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

Errors in computing the Crank-Nicolson Method

  1. Sep 2, 2011 #1
    Hi all!
    Having decided to return to coding some of the physical problems that I require solving(in lieu say, of Mathematica), I endeavored to adapt the Crank-Nicolson to my needs, but, in order to test my algorithm, I had first attempted to resolve the common and trivial heat equation, in one dimension, e.g [tex] \alpha\frac{\partial^2u}{\partial x^2} = \frac{\partial u}{\partial t} [/tex], with some plain boundary conditions, that is u(x, 0) = some trivial function(sine times cosine) and u(0, t) = 10(or some value), and at the end-point(10, t in this case), the same(essentially a rod kept at both ends at an equal temperature).
    The results however, are unsatisfactory; For sake of my equation, I used:
    [tex] \beta u^{i+1}_{j+1}+(1+2\beta)u^{i+1}_{j}+\beta u^{i+1}_{j-1}=u^{i}_{j} [/tex]
    Where beta is a constant derived from my step sizes(i.e, dx, dt) and i & j denote the step in time and displacement respectively.
    This, as is well known, creates a tridiagonal system, which is readily evaluated for all U(i+1)ns(in my case, I've got a grid with a size of 100 units).
    I've attached two graphs corresponding to the solutions; Image2 was created by Mathematica, and Image1 is done using my program. You can see the very disturbing groove that's created, that seems to wedge and ebb towards the margins of the computational region. This is, first-and-foremost, physically inaccurate, and should not occur, and in fact, using Mathematica, does not.
    In short, I am reliant on your aid to solve this discrepancy,
    I thank you for your attention,
    Beholden,
    Daniel
     

    Attached Files:

  2. jcsd
  3. Sep 10, 2011 #2

    AlephZero

    User Avatar
    Science Advisor
    Homework Helper

    One obvious difference between the plots is that the slope of the function at the left and right edges is different.

    In the right hand plot the slopes apparear to be constrained to be 0, in the left hand plot they are not.

    It's not very clear from your description of the problem or from the the plots (no labels on the axes) exactly what you were trying to do, but I would guess the boundary conditions are different in your code and in Mathematica.
     
  4. Sep 17, 2011 #3
    You're absolutely right on the matter, and indeed, correct, or rather, setting the proper Dirichlet boundary conditions aligned the two!
    Thank you for your help!!
    Daniel
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Errors in computing the Crank-Nicolson Method
  1. Crank Nicolson method (Replies: 0)

  2. Error in code (Replies: 4)

  3. F77 error (Replies: 3)

Loading...