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Advection equation and CrankNicolson 
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#1
May1411, 06:26 AM

P: 8

Hi, I want to numerically model the advection equation using the CrankNicolson scheme. Yes, I know that it is highly oscillatory but that is the point of the exercise as I want to highlight this. The problem I'm having is how do I apply the BC for grid point N in the scheme. The advection equation only needs one boundary condition at point 0 in the domain, but becuase of the centred space disretisation the scheme requires an artificial boundary condition at the other end. The basic equation is
[tex]\\frac{\sigma}{4}f_{i1}^{n+1}+f_{i}^{n+1}+\frac{\sigma}{4}f_{i+1}^{n+1}=\frac{\sigma}{4}f_{i1}^{n}+f_{i}^{n}\frac{\sigma}{4}f_{i+1}^{n}[/tex] So say I want the value at N+1 to be the same as N, that requires [tex]\left.\frac{\partial f}{\partial x}\right_{N}=\frac{f_{N+1}^{n}f_{N1}^{n}}{2\triangle x}=0[/tex] and hence [tex]f_{N+1}=f_{N1}[/tex] So if we sub that into the main scheme we get [tex] f_{N}^{n+1}=f_{N}^{n}[/tex] So according to this the final grid point always remains at the initial condictions, which is clearly wrong. Does anyone know what is wrong with my assumptions? Thanks for any info. 


#2
May1411, 12:49 PM

P: 659

Lots of ways, some better/easier than others.
You can try making your domain a cell too large at every boundary where you need to enforce a BC at the cell edge, then interpolate. You can also make the domain periodic, but then you get periodic solutions (which may or may not be acceptable). 


#3
May1511, 05:46 AM

P: 8

Unfortunately I can't use a periodic domain as I want to see the solution at the final time compared with the initial time. 


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