Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Solving the 3D Poisson Equation Using Finite Difference/Volume

  1. Nov 12, 2016 #1
    Hi,

    I'm attempting to solve the 3D poisson equation

    ∇ ⋅ [ ε(r) ∇u ] = -ρ(r)

    Using a finite difference scheme.

    The scheme is simple to implement in 3D when ε(r) is constant, and I have found an algorithm that solves for a non-constant ε(r) in 2D. But I am having trouble finding an algorithm that can handle a non-constant ε(r) in 3D.

    Here is the link for the 2D case: www.ece.utah.edu/~ece6340/LECTURES/Feb1/Nagel 2012 - Solving the Generalized Poisson Equation using FDM.pdf

    Would anyone be able to guide me to a 3D case, or help me determine the 3D form of equation 42 in the above link? Below is my guess

    a0 = ε(i,j,k) + ε(i,j-1,k) + ε(i,j-1,k-1) + ε(i,j,k-1) + ε(i-1,j,k) + ε(i-1,j-1,k) + ε(i-1,j-1,k-1) + ε(i-1,j,k-1)
    a1 = 1/4 [ ε(i,j,k) + ε(i,j-1,k) + ε(i,j,k-1) + ε(i,j-1,k-1) ]
    a2 = 1/4 [ ε(i,j-1,k) + ε(i-1,j-1,k) + ε(i-1,j-1,k-1) + ε(i,j-1,k-1) ]
    a3 = 1/4 [ ε(i-1,j,k) + ε(i-1,j-1,k) + ε(i-1,j,k-1) + ε(i-1,j-1,k-1) ]
    a4 = 1/4 [ ε(i,j,k) + ε(i-1,j,k) + ε(i-1,j,k-1) + ε(i,j,k-1) ]
    a5 = 1/4 [ ε(i,j,k) + ε(i-1,j,k) + ε(i-1,j-1,k) + ε(i,j-1,k) ]
    a6 = 1/4 [ ε(i,j,k-1) + ε(i-1,j,k-1) + ε(i-1,j-1,k-1) + ε(i,j-1,k-1) ]

    V(i,j,k) = 1/a0 [ a1V(i+1,j,k) + a2V(i,j-1,k) + a3V(i-1,j,k) + a4V(i,j+1,k) + a5V(i,j,k+1) + a6V(i,j,k-1) - Q(i,j)/ε0]

    Thanks
     
    Last edited by a moderator: May 8, 2017
  2. jcsd
  3. Nov 17, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Solving the 3D Poisson Equation Using Finite Difference/Volume
  1. Poisson Equation (Replies: 0)

Loading...