Solution of Poisson's equation

[[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: <br>
:<math>
{\partial  \over {\partial x_j }}\left( {\Gamma {{\partial \phi } \over {\partial x_j }}} \right) = q_\phi  
</math>

== The solution ==

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

We have, <math> \vec \alpha {\rm{ = }}\frac{{{\rm{\vec A}}}}{{{\rm{\vec A}} \bullet {\rm{d\vec s}}}} </math>

giving us the expression: <br>
:<math>
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]
</math> <br>
where <math> \bar \nabla \phi _f  </math> and <math> \Gamma _f   </math> are suitable face averages. <br>
When broken into implicit and explicit parts <br>
:<math>
D_{implicit}  = \Gamma _{\rm{f}} \left[ {\left( {\phi _1  - \phi _0 } \right)\vec \alpha  \bullet {\rm{\vec A }}    } \right]
</math> <br>

:<math>
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]
</math> <br>

This gives us the coefficient of matrix as:
:<math>
A_l  = A_l  + \Gamma _f \vec \alpha  \bullet \vec A
</math> <br>

:<math> 
A_P  = A_P  - \Gamma _f \vec \alpha  \bullet \vec A
</math> <br>

and for source matrix: <br>
<math> 
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]
 
</math> <br>

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

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<i> Return to [[Numerical methods | Numerical Methods]] </i>