Solution of Poisson's equation
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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. | 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> | ||
Revision as of 06:05, 3 October 2005
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:
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, 
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]](/W/images/math/7/7/2/772b96059e9032fc07b25c0dee76db6c.png)
where 
and 
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]](/W/images/math/e/b/6/eb6a1eedd86844e3b07754871a790a5e.png)
![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]](/W/images/math/d/8/f/d8fe36fb5c721a8e4f2b149de57925fb.png)
