[alinik]

Does anyone have any documentations discussing how pseudo-transient method helps convergence in fluent, from theoretical perspective?

Thanks,

[LuckyTran]

Do you have a specific question?

The pseudo-transient method is a form of implicit relaxation (as opposed to explicit relaxation). In explicit relaxation you solve the equation but limit the variable change a posteriori. In implicit relaxation, you don't solve the original equations but relaxed equations. The relaxed equations limit the variable change because you inject the current solution into the equations.

You can think of implicit relaxation as lowering the local cell flow courant number. You can think of a steady state solver as a transient solver with a very wonky but large (almost infinite) time-step and a transient solver as having a small finite time-step.

[alinik]

Thanks LuckyTran.
Yes, I had seen that, but as you can see it does not give a good discussion why this would help.
Reading you comment was helpful.

Can we think of it the following way too?
Adding transient term, makes the equation hyperbolic and it is no longer elliptic, hence it would be easier for the solver to reach convergence.
Is this reasoning correct?

[LuckyTran]

Originally Posted by alinik

Can we think of it the following way too?
Adding transient term, makes the equation hyperbolic and it is no longer elliptic, hence it would be easier for the solver to reach convergence

Not really. Hyperbolic/elliptic/parabolic is a classification referring to the characteristics in one dimension only (either only time or only space). Introducing a transient term doesn't change the characteristics in space.

Also the characteristics has nothing to do with convergence. A parabolic/hyperbolic/elliptic equation can all be solved. It also doesn't explain how under-relaxation relates to stability, which is what the pseudo-transient option does. The pseudo-transient solver applies implicit under-relaxation to a steady solver.

What's hidden-under-the-hood is that the solution update of a steady solver vs iteration has a correspondence in time. So that a steady solver can be interpreted as a transient solver with a very wonky time-step. It's not so wonky when you realize that it is related to the local flow Courant number of each cell. But rather than the entire solution advancing the same time-step, each cell advances at its own local time-step. You can experience this most readily if you use the COUPLED pressure-velocity coupling scheme which is available in steady or transient solvers and has a flow courant number as an input.


Implicit relaxation slows down the solution so that it advances in time slower. It reduces the effective time-step of the simulation. As I mentioned already, a steady solver is like transient solver with a wonky time-step. The pseudo-transient option makes this wonky time-step smaller.