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Poisson equation with periodic and zero gradient boundary conditions

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Old   August 25, 2022, 03:20
Default Poisson equation with periodic and zero gradient boundary conditions
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Divyaprakash
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Hi


I am trying to solve the following problem.


A 2D rectangular domain, with walls boundary condition at the top and bottom and periodic boundary conditions at left and right.




I started by first implementing the pressure-Poisson solver



The boundary condition applied at the walls is



Which gives me

at top and bottom walls respectively.



Applying the periodic BC gives me

at the left and right boundaries respectively.


Using these boundary conditions and discretizing using finite difference method, I obtain the following coefficient matrix on 3X3 grid shown below.
|```````|````````|``````|
| o (7)| o( 8) | o(9)|
|........| .........|.......|
|```````|````````|``````|
| o (4)| o(5) | o(6)|
|........| .........|.......|
|```````|````````|``````|
| o (1)| o( 2) | o(3)|
|........| .........|.......|


A = [
-3 1 1 1 0 0 0 0 0
1 -3 1 0 1 0 0 0 0
1 1 -3 0 0 1 0 0 0
1 0 0 -4 1 1 1 0 0
0 1 0 1 -4 1 0 1 0
0 0 1 1 1 -4 0 0 1
0 0 0 1 0 0 -3 1 1
0 0 0 0 1 0 1 -3 1
0 0 0 0 0 1 1 1 -3]



Q = [1 1 1 1 1 1 1 1 1];


I solved it in Matlab.

I am unable to get a solution for this. It says that the matrix is ill-conditioned.


I had earlier tried to solve this problem with Gauss-Siedel, however it did not converge. I thought an implcit method will surely give me result, but that too doesn't seem to work.

Is there an issue with me formulation?
Is it not possible to obtain a solution for the above set of boundary conditions?

Last edited by Divyaprakash; August 25, 2022 at 03:26. Reason: Formatting error
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Old   August 25, 2022, 05:28
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Filippo Maria Denaro
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You are on the wrong way... First, I suggest to search for similar posts where you can find the correct way.
However, think about the fact that the pressure equation is nothing but the continuisty equation. The divergence-free condition is transformed by means of the Hodge decomposition:


Div v = 0 -> Div (Grad phi - v*)=0


The correct equation to discretize is



Div (Grad phi) =q


where q = Div v*


The problem is closed by the Neumann BCs


v.n=(Grad phi - v*).n



As you have seen, the matrix is singular, if you want a solution for this problem you need to satisfy the compatibility condition.
In your case:


Int[S] dphi/dn dS = 0 = Int[V] q dV


is not satisfied and you cannot get a solution-
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Old   August 25, 2022, 11:33
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Thanks a lot. Can you please elaborate what you mean by the correct way?
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Old   August 25, 2022, 11:40
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Quote:
Originally Posted by Divyaprakash View Post
Thanks a lot. Can you please elaborate what you mean by the correct way?

The way I addressed in the equation I wrote. You have to discretize DivGrad operators, not the Laplacian. Then you have to substitute the correct Neumann BCs.
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boundary conditions, periodic, poisson, pressure, zero gradient

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