[ibopaul]

Hi,

I read reference paper 'Implicit solution of Preconditioned Navier-Stokes Equations Using Algebraic Multigrid' and found that CFL number was close to infinity when AMG used in a density-based algorithm for Euler and Navier-Stokes equations.

Should CFL numbers always be large when using algebraic multigrid for density based solver?

If CFL number is 1 or less than 1, there is no acceleration while using algebraic multigrid?

Thanks in advance.

[sbaffini]

CFL in that algorithm (Fluent one if I recall correctly) is basically an implicit under relaxation factor that affects the diagonal dominance of the system you are going to solve at each iteration. Still, it is also connected to a local time step.

What happens when you lower the CFL is that, for each iteration, you advance less but you do it more robustly (the system more and more resembles a diagonal one).

AMG is used to accelerate each of these single iterations or, better, solve the system in them. But, as with lower CFL the system becomes more diagonal and the solution advances less, the AMG acceleration becomes less useful. Indeed, with CFL 1 you could probably go full explicit and see little difference in the overall solution advancement during the iterations

[sbaffini]

Now I saw your previous post as well. Note that the capability to work at very large CFL numbers is not only linked to the proper coding of the AMG, but also depends from all the other parts in the code, that must be implicit. And, in any case, not all problems can work at infinite CFL, that is only true for linear problems.

[ibopaul]

Originally Posted by sbaffini

Now I saw your previous post as well. Note that the capability to work at very large CFL numbers is not only linked to the proper coding of the AMG, but also depends from all the other parts in the code, that must be implicit. And, in any case, not all problems can work at infinite CFL, that is only true for linear problems.

Thanks for the advice!

[ibopaul]

Originally Posted by sbaffini

Now I saw your previous post as well. Note that the capability to work at very large CFL numbers is not only linked to the proper coding of the AMG, but also depends from all the other parts in the code, that must be implicit. And, in any case, not all problems can work at infinite CFL, that is only true for linear problems.

Thank you for reply.

Since AMG is based on implicit time stepping code, i understand that your 'implicit' word as exact differentiation of flux jacobian in implicit operator.

May I ask you one more question? As reference paper used Barth & Jespersen limiter, i tried to use them either. however, since Barth & Jespersen limiter has Minimum, Maximum functions , i realized that i can't differentiate them. (because Minimum, Maximum functions are discontinuous)

Is there any method or reference paper which shows the way how to exactly differentiate flux jacobian with limiter used?

Thanks in advance.

[sbaffini]

I follow the route of the Fluent paper. In that case the implicit part only takes into account the first order part of a scheme, so no gradients and no limiters. It is not exact but it works and is easy

[ibopaul]

Originally Posted by sbaffini

I follow the route of the Fluent paper. In that case the implicit part only takes into account the first order part of a scheme, so no gradients and no limiters. It is not exact but it works and is easy

Thanks for reply!

I will follow the Fluent's work as you mentioned. Since my code solves only Euler equations, my code's verification case will be small bump case.

Thanks again!

[sbaffini]

I also suggest to test on the shock tube (as many variants as you can) and converging-diverging nozzle at least.

[ibopaul]

Thank you for suggestion.

I will test the cases that you mentioned as soon as my code completed!

Thanks again!