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March 1, 2016, 12:39 
Diffusion Equations coupled by Boundary Conditions

#1 
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Michael
Join Date: Mar 2016
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Hi, I'm new here and not sure whether my question is in the right place.
I have the following problem: I'd like to solve two diffusion equations, e.g. and which means the PDEs are coupled by their boundary conditions. The solution of u depends on w by a Dirichlet BC. And w depends on u by a Neumann BC. At the moment I first solve the equation for u, then compute g(u) and afterwards solve the equation for w, which allows me to compute f(w) for the next iteration, etc. .... This works very nice for the stationary case, i.e. for For the time dependent case however....the method is highly unstable. I'm searching now for a method to solve both equations simultaneously, somehow  instead of one after another. But I have no clue how to do this. If one searches for literature about this issue, it seems nobody has ever had the problem to solve PDEs coupled by boundary conditions. Any hint? Any literature? Ideas? BTW: I'm not working with any commercial software, but a own FEM software written in Fortran. Therefore please don't recommend commercial software. Greetings, AmigaFreak 

March 1, 2016, 13:06 

#2 
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Filippo Maria Denaro
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Quote:
numerical instability appears if you do not use a time step within the stability region. What criterion have you used? 

March 1, 2016, 13:38 

#3 
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Michael
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March 1, 2016, 15:29 

#4 
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Filippo Maria Denaro
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What about the functions for the bc.s ?


March 1, 2016, 15:57 

#5 
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Michael
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March 1, 2016, 17:07 

#6 
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It seems that your instability (I am saying this without knowing anything about the way you implement this numerically...) is due to the fact that while the equation system is elliptical (everything happens instantaneously everywhere at each timestep), you introduce a timelag between the first equation and the second.
Did you write the solver yourself? In that case, writing it to solve the two equations at the same time should be straightforward. A more elegant solution would be to write the solver as implicit. Compared to NavierStokes equation, your set is relatively simple and, above all, linear. This means that the implicit solver, even in 2D and 3D, remains quite simple. The advantage is that the boundary conditions are imposed at the same of the solution. If you did not write the code, you may "trick" the solver into solving both equations (they are the same) by using a vector that is twice as big {u  w} instead of each vector {u} and {w} 

March 1, 2016, 17:40 

#7 
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Filippo Maria Denaro
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The set of equations is parabolic in the unsteady case and classical numerical integrations, such as the CrankNicolson method, can be used.
The only care is to provide BC.s that ensure that the time derivatives do not blowup. For example, the Neumann BC.s for w can be critical, it is simple to see that by integration over the whole computational domain V: d/dt Int [V] w dV = D2 Int[S] g(u) dS Therefore, if g(u) is such that Int[S] g(u) dS > 0 then the integral of the w will blowup in time 

March 1, 2016, 20:53 

#8 
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Thanks for the correction, FMDenaro. I guess I overlooked the derivative over time.
I agree, parabolic > CrankNicholson Please, disregard my previous comment 

March 2, 2016, 05:26 

#9  
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Michael
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No, I'm a PhD student and the solver was written by other PhD students before me, several years ago.
Quote:
Maybe It's easy and I'm just dumb, but I don't see how to do this. 

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