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 Mike May 17, 2005 14:57

relaxation factors and time accuracy

I am wanting to create a time accurate model of an unsteady process. Every time step, the solution is changed by deltaV. If I understand relaxation factors correctly,

Vn+1 = Vn + (RF)*deltaV

where RF is the relaxation factor. Do all my relaxation factors need to be 1 for the solution to be time accurate?

Thanks,

Mike

 Jim_Park May 17, 2005 15:21

Re: relaxation factors and time accuracy

This doesn't seen right.

If you're doing an explicit technique,

V(n+1) = V(n) + dt*(partial V/partial t)(n), (1) where t is the time and dt is the time step,

dt = t(n+1) - t(n). (2)

The partial derivative of V wrt t (eqn 1) is evaluated by finite differences, finite volumes, or some such based on the values at t(n). That is explicit, no iteration, no relaxation.

If you use an implicit technique, the partial derivative of V wrt t is evaluated at some time between t(n) and t(n+1). Then the right hand side (rht) of eq. 1 depends on the solution V(n+1):

V(n+1) = V(n) + dt*(partial V/partial t)(n+?), (3)

Eq. 3 may be solved iteratively, perhaps by relaxation. In that case, the accuracy of the solution shouldn't be effected by the relaxation factor chosen [IF the process converges], but the speed of solution is greatly impacted by the value chosen. Often, the process will not converge to a solution if the relaxation factor is too large!

You might thoroughly sort out first the concepts of explicit and implicit methods, then look into solution techniques for the implicit methods.

 Mani May 17, 2005 16:08

Re: relaxation factors and time accuracy

Here's what Jim said in short form:

If your index 'n' denotes a real-time level, then you're doing explicit time marching, and your relaxation factor must be 1 (i.e. no relaxation).

If the index 'n' denotes one of many iterations within one real-time step in an implicit method, then you can (and should) apply relaxation.

It really sounds like you might be confusing both methods right now.

 Mike May 17, 2005 17:53

Re: relaxation factors and time accuracy

Thanks guys,

I'm using Fluent, with an implicit solver. The help files tell me nothing about time accuracy. I'm currently using underrelaxation, and 30 iterations per time step. The iterations per time step make me think that

If the index 'n' denotes one of many iterations within one real-time step in an implicit method, then you can (and should) apply relaxation.

Will this work?

Thanks for you time,

Mike

 Jim_Park May 18, 2005 08:44

Re: relaxation factors and time accuracy

Thanks Mani,

Very nice clarification of what I tried to say!

Jim

 Mani May 18, 2005 12:30

Re: relaxation factors and time accuracy

So you are using Fluent. I would like to assume that their code is doing the right thing, so I don't see whyrelaxation of the implicit scheme doesn't work. You should check if 30 iterations are enough for convergence within each time step. Underrelaxation may increase stability at the expense of convergence, so don't use it excessively, and increase the number of iterations.

What was the reason for applying underrelaxation? Stability problems?

 Mike May 18, 2005 14:03

Re: relaxation factors and time accuracy

I used underrelaxtion to stop the residuals from blowing up. I also tried reducing the time step, but it needed to be excessively small to work without underrelaxation. I have enough iterations per time step that my code does converge every time step.

Thanks for the help,

Mike

 Larry Clark May 21, 2005 12:41

Re: relaxation factors and time accuracy

I have just an experience answer for the time accurate question. I have a finte volume implicit Patankar type code(compressible) and have done turbo engine stall inlet inlet unstart problems with it to good effect. I found that if I under relaxed the pressure the result would be to introduce a time lag in the pressure pulse. If I kept the relaxation of the pressure at 1.0 phase would match experiment(I also did studies with shock reflection problems to develop the approach). The relaxation of velocity and temperature did not seem to affect the result. I have no theory for this, just observation.

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