Is there more than one CrankNicolson scheme?by gjfelix2001 Tags: crank nicolson, diffusion, heat, numerical, pde 

#1
Aug711, 06:40 PM

P: 19

Hi everybody...
I want to solve the diffusion equation in 1D using the CrankNicolson 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_{i1,j}+w_{i+1,j+1}2w_{i,j+1}+w_{i1,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,j1}}{k}\frac{\alpha^2}{2h^2}\Big[w_{i+1,j}2w_{i,j}+w_{i1,j}+w_{i+1,j1}2w_{i,j1}+w_{i1,j1}\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,j1}[/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_{i1,j+1}[/itex] and [itex]w_{i+1,j1}2w_{i,j1}+w_{i1,j1}[/itex]. Are both schemes named CrankNicolson? Can somebody help me with this?? Thanks!! 



#2
Aug711, 07:31 PM

Engineering
Sci Advisor
HW Helper
Thanks
P: 6,388

It is just a difference in notation.
If you replace j+1 by j and j by j1 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 j1 terms. 


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