Hi All,
My experience with relaxation factors comes from requiring the simulation to converge sufficiently in exchange for simulation time.
Sometimes a complex mesh or meshes for highly complex phenomena is very difficult to refine and even after significant iterations of mesh refinement, there will still be cells that will cause significant numerical fluctuations in the residuals. That’s where the relaxation factors come in. With relaxation factors you are able to influence either the final convergence or speed of the convergence. In exchange, you sacrifice either simulation time (under-relaxation) or convergence stability (over-relaxation).
So all in all, it really depends on what you need. Say if your simulation is very complex, difficult to refine but you have the computing time and resources available, usage of under-relaxation factors to allow the simulation to converge might be what is just needed to get the results you want.
Similarly, if you have a relatively simple case with a good mesh that is taking significantly longer to converge, you can apply over-relaxation to allow it to converge faster provided the instability in residuals does not cause the solver to prompt and error.
In use cases, as Ric has pointed out, one would leave the relaxation factors as default and then apply them as needed. Relaxation factors should not affect the result, merely the convergence. Of course then, a poor convergence will mean inaccurate results so keep that in mind.
As for the math, I went to do some searching and this simple equation poped out from the CFD forums.
Under and over relaxation factors control the stability and convergence rate of the iterative process.The under relaxation factor increases the stability while over relaxation increases the rate of convergence.
xk+1 = w.xcal + (1-w).xk
yk+1 = w.ycal + (1-w).ykFor 0<w<1, the method is known as successive under relaxation.
For 1<w<2, the method is known as successive over relaxation.where xcal is the value calculated based on xk
If anyone can find a proper mathematical description and break it down that would be great.
Cheers.