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Old   March 4, 2003, 04:55
Default I am suspecious about DNS
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kenn
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I am suspecious about DNS. I will get a chance to investigate it after I investigate reliabilities of numerical methods for incompressible laminar flow, one by one.

my suspecions are based on two observations:

1. if we try to use finite difference(three points) to calculate second derivatives of a third-order polynomial at a specific point, we need to choose dx. now, we might be enticed to choose a smaller dx. because according to our commonsense, the small dx, the better results. but, the fact is simply to the opposite. you can try. as a matter of fact, this is a homework problem for students studying numerical analysis.

now, the story behind is: in this case, there is no truncation errors. all errors comes from round-off errors. when we decrease dx, truncation errors decrease in many other situations but irrelevant in this special case. on the other hand, smaller dx, more operations, possibly bigger round off error. guess how severe round off error is? suppose you have exact solution 10, your round off error could be as big as 10 when you choose, for example, dx = 0.00001.

now, with such a phenomena, am I ligitimate to doubt DNS?

2. as in another post of mine, in the cavity flow using FEM, when I use 10x10, I get better solution. 20x20 mesh, better, 40x40, even better. however, 100x100. what I got is ridiculous.

people assume, we can refine mesh, refine mesh. it's not true at all. I wish I am wrong, but the fact is every numerical method has a mesh bandwidth ( which I am going to actively advocate in the next a couple of years), that is, only within this bandwidth, the solution is reliable. a good method has a wider bandwidth, and a bad method has a narrow one.

as a matter of fact, I investigated the popular projection FD on staggered mesh for a periodic square ( with periodic BC and periodic bodyforce), I found the method has the same disease.

if what I said about FEM is true, should I doubt DNS?

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Old   March 4, 2003, 07:21
Default Re: I am suspecious about DNS
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Tom
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Kenn,

I think your arument about the refining of the grid is a bit misleading when it comes to pdes (I agree that if you use the finite difference formula to calculate the derivative you will get nonsense due to the rounding error if dx is reduced - this is one of the reasons for using double precision since it delays this effect).

When you solve a pde using finite differences you write down the equation assuming infinite accuracy as dx-->0 and then multiply through, by dt or dx^2 depending on your problem. The multiplying through by the small factor helps reduce the problem with the rounding error since you end up with ratio of small numbers which can be maintained at moderate values as the grid is refined - you calculate the differences and not the derivative.

Depending on the type of DNS you are talking about then you should be suspicous - I've seen papers on DNS of fully turbulent pipe flow with Reynolds numbers below transition value (the so-called fully turbulent flow is actually being driven by the inflow bc, and possibly numerical noise, and is not being generated naturally).

Tom.
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