CFD Numerics: Relaxation Factors

Relaxation factors control the under-relaxation of the solution. This is a technique used for improving the stability of a calculation, especially in solving steady-state analysis, where the first iterations are critical.

Manual Relaxation

The best choice of the relaxation factor is one that is small enough to ensure stability but large enough to move the iterative process to convergence at a decent speed. For example, if $\alpha$ is the relaxation factor then a value of $\alpha$ that is: 

• < 0.15 is not advisable as it will slow down the solution too much.
• > 0.7 can cause unstable solution.
• > 0.9 is not recommended as it can cause divergence.

In the case of manual relaxation, values between 0.3 < $\alpha$ < 0.7 are recommended.

Auto-Relaxation

The auto-relaxation feature is designed to dynamically change the factor $\alpha$ for the flow variables to speed up convergence while trying to keep the solution stable. In some cases, if this causes divergence, it is recommended to try with manual relaxation factors that are fixed over the iterations. 

Under-Relaxation

The method of under-relaxation is basically limiting the amount by which a variable changes from the previous iteration to the next one. Due to the nonlinearity in the equations, it is important to control the change of the variable.

Because of the non-linearities of the equations being solved, it is necessary to control the change of variable $\phi$. This is achieved by under-relaxation as follows:

$${{\phi }_P}^n={{\phi }_P}^{n-1}+\alpha \times ({{\phi }_P}^{n\ast }–{{\phi }_P}^{n-1})$$

Where $\alpha$ is the factor that defines the relaxation such that:

• $\alpha$ < 1 means under-relaxation. This will slow down the convergence rate but increase the stability.
• $\alpha$ = 1 means no relaxation at all. The predicted value of $\phi$ is simply used.
• $\alpha$ > 1 means over-relaxation. It can sometimes be used to accelerate the convergence rate but will decrease stability.

And:

• $n$ refers to the new, used value of ${\phi }_P$;
• $n-1$ refers to the previous value of ${\phi }_P$;
• $n\ast$ refers to the new, predicted value of ${\phi }_P$.

This means that the new value of the variable $\phi$ depends upon the old value, the computed change of $\phi$, and $\alpha$.

The under-relaxation factor $\alpha$, specifies the amount of under-relaxation, such that:

• If $\alpha$ decreases, the under-relaxation increases.
• If $\alpha$ < 1 means the solution is under-relaxed. The specified fraction of the predicted value change is used. This may slow convergence but increases stability.
• If $\alpha$ = 1 there is guaranteed matrix diagonal equality, and no under-relaxation. The predicted value is simply taken.
• If $\alpha$ = 0 the solution does not change with successive iterations.

Last updated: July 6th, 2022