Relaxation and Convergence Criteria

Numerical methods used to solve the equations for fluid flow and heat transfer most often employ one or more iteration procedures. By their nature, iterative solution methods require a convergence criteria that is used to decide when the iterations can be terminated.

In many cases, iteration methods are supplemented with relaxation techniques. For example, over-relaxation is often used to accelerate the convergence of pressure-velocity iteration methods, which are needed to satisfy an incompressible flow condition. Under-relaxation is sometimes used to achieve numerically stable results when all the flow equations are implicitly coupled together.

Choosing Relaxation Criteria

The amount of over or under-relaxation used can be critical. Too much leads to numerical instabilities, while too little slows down convergence. Similarly, a poorly chosen convergence criteria can lead to either poor results (when too loose) or excessive computational times (when too tight).

Selecting proper relaxation and convergence criteria can be a difficult and frustrating experience for users of computational fluid dynamics software. The criteria depend on the specifics of the problem being solved, which may change during the evolution of a problem. Unfortunately, there are no universal guidelines for selecting criteria because they depend not only on the physical processes being approximated, but also on the details of the numerical formulation. Many CFD programs have a standard set of recommended criteria, but users must often resort to trial-and-error adjustments to get good results.

Relaxation and Convergence Criteria

Numerical methods used to solve the equations for fluid flow and heat transfer most often employ one or more iteration procedures. By their nature, iterative solution methods require a convergence criteria that is used to decide when the iterations can be terminated.

In many cases, iteration methods are supplemented with relaxation techniques. For example, over-relaxation is often used to accelerate the convergence of pressure-velocity iteration methods, which are needed to satisfy an incompressible flow condition. Under-relaxation is sometimes used to achieve numerically stable results when all the flow equations are implicitly coupled together.

Choosing Relaxation Criteria

The amount of over or under-relaxation used can be critical. Too much leads to numerical instabilities, while too little slows down convergence. Similarly, a poorly chosen convergence criteria can lead to either poor results (when too loose) or excessive computational times (when too tight).

Selecting proper relaxation and convergence criteria can be a difficult and frustrating experience for users of computational fluid dynamics software. The criteria depend on the specifics of the problem being solved, which may change during the evolution of a problem. Unfortunately, there are no universal guidelines for selecting criteria because they depend not only on the physical processes being approximated, but also on the details of the numerical formulation. Many CFD programs have a standard set of recommended criteria, but users must often resort to trial-and-error adjustments to get good results.

Thanks for your through write up.

My question is why there a slow convergence while you reduce the underrelaxation factors?

Because low relaxation factor means low residual, means quick convergence.

Please explain this.

Thanks

Vivek

You only need relaxation factors less than one (w < 1) if you're having issues with instabilities where your solutions are likely to overshoot their "true" solutions.

As Pedro stated earlier, the updated value is computed as a portion of the new and old values:

uk+1 := w*uk+1 + (1-w)*uk

In doing so, the final value on the left-hand side (uk+1) will more slowly approach the "true" target solution, and the residuals (uk+1 - uk) will decrease proportionally to the relaxation factor (w). Therefore, convergence may actually take longer, but at a reduced risk of overshooting/oscillating around the "true" solution.

Conversely, if your simulation is very stable and you want to speed up the convergence, use over-relaxation (w>1) -- this will increase the residuals.

Relaxation factors are primarily used for maintaining solution stability and as everyone pointed out, relaxation factors less than 1 may keep your solution stable and prevent overshooting the true solution.  For this reason I generally set up additional surface monitors such as a flux or resulting pressure force to monitor convergence.  I also may start at recommended relaxation values or start low and work up if the solution appears to be stable.  But as Franco pointed out, many times it is trial and error and problem dependent.

The theory behind under-relaxation, residuals and convergence has been well covered by other contributors to this thread. From a practical point of view, I can share some tips from my experience. 

Firstly, the use of residuals to judge convergence is often not very useful and can even result in substantial errors. Personally, I never use residuals to judge convergence, but rather monitor the one or two most important simulation outputs to see when a converged solution is reached. 

For transient simulations, you can get a fairly good approximation of the necessary number of iterations per timestep by using the equation provided by Dennis. For example, if your numerical problem is fairly simple and your lowest under-relaxation factor can be as high as 0.7, you can get away with only about 4 iterations per timestep to achieve 99% convergence, but if your problem requires under-relaxation factors as low as 0.2, you will need a full 21 iterations to achieve the same level of convergence. From my experience, this rule of thumb generally leads to good results.

Transient simulations typically take a lot of time, so the best practice here is to start with very high under-relaxation factors and only reduce them if your simulation crashes. Just make sure you always have a saved data file to return to in the case of a diverged solution. 

For steady simulations, you should also use the highest under-relaxation factors that your simulation can handle without crashing. However, in this case, it is good practice to see if the under-relaxation factors have an influence on the final converged solution. In some cases, a steady solution obtained with high under-relaxation factors can be different from one obtained with lower under-relaxation. For this reason, once the simulation output you monitor to judge convergence becomes stable, you should reduce the under-relaxation factors and run a few more iterations to see if the solution changes significantly. 

Is there a relationship between under-relaxation factors and time step? For example, when I reduce my under-relaxation factors (under 1) and also reduce my time step (0.05 delta/alpha degree), would that affect my convergence? I notice my case diverging when the under-relaxation factors along with time step are very low. Please advice. Thanks

Rahul.S