How to Derive the Finite Difference Laplacian in Various Coordinates?

[mojomike]

Updated: Finite difference of Laplacian in spherical

Homework Statement

I understand the problem a little better now and am revising my original plea for help. I don't actually need to integrate the expression. Integration was just one technique of arriving at a finite difference expression for the Cartesian Laplacian. I'm now convinced there is a simpler way to do it, possibly with some kind of expansion.

Here's what I'm seeking: A finite difference, 1-dimensional expression of the Laplacian, expressed so that it can be made to work in cartesian(n=0), cylindrical(n=1), or shperical(n=2) coordinates.

∇² φ(r) dr =

$$\frac{1}{r^{n}}\frac{\partial}{\partial r}r^{n}\frac{\partial}{\partial r}\Phi(r) dr$$

Homework Equations

see above

The Attempt at a Solution

I have solved the finite difference laplacian in Cartesian coordinates, which is pretty easy. I just don't know how to express that double derivative with the rⁿ factor in the middle in finite difference form. I'll continue to update my request as I make progress, so nobody thinks I'm loafing here. Thanks again.

,Mike S

 

[mojomike]

Problem resolved. Thanks anyway!

 

[jamesavery]

Dear Mike,

Since you've resolved the problem, perhaps you could also post the solution? This way, when someone with a similar problem searches for it, they will find an answer instead of just the question.

Cheers,