5.2 GaussSeidel method
Finite volume numerics generally uses iterative methods to solve each matrix equation. These methods calculate approximate solutions for , which become more accurate with successive iterative solutions.
Iterative methods are preferred because they are more eﬃcient than direct methods, which solve a matrix equation exactly. Gaussian elimination, which is the numerical basis for direct solution methods, has a computational cost . This is prohibitive for many sizes of mesh in ﬁnite volume CFD.
GaussSeidel^{1} is a simple, iterative method which is generally eﬀective for solving transport equations such as the example in Sec. 5.1 . The method is illustrated by a sample equation

(5.2) 
Starting with, , new values of are calculated by Eq. (5.3a ), Eq. (5.3b) and Eq. (5.3c ) in sequence, where the notation “” denotes “ is assigned the value of ”.
The ﬁrst solution of Eq. (5.3a) is . The updated is substituted in Eq. (5.3b), whose solution is . Both updated values are substituted in Eq. (5.3c ) to give .
The process is then repeated and through successive sweeps over the equations the solution converges as shown below.
Variable  Start  Sweep 1  …2  …3  …4 






0.0000  2.0000  2.4167  2.7431  2.8821  
0.0000  0.0000  0.5833  0.8069  0.9121  
0.0000  1.2500  1.6458  1.8392  1.9266  
Variable  Sweep 5  …6  …7  …8  …9 






2.9462  2.9755  2.9888  2.9949  2.9977  
0.9599  0.9817  0.9916  0.9962  0.9983  
1.9665  1.9847  1.9930  1.9968  1.9985 
The error is , i.e. the diﬀerence between the approximate and exact values , and . After 9 sweeps for all variables, i.e. within 0.2% of the exact solution.
In summary, the GaussSeidel method is the following sequence of calculations for , repeated until convergence:

(5.4) 
Convergence of the method, and convergence measures for iterative methods in general, are discussed in the following sections.