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Is there more than one Crank-Nicolson scheme?

  1. Aug 7, 2011 #1
    Hi everybody...

    I want to solve the diffusion equation in 1D using the Crank-Nicolson scheme. I have two books about numerical methods, and the problem is that in "Numerical Analysis" from Burden and Faires, the differences equation for the diffusion equations is:


    [itex]\frac{w_{i,j+1}-w_{i,j}}{k}-\frac{\alpha^2}{2h^2}\Big[w_{i+1,j}-2w_{i,j}+w_{i-1,j}+w_{i+1,j+1}-2w_{i,j+1}+w_{i-1,j+1}\Big]=0[/itex]

    On the other hand, in "Numerical and analytical methods for scientists and engineers using mathematica", the same equation is expressed as:

    [itex]\frac{w_{i,j}-w_{i,j-1}}{k}-\frac{\alpha^2}{2h^2}\Big[w_{i+1,j}-2w_{i,j}+w_{i-1,j}+w_{i+1,j-1}-2w_{i,j-1}+w_{i-1,j-1}\Big]=0[/itex]

    [itex]i[/itex] represents the space steps, [itex]j[/itex] the time steps, [itex]k[/itex] is [itex]\Delta t [/itex], [itex]h[/itex] is [itex]\Delta x[/itex]

    Should this schemes yield the same results? Why the differences?

    I mean, in the first term of the first scheme, the numerator is [itex]w_{i,j+1}-w_{i,j}[/itex], but in the second scheme is [itex]w_{i,j}-w_{i,j-1}[/itex].

    In addition to this, the last 3 terms of the equations (inside the brackets) are [itex]w_{i+1,j+1}-2w_{i,j+1}+w_{i-1,j+1}[/itex] and [itex]w_{i+1,j-1}-2w_{i,j-1}+w_{i-1,j-1}[/itex].

    Are both schemes named Crank-Nicolson?

    Can somebody help me with this?? Thanks!!
     
    Last edited: Aug 7, 2011
  2. jcsd
  3. Aug 7, 2011 #2

    AlephZero

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    It is just a difference in notation.

    If you replace j+1 by j and j by j-1 in the first equation, you get the second equation (but with the terms in the [ ] written in a different order).

    The method described in the first book is going to solve for the j+1 terms using the j terms. The second book is going to solve for the j terms using the j-1 terms.
     
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