A question about FDM in cylindrical coordinates.
 

Hello everybody!

My question is about solving equations (for example Laplass or Poisson equations) in cylindrical coordinates using the finite difference method.

Actually I dont understand which boundary conditions should I apply for this finite-difference scheme in the point r=0 (or r = "very small dr", one dr step for example).

In the point r=R I use (for example) F(r,fi) = 0; In fi=0 and fi= 2pi I use F(r,0) = F(r,2pi); These are the 3 of 4 sides of the corresponding rectangualar.

But which boundary condition should i write for the point r=0 (or r=dr) ? May be, F(0,fi) = F(o,fi+pi)? Or F(0,fi) = F(0, any fi)?

Please, share your experience.

Re: A question about FDM in cylindrical coordinate
 

It depends on how discretize your domain, for example, cell-centered or cell-edged, etc.

Re: A question about FDM in cylindrical coordinate
 

Sorry, can you clarify your answer?

I discretize my domain using the standart 5-point Laplass operator approximation. Without coefficients it is like that: d2F/dr2 = F(r+1,fi) - 2F(r,fi) + F(r-1,fi) dF/dr = F(r+1,fi) - F(r-1,fi) F = F(r,fi) dF/dfi = F(r,fi+1) - 2*F(r,fi) + F(r,fi-1) ... and so on.

So, we have 5-diagonal matrix (except for boundary conditions for fi) with F(r,fi) on the main diagonal.

Re: A question about FDM in cylindrical coordinate
 

See page 67 of this thesis:

http://ctr.stanford.edu/thesis.pdf

Re: A question about FDM in cylindrical coordinate
 

Use dF/dR = 0 at r=0.

Re: A question about FDM in cylindrical coordinate
 

See paper by Morinishini et al. (2004) in journal of computational physics (and references there). Remember that the centreleine in a non-physical (grid generated) singularity and special treatment is needed there.