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Solution of Poisson's equation

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Poisson equation occurs in many forms in CFD. Efficient and fast solution to Poisson equation is important aspect of CFD.

In CFD the Poisson equation occurs mainly in the form:


{\partial  \over {\partial x_j }}\left( {\Gamma {{\partial \phi } \over {\partial x_j }}} \right) = q_\phi

The solution

The left hand side can be discretised in the manner explained in section related to diffusive term . The diffusive term can be broken into two explicit and implicit parts.

We have,  \vec \alpha {\rm{ = }}\frac{{{\rm{\vec A}}}}{{{\rm{\vec A}} \bullet {\rm{d\vec s}}}}

giving us the expression:


D_f  = \Gamma _f \nabla \phi _f  \bullet {\rm{\vec A = }}\Gamma _{\rm{f}} \left[ {\left( {\phi _1  - \phi _0 } \right)\vec \alpha  \bullet {\rm{\vec A + }}\bar \nabla \phi  \bullet {\rm{\vec A - }}\left( {\bar \nabla \phi  \bullet {\rm{d\vec s}}} \right)\vec \alpha  \bullet {\rm{\vec A}}} \right]

where  \bar \nabla \phi _f  and  \Gamma _f   are suitable face averages.
When broken into implicit and explicit parts


D_{implicit}  = \Gamma _{\rm{f}} \left[ {\left( {\phi _1  - \phi _0 } \right)\vec \alpha  \bullet {\rm{\vec A }}    } \right]

D_{explicit}  = \Gamma _f \left[ {\bar \nabla \phi  \bullet \vec A - \left( {\bar \nabla \phi  \bullet d\vec s} \right)\vec \alpha  \bullet \vec A} \right]

This gives us the coefficient of matrix as:


A_l  = A_l  + \Gamma _f \vec \alpha  \bullet \vec A
 
A_P  = A_P  - \Gamma _f \vec \alpha  \bullet \vec A

and for source matrix:
 
S_U  = S_U  + q_\phi   - D_{explicit} = S_U  + q_\phi -  \Gamma _f \left[ {\bar \nabla \phi  \bullet \vec A - \left( {\bar \nabla \phi  \bullet d\vec s} \right)\vec \alpha  \bullet \vec A} \right]

Once the coefficient and source matrix is constructed, the system could be solved with the help of iterative solvers.


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