[HectorRedal]

Hi,


I would like to raise the following question from my side:
Does relaxation factor apply in transient problems?


I am currently studying the CBS algorithm for transient problem in fluids:
https://onlinelibrary.wiley.com/doi/...3E3.0.CO%3B2-7


The algorithm uses two factors: theta1 and theta2.
They are used in the follwoing manner:
Velocity: V(t=n+theta1) = V(t=n) * (1-theta1) + V(t=n+1) * theta1
Pressure: P(t=n+theta2) = P(t=n) * (1-theta2) + P(t=n+1) * theta2


Which value of theta1 and theta2 should I use for transient problems?
Is there any recommended value?
Can these parameters be considered "relaxation factors"?


Thanks for your kind support.


Best regards
Hector

[FMDenaro]

Originally Posted by HectorRedal

Hi,


I would like to raise the following question from my side:
Does relaxation factor apply in transient problems?


I am currently studying the CBS algorithm for transient problem in fluids:
https://onlinelibrary.wiley.com/doi/...3E3.0.CO%3B2-7


The algorithm uses two factors: theta1 and theta2.
They are used in the follwoing manner:
Velocity: V(t=n+theta1) = V(t=n) * (1-theta1) + V(t=n+1) * theta1
Pressure: P(t=n+theta2) = P(t=n) * (1-theta2) + P(t=n+1) * theta2


Which value of theta1 and theta2 should I use for transient problems?
Is there any recommended value?
Can these parameters be considered "relaxation factors"?


Thanks for your kind support.


Best regards
Hector

If I understand, this form is generally related to an implicit class of time integration, for theta=1/2 the well known Crank-Nicolson time integration resulting in the scheme. It is a second order accurate quadrature formula for the time integral.

[HectorRedal]

Originally Posted by FMDenaro

If I understand, this form is generally related to an implicit class of time integration, for theta=1/2 the well known Crank-Nicolson time integration resulting in the scheme. It is a second order accurate quadrature formula for the time integral.

I understand it now.
Thanks for your clarification.

[HectorRedal]

Hi,


One question from my side: If I understand it correctly, this means tha the error for the Crank-Nicolson scheme would be in the order of c * (delta T)^3.
How big is that c, constant? My understanding is that this c would be in the order of the third derivate, right?


Now, I would like to comment that I am using different schemes in my simulation (Backward Euler, forward Euler and Crank-Nicolson), but I do not see much difference in the simulation.


Additionally, I am reducing a lot the delta T for the simulation, and the results does not change.



This is puzzling me a bit.


Best regards,
Hector.

[FMDenaro]

Originally Posted by HectorRedal

Hi,


One question from my side: If I understand it correctly, this means tha the error for the Crank-Nicolson scheme would be in the order of c * (delta T)^3.
How big is that c, constant? My understanding is that this c would be in the order of the third derivate, right?


Now, I would like to comment that I am using different schemes in my simulation (Backward Euler, forward Euler and Crank-Nicolson), but I do not see much difference in the simulation.


Additionally, I am reducing a lot the delta T for the simulation, and the results does not change.



This is puzzling me a bit.


Best regards,
Hector.

No, the CN scheme is second order accurate, is not a third order scheme.
You can easily see the local truncation error of the scheme by applying the trapezoidal rule to the time integral:


Int [tn,tn+1] f dt - dt *[f(tn)+f(tn+1)]/2 = LTE