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• # Regarding the convergence of the results

Posted by on October 1, 2023 at 9:22 pm

I have 2 doubts regarding the covergence of the results during each iteration :

1. Is the Converging nature (converging of results) of any numerical solution is if the change happening in its plot (profile) after each iteration is decreasing irrespective of whatever its profile would be ?

2.How we are knowing that the results of the iteration has converged after t=0.2s (In 1-D unsteady state heat conduction),is it because of the fact that we know that for steady state(asssuming steady state has reached by t= 0.2 s) heat conduction the profile of temperature is linear .I mean should we know the exact solution of the PDEâ€™s or exact solution of the problem beforehand to reach at the conclusion of convergence ?

replied 6 months, 2 weeks ago 2 Members · 6 Replies
• 6 Replies
• ### Sandeep

October 1, 2023 at 9:49 pm

Good Question Abhishek!
1. In a steady state problem we are iterating towards the truth or the final result, but how do you know that we are closer to the truth? For that, we implement the convergence criteria. We will check the results of the previous iteration and the current iteration will be continuously monitored. If we don’t see any change from the previous iteration to the current iteration then there is no point in further iterations as well, which means we have reached the final result. (There are multiple ways of checking the results from previous to current like using rms values etc., you can explore it. it is also discussed in the cfd foundation course).

2. In transient simulation the situation is slightly different though similar. In transient simulations, you are marching in time through timesteps. For example, to reach 1 sec of simulation, if you can have a delta t of 0.1s then you need to march 10 times to reach 1 sec. During each time step of the calculation you need to check for convergence, so for each time step you have inner iterations and you need to check if your previous iteration result and current iteration results are now almost the same. if yes, then it is converged at that time step and then you march to the next time step and again check for convergence and then move to the next time step.

• ### abhishek_k

Member
October 1, 2023 at 10:04 pm

Okk ..got ur point. But like should we know the exact solution to tell that the solution has converged like in unsteady heat transfer we were knowing that at equilibrium( steady state) the temperature profile is linear ,so we waited till the temperature profile became linear.But what if we don’t know the exact solution , In that case how would we know that the solution has converged? Whether in that case we will wait until the profile (let say of temperature)has become constant ?

• ### Sandeep

October 1, 2023 at 10:57 pm

This is exactly what I wrote above.
The convergence is checked at the inner loop iteration levels and not at the time step levels.

Now for transient problems which are highly fluctuating like the vortex shedding behind a cylinder or stall on an airfoil etc. To check the capturing of physics is real (convergence), not just the numerical check is enough, but you should be able to observe the repetition of signals. Once you see the repetition or several cycles, then you can say it is converged and then you can take an average (time average of your quantities of interest like drag force etc. for a few cycles).

• ### abhishek_k

Member
October 6, 2023 at 5:10 pm

As u said that in transient problems the convergence is checked at the inner loop iteration levels and not at the time step levels ,I got this . Like u said that to reach 1 sec of simulation with a delta t of 0.1 s we have to march 10 times, now let suppose at each time time step( at inner iteration levels) the solution converged and we’ve reached the 10th time step. Then how can we reach to the conclusion that after 10th time step the solution will not converge further (at inner iteration levels), i.e. what if in case we have few more time steps and at inner iteration levels of each time step the results are converging then how we would be knowing that equilibrium (steady state) is attained after end of a particular time step because results are still converging ?

• ### abhishek_k

Member
October 6, 2023 at 5:12 pm

?