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Old   September 5, 2006, 14:27
Default Need your help!
  #1
hawk
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Hi! All.

I am quite confuesed by the argument from the following link.

http://www.flow3d.com/Cfd-101/impvexp.htm

It is on the accuracy of Implicit and Explicit Methods.

In my opinion, if both are the first order Euler scheme, either implicit or explicit will have the error of the same order. But from the statement in the above webpage, it seems that implicit is more inaccuracy even with the same time step, which, of course, satisfies the stable condition for explicit method.

Could you give me more explanation?

Thank you very much.

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Old   September 5, 2006, 22:42
Default Re: Need your help!
  #2
deepak
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hi,

a very crude idea reg this,we can take explicit method as direct method for solving eqns resulting in exact solution like solving system of eqns using some analytical method.

And implicit method as an iterative process,which as a result of convergence, results to solution nearer to exact soln. (pls correct me if my intuition is wrong)..

But implicit has its own advantages compared to explicit..u can refer to our previous discussions reg the same.

hope this will help u..

thanks and with regards, Deepak Thirumurthy
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Old   September 6, 2006, 02:34
Default Re: Need your help!
  #3
Ahmed
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Accuracy, is not only a function of the order of the scheme but it depends also on the number of mathematical operations needed to arrive to a solution. each operation adds a rounding error (unavoidable), I leave thr rest of the sentence to you
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Old   September 6, 2006, 07:37
Default Re: Need your help!
  #4
Mani
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The argument rests on the supposedly necessary under-relaxation in the implicit method. I cannot comment on that as I have never used the first order Euler scheme (and I don't know who would). There are some other arguments in this article that make me question the experience of the author. For example:

"Another general rule is that the time-step sizes for explicit stability and accuracy are usually equivalent."

This is complete nonsense. I can imagine cases like that but it's certainly not a "general rule" and I wouldn't say it's "usually" the case. There are plenty of unsteady flows where the time scales of interest are way larger than the explicit stability limit. Explicit methods as useless for those cases.
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Old   September 9, 2006, 14:57
Default Re: Need your help!
  #5
Jitendra
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Mani,

I believe what the Author is trying to say that "If Explicit Scheme is stable for a particular time step, its usually accurate too... " This is not the case with Implicit case, as you rightly said due to under-relaxation paramter. It makes pretty much sense to me, with my limited experience with these schemes.

As far as unsteady flows are concerned if we have time step for which explicit scheme is stable (and is small as compare to time scale of interest) you can still use the scheme. In fact for a Vortex Shedding case I solved both with explicit and implicit schemes. I found explicit scheme to give better accuracy although at the compromise of CPU time reqd.

Regards,

Jitendra
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Old   September 11, 2006, 08:47
Default Re: Need your help!
  #6
Mani
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The author is saying that the requirements for accuracy and explicit stability are equivalent, and that's not generally the case.

In cases like your vortex shedding example, the time steps required for accuracy are much larger (somewhat depending on Reynolds number, laminar versus turbulent flow) than the time step required for explicit stability on the very fine grid used for this type of flow. Of course, it's possible to use an explicit scheme. The question is: Is it efficient? If you could use a time step of 1 with an implicit scheme, because that's all you need for accuracy, but you have to go down to 1e-3 with your explicit scheme, your explicit approach is likely less efficient than the implicit one (that's what I meant by 'useless' in such cases).

I cannot say why you got different accuracy using the explicit vs implicit scheme. That depends on a lot of parameters such as type and order of the scheme, size of time step, convergence, grid resolution, a.s.o. The proper way to compare both is to

a) compare the computational runtimes of both methods required to get to the same level of accuracy

b) compare the accuracies of both methods running the same amount of clock-time per vortex shedding period
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