5.4 Residual
In Sec. 5.3 , we established a criterion for convergence of the GaussSeidel method. We now need a way to estimate a level of convergence to determine when to stop iterating.
The analysis of convergence centred on the solution error , introduced in Sec. 5.2 . In practice, cannot be determined since the exact solution is unknown. Instead the residual provides a measure of the accuracy of the solution. The residual vector represents the change to the solution of the equation, required to make exact, according to

(5.10) 
The vector (of size ) provides one value per matrix row, with both positive and negative values. A measure of residual given by a single value, is deﬁned as

(5.11) 
The residual provides a measure of error in the solution of , rather than the absolute error . It is divided by the norms of and to reduce its dependency on the scale of the geometry and solution variable. By reducing its scaledependency, can be used to compare the level of error equitably between simulations at diﬀerent scales.
The ﬁgure above shows calculated from Eq. (5.11 ) following successive sweeps of the GaussSeidel method (starting from the initial ). The graph uses a logarithmic vertical scale since the values of extend over 4 orders of magnitude.
Tolerance
CFD software generally provides the following controls to stop the iterative solver:
Sweeping ceases if either tolerance condition is satisﬁed: ; or , where is the initial residual within the particular solution step. The criterion is often deactivated by setting , especially for transient simulations when suﬃcient accuracy is required at every solution step.