- #1
debilo
- 6
- 0
- TL;DR
- I try to solve 3D Poisson equation with boundaries condition at infinity.
Hello !
I want to solve the 3D Poisson equation using spherical coordinates and spherical harmonics.
First I must solve this : ##d^2\phi/dr^2 + 1/rd\phi/dr - l*(l+1)/r^2 = \rho (r)## with ##\phi (\infty ) = 0## (here ##\phi## is the gravitationnal potential and ##\rho## is the mass density).
To deal with the infinite grid I tried to use the hyperbolic transformation ##r = X*arctanh(R)## with ##X## a constant and ##R## between 0 and 1.
I use a LU-decomposition method to solve this. For ##\phi (0)## I apply symmetrical conditions.
I tried to solve the problem in the case of a spherical mass distribution. I compared the calculated potential with the theoretical potential.
I find a pretty good result when I fix the value in rmax.
But for ##\phi (\infty )## I have a problem. In fact I obviously get in trouble when I let R go to 1 (or 0) ; so I take ##Rmin = 0+\varepsilon## and ##Rmax = 1-\varepsilon##.
My question is : do you know a trick to really set the value of the potential to zero at infinity ?
I want to solve the 3D Poisson equation using spherical coordinates and spherical harmonics.
First I must solve this : ##d^2\phi/dr^2 + 1/rd\phi/dr - l*(l+1)/r^2 = \rho (r)## with ##\phi (\infty ) = 0## (here ##\phi## is the gravitationnal potential and ##\rho## is the mass density).
To deal with the infinite grid I tried to use the hyperbolic transformation ##r = X*arctanh(R)## with ##X## a constant and ##R## between 0 and 1.
I use a LU-decomposition method to solve this. For ##\phi (0)## I apply symmetrical conditions.
I tried to solve the problem in the case of a spherical mass distribution. I compared the calculated potential with the theoretical potential.
I find a pretty good result when I fix the value in rmax.
But for ##\phi (\infty )## I have a problem. In fact I obviously get in trouble when I let R go to 1 (or 0) ; so I take ##Rmin = 0+\varepsilon## and ##Rmax = 1-\varepsilon##.
My question is : do you know a trick to really set the value of the potential to zero at infinity ?