Timeline for Applying Dirichlet boundary conditions to the Poisson equation with finite volume method
Current License: CC BY-SA 3.0
15 events
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| Sep 21, 2013 at 9:48 | history | edited | Jan | CC BY-SA 3.0 |
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| Sep 21, 2013 at 9:44 | comment | added | boyfarrell | OK I understand! Thank you. There is a typo. in the 2nd paragraph "Thus, if in your setup, the approach [eqn] is unstable, this is no contradiction to known stability results." The "no" should be "in". This flips the meaning of the sentence to mean the opposite of what you want (I think)! | |
| Sep 21, 2013 at 9:26 | history | edited | Jan | CC BY-SA 3.0 |
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| Sep 21, 2013 at 2:41 | comment | added | boyfarrell | I still don't follow how to shift the cell, but don't worry I will ask another equation about that if I ever need to do it. I should get a copy of Grossmann & Roos it seem useful. There is an update above which is how I will implement your suggestions, would you mind giving me you opinion? Is this a stable way of introducing Dirichlet conditions into the finite volume method? | |
| Sep 21, 2013 at 1:06 | vote | accept | boyfarrell | ||
| Sep 20, 2013 at 18:30 | comment | added | Jan | Yes, in practice, the ghost cell equation is eliminated right away, and the boundary value appears in the right hand side. No, I wouldn't say it is a finite difference. It is rather a finite volume average of the boundary value. But, yes it is accurate of 1st order as shown by Grossmann&Roos. | |
| Sep 20, 2013 at 15:22 | comment | added | boyfarrell | Can the dependence on the value of the ghost cell be removed with this approach? I guess is must not be included into the equations but only used a tool to write the boundary conditions. Regarding the "shifted" boundary cell. It looks like that point uses finite difference rather than the finite volume method. Would that be accurate? | |
| Sep 20, 2013 at 15:10 | history | edited | Jan | CC BY-SA 3.0 |
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| Sep 20, 2013 at 14:59 | history | edited | Jan | CC BY-SA 3.0 |
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| Sep 20, 2013 at 14:58 | comment | added | Jan | Would be interesting to see what happens as $h_\Gamma \to 0$, as this limit is the discretization approach (**). (I have assumed you use the linear interpolation between $\phi_1$ and $\phi_0$.) | |
| Sep 20, 2013 at 14:56 | comment | added | Jan | Yes, if you introduce a ghost cell, then you need not change the grid of your example picture. Regarding the shift you mentioned to establish the situation of my drawing. No, it is not a degenerated cell! The offset $h_\Gamma$ really enters the equations in so far as this strip does not appear in the integrals, taken, e.g., of the right hand side. | |
| Sep 20, 2013 at 14:42 | comment | added | boyfarrell | Thank you Jan, that's really interesting. That would certainly mimic my experience with certain approaches being unstable. Am I right, if I use a ghost cell approach I don't need to shift the last cell so that the centre is on the boundary? I also have a problem with the concept of shifting the boundary cell; doesn't it imply that that cell has zero volume? | |
| Sep 20, 2013 at 13:42 | history | edited | Jan | CC BY-SA 3.0 |
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| Sep 20, 2013 at 13:25 | history | edited | Jan | CC BY-SA 3.0 |
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| Sep 20, 2013 at 12:53 | history | answered | Jan | CC BY-SA 3.0 |