Is there more than one Crank-Nicolson scheme?

gjfelix2001

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!!

AlephZero

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.