[Simbelmynë]

Originally Posted by FMDenaro

your criterion should account for the fact that you have a multidimensional case...for example the case of pure diffusion would give

ni*dt* (1/dx^2 + 1/dy^2) <= 1/2

or for the upwind case applied on a pure convective equation

dt*(u/dx + v/dy) <=1

when you have both convection and diffusion the stability region involves a functional relation of them

Yes, you are correct. However in the Poiseuille case I would guess that the stability criteria for convection should reduce to a 1d case since v=0. Also, since I have dX=dY then the diffusion case would reduce to half the expression I used. Since I have the "safety" parameter beta=0.33 then I would assume that I am within the stability region?

My experience with stability is that it is usually catastrophic to be outside the stability region (i.e. quick divergence). As I understand from your previous posts (other threads) this might not be the case. Admittedly I have a hard time seeing that the solution would blow up really slow but this is the tricky part of CFD

[Alex C.]

Originally Posted by robertreed

I would guess that a low order time-step discretization have the possibility to pollute the entire solution

That's not really what is going on here. For the coarse time step, it seems like the stability constraint of the numerical scheme is really on the edge. If the scheme is unstable, it is diverging slowly (in opposition to a quick divergence) and if it is stable, it is not converging quickly enough. It is my belief that if enough time steps are given the solution will either blow up or converge to the solution. 'enough' here is until the time derivative is close to zero (ideally machine zero).

I also believe that a higher order time discretization could exhibit the same behavior.

[Simbelmynë]

Now I have tested many different time-steps for Re=1000 and if the time-step is close to the stability limit (my idea of stability limit, perhaps not the true one) the average L1 norm might go down to some low value (say 1e-8) and then begins to oscillate. If this is the onset of a slow divergence or not is something that I have not been able to prove.

Small time-steps yield machine precision results.

Thanks for all input, I think this might be related to the convergence criteria, although I am not certain how to to correctly define it. I will try to test this on some commercial codes I have access to in order to understand the limits.

[Simbelmynë]

Sorry to necro this old thread, but this topic bugs me.

I have updated my code to run a second order (still explicit) time stepping and for different time steps I get different results (very small differences though). The convergence criteria is met (1e-10) and convergence is monotone in all cases.

It seems that this behavior is also present in SIMPLE formulations. Ferziger and Peric tested different relaxation parameters and found that they have an effect on the steady state solution. Seeing that the time-step in an explicit method is very similar to the relaxation factors in the SIMPLE-type methods, isn't is possible that the results are correct (i.e. that they can actually vary with the time-step)?

regards

[FMDenaro]

The local truncation error explains why, it has the time step therefore the magnitude has effect on the discretization error you measure

[Simbelmynë]

Originally Posted by FMDenaro

The local truncation error explains why, it has the time step therefore the magnitude has effect on the discretization error you measure

I do not know the discretization error, but I measure convergence based on Linf or average L1 norm of the time derivatives.

So do you suggest that I should always perform a time-step sensitivity analysis for the steady-state solution?

[FMDenaro]

using explicit time marching scheme, the time step has effect on the solution

[Simbelmynë]

Originally Posted by LuckyTran

In specific contexts it can be true but in general this is not true.

For 1st order systems (systems that don't have inherent oscillations), the time-step size is arguably arbitrary, up to stability limits. If you have a scheme that is numerically stable then sure you can use any time-step to arrive at the steady-state point. But you can have schemes that are numerically unstable and the time-step is therefore limited. I would consider guaranteed divergence to be "some impact" on the steady state solution. This is a stability problem and not really an accuracy problem as the system monotonically approaches the steady state value.

Originally Posted by FMDenaro

using explicit time marching scheme, the time step has effect on the solution

Thank you Filippo, this seems reasonable when viewing my own results. There seem to be some controversy about the issue though, as is evident in the post by LuckyTran for instance.

Many papers have grid sensitivity studies for steady state flows, but very few have time-step (or relaxation factor) sensitivity studies. Sure, the "relaxation error" might be several orders of magnitude smaller than the discretization error. Might be.

[FMDenaro]

I give you an example, very simple, in which the time step has effect in the solution using an explicit first-order accurate discretization.

Consider the equation d phi/dt + u d phi/dx = 0, u= constant>0. Such equation say that the solution is steady along the pathline (D phi /Dt = 0). Depending on the BC.s, you can have also steady solution in all the space.
Using the FTUS discretization, the modified equivalent equation is

d phi/dt + u d phi/dx = u*h*(1-c)/2 d2 phi/dx2 + ...

and you can see that a steady state (d phi/ dt = 0) depends on c=u*dt/h, that is on the time step. For fixed h, the dt provide the slope of the numerical path-line that approximates the exact one.

The same analysis is valid if you add the diffusive term to the equation.

In general, the convergence analysis is performed for dt/h = constant, so that when you refine the spatial grid, the dt is diminuished according to the stability constraint.