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

Finite difference approximation for third order partials?

  1. Nov 12, 2011 #1
    I'm attempting to perform interpolation in 3 dimensions and have a question that hopefully someone can answer.

    The derivative approximation is simple in a single direction:

    df/dx(i,j,k)= [f(i+1,j,k) - f(i-1,j,k)] / 2

    And I know that in the second order:

    d2f/dxdy(i,j,k)= [f(i+1,j+1,k) - f(i+1,j,k) - f(i,j+1,k) + f(i-1,j-1,k)] / 4

    The final item I need is the third order approximation, and I'm not sure how to scale the first two into a third variable.

    d3f/dxdydz(i,j,k)= ?

    Can anyone shed some light on this?

    Thanks in advance!
     
  2. jcsd
  3. Nov 13, 2011 #2

    hunt_mat

    User Avatar
    Homework Helper

    can't you just apply Taylor series? You want
    [tex]
    \frac{\partial^{3}f}{\partial x\partial y\partial z}
    [/tex]
    Right?
     
  4. Nov 13, 2011 #3
    Yes but I don't have the function itself; I only have its value at various points i-2, i-1, i, i+1, i+2, etc.
     
  5. Nov 13, 2011 #4

    hunt_mat

    User Avatar
    Homework Helper

    Just apply Taylors theorem in 1D to each variable in turn.
     
  6. Nov 14, 2011 #5
    You already have
    d2f/dxdy(i,j,k) = [f(i+1,j+1,k) - f(i+1,j,k) - f(i,j+1,k) + f(i-1,j-1,k)] / 4
    so you have to do the first order derivation for z, which means for indexes in short-hand
    notation

    [(k->k+1) - (k->k-1)]/2

    By the way, congratulations for having chosen the symetrical representation of the first derivative, it is much more accurate than [(k->k+1) - ()].

    So lets do it:
    d3f(x,y,z)/dxdydz = {[f(i+1,j+1,k+1) - f(i+1,j,k+1) - f(i,j+1,k+1) + f(i-1,j-1,k+1)]
    - [f(i+1,j+1,k-1) - f(i+1,j,k-1) - f(i,j+1,k-1) + f(i-1,j-1,k-1)]}/8 =
    [f(i+1,j+1,k+1) - f(i+1,j,k+1) - f(i,j+1,k+1) + f(i-1,j-1,k+1)
    - f(i+1,j+1,k-1) + f(i+1,j,k-1) + f(i,j+1,k-1) - f(i-1,j-1,k-1)]/8
    That's it.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Finite difference approximation for third order partials?
Loading...