[Beans8]

The leading term of the truncation error of the Crank Nicolson scheme is four times lower than the one of the second-order backward differencing scheme. In addition, the Crank Nicolson scheme is bounded, while the second-order backward differencing scheme is not. Ansys Fluent has a ''bounded second order implicit'' scheme, but there is no reference about that method.

Based on the above, it seems that the Crank Nicholson scheme is superior. Am I missing something? If my conclusion is right, why Ansys Fluent does not incorporate it?

[FMDenaro]

Originally Posted by Beans8

The leading term of the truncation error of the Crank Nicolson scheme is four times lower than the one of the second-order backward differencing scheme. In addition, the Crank Nicolson scheme is bounded, while the second-order backward differencing scheme is not. Ansys Fluent has a ''bounded second order implicit'' scheme, but there is no reference about that method.

Based on the above, it seems that the Crank Nicholson scheme is superior. Am I missing something? If my conclusion is right, why Ansys Fluent does not incorporate it?

The analysis cannot be just focused on the time integration. Bounded schemes are a result of a fully space and time integration method.
Do not forget that the Godunov theorem says that monotone linear scheme are only first order accurate.

[LuckyTran]

Firstly, CN is quite unstable once you actually implement it. In practice a blended scheme is used which is the same idea as the "bounded second order upwind" scheme in Fluent.

Crank Nicolson "is bounded" but not oscillation free. For large time-steps, the undershoots and overshoots can get quite wild. Most CFD calculations do not have small enough timesteps to take advantage of the benefits of CN.

What you should consider is how a practical calculation with CN and second order upwind behaves at small time steps versus large time steps, i.e. a Courant number of 0.5 versus 100. Which one blows up first? Preferably you would be comparing their limited versions in this exercise as well.

[Beans8]

Originally Posted by FMDenaro

The analysis cannot be just focused on the time integration. Bounded schemes are a result of a fully space and time integration method.
Do not forget that the Godunov theorem says that monotone linear scheme are only first order accurate.

Thank you for your answer. I forgot to say that I was focused on the time integration. I understand that even if I use an Euler Implicit scheme, if the spatial discretization is done with a with a Centaal Differencing scheme, then the resulting scheme will not be bounded.

Originally Posted by LuckyTran

Firstly, CN is quite unstable once you actually implement it. In practice a blended scheme is used which is the same idea as the "bounded second order upwind" scheme in Fluent.

Crank Nicolson "is bounded" but not oscillation free. For large time-steps, the undershoots and overshoots can get quite wild. Most CFD calculations do not have small enough timesteps to take advantage of the benefits of CN.

What you should consider is how a practical calculation with CN and second order upwind behaves at small time steps versus large time steps, i.e. a Courant number of 0.5 versus 100. Which one blows up first? Preferably you would be comparing their limited versions in this exercise as well.

Thank you for your answer. I understand that the numerical issues of the CN method are related to the spatial discretization of the equations in the FVM (in the like of what JMDenaro said).

Are there any references of these blended schemes like the "bounded second order implicit" scheme? Fluent gives a couple of formulas about the "bounded second order implicit" but without any additional theoretical background.

Thank you

[sbaffini]

The 2nd order backward differencing has 2 very important advantages over any other scheme:

1) It is, to the best of my knowledge, the only second order implicit A-stable scheme

2) Terms at previous times do not involve spatial terms, but are just like simple source terms, which is extremely useful for moving/changing meshes

[Beans8]

Originally Posted by sbaffini

The 2nd order backward differencing has 2 very important advantages over any other scheme:

1) It is, to the best of my knowledge, the only second order implicit A-stable scheme

2) Terms at previous times do not involve spatial terms, but are just like simple source terms, which is extremely useful for moving/changing meshes

Thank you for your answer. Where can I find more information about the A-stability of the CN and 2nd order backward differencing schemes? The books that I have read are very general and do not provide much information about these schemes apart from being bounded/unbounded and their accuracy.

Thank you

[sbaffini]

The last book where I've read about it is the Jameson book, Computational Aerodynamics. BUT, it says that CN is A-stable while BDF2 is L-stable.

I have alsways taken this matter for granted, as several books mention instead that BDF2 is the only second order A-stable scheme.

The general matter is the Dahlquist barrier, if you want to investigate. And the A vs L stability mostly regards large time steps for stiff problems.

As a practitioner, the very point is that BDF2 is more stable than CN and discretizes the time derivative at n+1.

[Beans8]

Originally Posted by sbaffini

The last book where I've read about it is the Jameson book, Computational Aerodynamics. BUT, it says that CN is A-stable while BDF2 is L-stable.

I have alsways taken this matter for granted, as several books mention instead that BDF2 is the only second order A-stable scheme.

The general matter is the Dahlquist barrier, if you want to investigate. And the A vs L stability mostly regards large time steps for stiff problems.

As a practitioner, the very point is that BDF2 is more stable than CN and discretizes the time derivative at n+1.

Thank you for the suggestion.