10.3.1 – Example
Consider the two different levels of Multigrid
Writing the Equation 10.3.1 for each one of the control volumes of the cell 05 to the right
 |
(10.3.1.1a) |
 |
(10.3.1.1b) |
 |
(10.3.1.1c) |
 |
(10.3.1.1d) |
 |
(10.3.1.1e) |
 |
(10.3.1.1f) |
 |
(10.3.1.1g) |
 |
(10.3.1.1h) |
 |
(10.3.1.1i) |
The corrected values of the variables can be written in the form
 |
(10.3.1.2a) |
 |
(10.3.1.2b) |
 |
(10.3.1.2c) |
 |
(10.3.1.2d) |
 |
(10.3.1.2e) |
 |
(10.3.1.2f) |
 |
(10.3.1.2g) |
 |
(10.3.1.2h) |
 |
(10.3.1.2i) |
It should be guaranteed that the values of 
obey the conservation equations, therefore
obey the conservation equations, therefore  |
(10.3.1.3a) |
 |
(10.3.1.3b) |
|
(10.3.1.3c) |
 |
(10.3.1.3d) |
 |
(10.3.1.3e) |
 |
(10.3.1.3f) |
 |
(10.3.1.3g) |
 |
(10.3.1.3h) |
 |
(10.3.1.3i) |
Substituting now the Eq. 10.3.2
 |
(10.3.1.3a) |
 |
(10.3.1.3b) |
 |
(10.3.1.3c) |
 |
(10.3.1.3d) |
 |
(10.3.1.3e) |
 |
(10.3.1.3f) |
 |
(10.3.1.3g) |
 |
(10.3.1.3h) |
Analyzing the residues
 |
(10.3.1.5a) |
 |
(10.3.1.5b) |
 |
(10.3.1.5c) |
 |
(10.3.1.5d) |
 |
(10.3.1.5e) |
 |
(10.3.1.5f) |
 |
(10.3.1.5g) |
 |
(10.3.1.5h) |
 |
(10.3.1.5i) |
Rewriting the Eq. 10.3.7
 |
(10.3.1.6) |
or
 |
(10.3.1.7) |
Where
 |
 |
(10.3.1.8a) |
 |
(10.3.1.8b) |
|
 |
(10.3.1.8c) |
|
 |
(10.3.1.8d) |
|
 |
(10.3.1.8e) |
|
 |
(10.3.1.8f) |
|
 |
(10.3.1.8g) |
|
 |
(10.3.1.8h) |
|
 |
(10.3.1.8i) |
|
 |
(10.3.1.8j) |
|