Mathematicians Find Wrinkle in Famed Fluid Equations • Hacker News Board

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#111

Post by geezerjay » Fri, Dec 22, 2017, 7:59 PM UTC

pge wrote: post #74

I'm struggling to understand the significance of this, at least as the N-S equations are used in the real world. Many years ago, I interned with the Navy writing fortran code for fluid dynamics simulations on submarine hulls, and IIRC there were flow dynamics we observed consistently in the real world (e.g. oscillating vortices) that were fundamentally inconsistent with the results coming from our N-S calculations (w…

> That's only one data point, but it certainly seemed to me that no one was relying on N-S as an accurate predictor of motion

You're confusing how models are used by (some) engineers (in some applications) with how models are used by physicists and mathematicians. Engineers have to deal with all kinds of uncertainties, from matetial parameters to use cases to limit states to wear and fatigue and geometrical deviations etc etc etc. Therefore, engineers develop robust designs to comply with all design requirements under any plausible and probable scenario given a design life. To accomplish this, engineers use models to provide approximate but accurate results that are on the safe side of any limit state. Yet, eventhough designs need to be robust, simulations do need to be accurate.

These findings suggest that low-resolutoon Navier-Stokes simulations that were believed to be on the safe side may actually not be on the safe side. These finding are important, as they will illicit significant changes on how Navier-Stokes simulations are used in cases where accuracy matters.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#112

Post by geezerjay » Fri, Dec 22, 2017, 8:05 PM UTC

incidentnormal wrote: post #78

eadmund wrote: post #75

Earlier quoted context omitted.

What I got from the article is that there's a possibility of a chaotic result: it may not be possible to get computed results which are arbitrarily close to the real-world results. I was reminded a bit of the famous Lorenz system (originally for weather prediction, IIRC), which turns out to be chaotic under some conditions.

I think it's not a chaotic result as such (as the Lorenz weather prediction models you're referencing are) - which would mean large differences in outcome from very small differences in input, but a case where multiple different outcomes can arise from identical inputs.

> but a case where multiple different outcomes can arise from identical inputs.

If they are identical but not equal and these small differences in the input lead to large deviations in the output then that's pretty much the definition of a chaotic system.

Edit: I've just browsed through the paper in question and it actually demonstrates thar an approximate (weak) solution is not unique, which means that the exact same inputs in may have multiple weak form solutions.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#113

Post by btrettel » Fri, Dec 22, 2017, 9:00 PM UTC

jerf wrote: post #103

ericbowman wrote: post #13

That is fascinating. Is there any sort of immediate real-world impact (like to weather forecasting)?

In addition to the other fine replies, weather forecasting's inaccuracies are dominated by the lack of information, then by lack of processing power. Lack of closed-form solutions to NS or better solutions rates quite a ways down the list of issues it has, or put another way, even if we had a magic box that completely accurately solved NS for weather forecasting, it would not get that much accurate. (My suspicion is…

It seems popular to believe that for fluids in general forecast accuracy is dominated by errors in the initial conditions (ICs), but my own look at the problem suggests that's not so clear. I recall skimming a book on forecasting by a weather forecaster and he addressed this misconception. It appears that there are multiple sources of error, from errors in the ICs to numerical integration to the fact that the models they use are approximate (i.e., they don't solve NS; they solve a filtered version of NS with a turbulence model and additional models for other physics like chemistry), etc. My impression is that the dominant two are model inadequacy (the models are approximate) and compounded errors due to IC errors and non-linearity, but which is larger likely depends on the problem, and I am not particularly confident about this in general as I don't have hard data. (Certain types of turbulence models get more accurate as the resolution/computational cost increases, but I can't speak for other models. This fits with what you said about lack of computational power.)

The right way to do this is through uncertainty quantification techniques, and I don't know a lot about those at the moment. Until then, all I can say is that there are multiple sources of error.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#114

Post by Gibbon1 » Fri, Dec 22, 2017, 9:20 PM UTC

eru wrote: post #45

Gibbon1 wrote: post #33

Earlier quoted context omitted.

Navier Stokes is probably Turing Complete.

Probably, yes. But the gymnastics required to make it Turing complete are unlikely to be in the regime that describes real fluids.

In the sixties some people experimented with fluidic logic gates.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#115

Post by labster » Fri, Dec 22, 2017, 10:14 PM UTC

thethirdone wrote: post #46

tanderson92 wrote: post #43

Earlier quoted context omitted.

Convergence, or at least consistency and order, of Runge-Kutta methods is shown via Taylor series, so I don't see the problem or why the GP is only "Almost" correct.

The Runge-Kutta methods are not a Taylor series though. So "almost" is apt.

Geez guys, I chose a term that I thought most people would be more likely to understand. I realize that some of us have worked a lot more on numerical methods, but other hackers never made it past Calculus 2.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#116

Post by cocoablazing » Fri, Dec 22, 2017, 11:18 PM UTC

qubex wrote: post #64

rusk wrote: post #55

Earlier quoted context omitted.

Well general relativity breaks down at very small scales, and imparts the need for quantum mechanics ...

If you are really interested, somebody has bothered to make a special-relativity & quantum-indeterminacy compatible formulation of Navier-Stokes suitable for calculating shockwaves in extremely dense mediums such as the neutronium neutron stars are thought to be made of (where the speed of sound is comparable to the speed of light in a vacuum, hence the relativity, and the matter is degenerate and in states of superp…

Was this for starquake or GRB modeling?

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#117

Post by m_mueller » Sat, Dec 23, 2017, 1:41 AM UTC

labster wrote: post #115

thethirdone wrote: post #46

Earlier quoted context omitted.

The Runge-Kutta methods are not a Taylor series though. So "almost" is apt.

Geez guys, I chose a term that I thought most people would be more likely to understand. I realize that some of us have worked a lot more on numerical methods, but other hackers never made it past Calculus 2.

Did you take offense from my comment? All I wanted is to set people on the right track if they want to find more information about how NWP is implemented.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#118

Post by R0b0t1 » Sat, Dec 23, 2017, 2:07 AM UTC

qubex wrote: post #81

R0b0t1 wrote: post #77

Earlier quoted context omitted.

Sir, would you please post a link to this code?

It wasn’t my code to link to and I probably last saw it in 1996, back in the era of dot-matrix printouts on green-and-white continuous paper (on a Windows NT 3.5 machine running FORTRAN PowerStation, if you care to commiserate). I think I can dig up the paper where the equation was published, though. EDIT: This isn't the paper I had in mind, and the equation presented is ‘ merely ’ relativistic, but it gives you a fe…

Thank you! If you have more time to look for the original, I would find it exceptionally interesting; however I understand if the keywords are lost to you.

Can you explain roughly what the quantum corrections were?

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#119

Post by fspeech » Sat, Dec 23, 2017, 2:12 AM UTC

geezerjay wrote: post #112

incidentnormal wrote: post #78

Earlier quoted context omitted.

I think it's not a chaotic result as such (as the Lorenz weather prediction models you're referencing are) - which would mean large differences in outcome from very small differences in input, but a case where multiple different outcomes can arise from identical inputs.

> but a case where multiple different outcomes can arise from identical inputs. If they are identical but not equal and these small differences in the input lead to large deviations in the output then that's pretty much the definition of a chaotic system. Edit: I've just browsed through the paper in question and it actually demonstrates thar an approximate (weak) solution is not unique, which means that the exact sam…

"Weak" here does not mean numerical approximations. These are mathematicians so any quantitative approximation would be bounded o(1) otherwise the work would be meaningless. One should think of "weak" as in constraints, for example constraints at lower spatial resolutions (but precise).

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#120

Post by thethirdone » Sat, Dec 23, 2017, 3:49 AM UTC

labster wrote: post #115

thethirdone wrote: post #46

Earlier quoted context omitted.

The Runge-Kutta methods are not a Taylor series though. So "almost" is apt.

Geez guys, I chose a term that I thought most people would be more likely to understand. I realize that some of us have worked a lot more on numerical methods, but other hackers never made it past Calculus 2.

I wouldn't have corrected you. I think your comment was fine.

> I realize that some of us have worked a lot more on numerical methods, but other hackers never made it past Calculus 2.

While that didn't occur to me, I do appreciate trying to keep things accessible.