10.3 – Formulation

The conservation equations in your discretizated form, obtained as a result of the application of the finite volumes method, they can be written in the form [MALISKA 2004]:
 


(10.3.1)

It is desired to obtain correction equations for the variable in the form
 




(10.3.2)


Where

it is the corrected solution, and
it is the correction calculated for the block I of the rude mesh that contains the control volume i.

It is desired that the corrected solution respects the conservation equations, in other words, present residual zero.

Adding the residues of the corrected equations ( ) for all the control volumes that are part of the control volume I of the rude mesh, so:
 


(10.3.3)

Where the residuals for each control volume can be written in the form
 

(10.3.4)

Substituting now Eq. (10.3.2) in the Eq. (10.3.4) we have
 

(10.3.5)

That can be written in the form
 


(10.3.6)

The term represented by ri represents the residual obtained with the original equation, before the application of the correction equation (Eq. (10.3.2)). 

Applying now Eq. 10.3.3, we would have
 



(10.3.7)

The Eq. 10.3.7 is the determination corrections equation ().

 

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