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Problem with Poisson equation with Neumann BC

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Old   March 27, 2011, 11:23
Default Problem with Poisson equation with Neumann BC
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Jeff
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Hi everybody !

In my CFD code, I use the pressure correction method and the package Fishpack to solve the pressure poisson equation. I tried to make my own solver for 2D Poisson equation with non homogeneous Neumann BC (FD scheme with dx=dy)

\frac{\partial^2 p}{\partial x^2}+\frac{\partial^2 p}{\partial y^2}=f(x,y) \qquad \forall (x,y) \in [a,b]\times[c,d]
\frac{\partial p}{\partial x}=h \qquad \forall y \quad x=a,b
\frac{\partial p}{\partial y}=g \qquad \forall x \quad y=c,d


I realized that it was not as easy as Dirichlet BC : I don't really know how to construct my matrix A. For example for a grid of 3x3 points, here is my matrix A : ( in f_{i,j} i represtents x and j represents y)

\begin{pmatrix}
1 & -\frac{1}{2} & 0 & -\frac{1}{2} & 0 & 0 & 0 & 0 & 0\\
-\frac{1}{2} & 2 & -\frac{1}{2} & 0 & -1 & 0 & 0 & 0 & 0\\
0 & -\frac{1}{2} & 1 & 0 & 0 & -\frac{1}{2} & 0 & 0 & 0\\
-\frac{1}{2} & 0 & 0 & 2 & -1 & 0 & -\frac{1}{2} & 0 & 0\\
0 & -1 & 0 & -1 & 4 & -1 & 0 & -1 & 0\\
0 & 0 &-\frac{1}{2} &0 & -1 & 2 & 0 & 0 & -\frac{1}{2}\\
0 & 0 & 0 & -\frac{1}{2} & 0 & 0 & 1 & -\frac{1}{2} & 0\\
0 & 0 & 0 & 0 & -1 & 0 & -\frac{1}{2} & 2 & -\frac{1}{2}\\
0 & 0 & 0 & 0 & 0 & -\frac{1}{2} & 0 & -\frac{1}{2} & 1\\
\end{pmatrix}

My vector of unknowns p :
\begin{pmatrix}
p_{1,1}\\
p_{2,1}\\
p_{3,1}\\
p_{1,2}\\
p_{2,2}\\
p_{3,2}\\
p_{3,1}\\
p_{3,2}\\
p_{3,3}\\
\end{pmatrix}

and the vector B :
\begin{pmatrix}
\frac{1}{4}(-dx^2f_{1,1}+g+h)\\
\frac{1}{2}(-dx^2f_{2,1}+g)\\
\frac{1}{4}(-dx^2f_{3,1}+g+h)\\
\frac{1}{2}(-dx^2f_{1,2}+h)\\
-dx^2f_{2,2}\\
\frac{1}{2}(-dx^2f_{3,2}+h)\\
\frac{1}{4}(-dx^2f_{3,1}+g+h)\\
\frac{1}{2}(-dx^2f_{3,2}+g)\\
\frac{1}{4}(-dx^2f_{3,3}+g+h)\\
\end{pmatrix}

What do you think ? Is that correct ? What method should I use to solve this system ? SOR method ?
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