# Rhie-Chow interpolation

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we have at each cell descretised equation in this form,

$a_p \vec v_P = \sum\limits_{neighbours} {a_l } \vec v_l - \frac{{\nabla p}}{V}$ ;

we have

$\vec v_P = \frac{{\sum\limits_{neighbours} {a_l } \vec v_l }}{{a_p }} - \frac{{\nabla p}}{{a_p V}}$

For continuity :

$\sum\limits_{faces} {\vec v_f \bullet \vec A} = 0$

so we get:

$\left[ {\frac{{\sum\limits_{neighbours} {a_l } \vec v_l }}{{a_p }}} \right]_{face} - \left[ {\frac{{\nabla p}}{{a_p V}}} \right]_{face} = 0$

this gives us:

$\left[ {\frac{{\sum\limits_{neighbours} {a_l } \vec v_l }}{{a_p }}} \right]_{face} = \left[ {\frac{{\nabla p}}{{a_p V}}} \right]_{face}$

defining $H = \sum\limits_{neighbours} {a_l } \vec v_l$

$\left[ {\frac{1}{{a_p }}H} \right]_{face} = \left[ {\frac{1}{{a_p }}\frac{{\nabla p}}{V}} \right]_{face}$

from this a pressure correction equation could be formed as:

$\left[ {\frac{1}{{a_p }}H} \right]_{face} - \left[ {\frac{1}{{a_p }}\frac{{\nabla p^* }}{V}} \right]_{face} = \left[ {\frac{1}{{a_p }}\frac{{\nabla p^' }}{V}} \right]_{face}$

This is a poisson equation.

Here the gradients could be used from previous iteration.