way and that bizarrely was even worse.

Mat

 

[Geofleur]

It might be helpful to look at the actual matrix you get for a case that causes trouble. It's probably also a good idea to do this for the smallest system possible. If you can see the problem play out on a system you could compute by hand, it might well give you an idea about what's going wrong.

 

[hunt_mat]

Possibly, but I have tested the 7 point stencil on simple examples and have not got the answers which I should have. The five point stencil however works relative well for first order derivatives but not on higher derivatives (I think)