The folks around here, even the scientific-computing folks, seem to misunderstand what "Agile" software means... I think they are under the impression it means we do "sprints" between capability demonstrations but keep on sprinting to new topics before people realize our previous capability doesn't work right
My only proof that it was implemented correctly and well-suited to the problem were some pretty pictures of stuff flowing around, but 0 actual quantitative data
But since I was using a newly-developed code feature that was done during a sprint to "demonstrate capability," it had no V&V done so I spent forever trying to figure out why my code was wrong. And it's for a problem nobody has ever done before so we don't have a guide for how to do it, so I spent the next forever trying to figure out why my models were all wrong
In a too-close-to-home scenario, the whole hardware_v8_final_final_yesterday_today.cad thing bit me last week. I have been trying to get a simulation working for the past 6 months or so but it turns out I was given geometry that wasn't real and operating conditions that would never work
The third is really related to the first, which is removal of noise from the signal. This is part of compression as well, but if I have a set of 10,000 samples, depending on the fit, you can reduce the impact of outliers and so on. In a lookup table, it's harder to identify whether a region of the table is reliable or whether it's noisy
@antimony There's several benefits -- the first is compression. I can take a large number of data points and fit it into N terms for an Nth order equation. The second is super-resolution. With a fit, I can find values anywhere I want between the data points quite easily (which could be done with a lookup table also, but that's just fitting on a subset of the data so it's quite similar anyway)
Although I haven't actually derived the dispersion relation on paper, just in my head quickly, so I could be wrong about the imaginary k's floating around
@ZeroTheHero If the powers of k are odd, then they are also imaginary no? So it would be something like \omega^2 = -i*k^3 for example. I don't think there's anything that prohibits it mathematically, but I don't know how to interpret it physically (or "physically" meaning the impact of numerical discretization on the solution). Is the vector space in which you are working giving you second-order-in-time equations where this becomes important to figure out?
Probably need to work through the Fourier transform to find the relationship between frequency and wavenumber to get a better idea of what's going on there
On the other hand, u_tt + u_xx + u_xxx = 0 can be factored into (d/dt + sqrt(1+d/dx)d/dx)*(d/dt - sqrt(1+d/dx)d/dx) u = 0, which becomes a system of two equations: u_t + sqrt(1+d/dx)u_x = v and v_t - sqrt(1+d/dx)v_x = 0. And I don't really know how to wrap my head around what a wave with a fractional-derivative wavespeed behaves
@ZeroTheHero So thinking about the second-order wave equation makes my head spin a bit, but here's what I think we've figured out... for the second-order wave-like equation u_tt = something, spatial derivatives of order 2^n for n > 1 are dissipative. All other even orders are dispersive. Odd-order derivatives result in fractional derivatives in u_t and I don't have any idea what that means
They have prose-type articles as well as research articles in Science. Scientific American is all prose, but kind of sits between pop-sci and what would show up in something like Science
@ZeroTheHero No problem! In a funny coincidence, my co-author just sent me the revised draft of the journal version of that paper about 2 minutes ago... heh. I asked him the question about d^2u/dt^2 and we're working through it now. He's the more math/analysis person, I try to keep him grounded into applications
Although if one were to use a purely central scheme (no dissipation terms in the truncation error, only dispersive ones) and one did not include any artificial dissipation, it would show plenty of oscillations and blow up. It's those disperisve truncation terms without any dissipative truncation terms that make forward-time, central-space unconditionally unstable
If I remember correctly, if I go through and actually define the Gaussian profile in the characteristic equation and convert that back into the conservative equations, then the numerical dispersion is much much smaller and it doesn't steepen.
So the wave front steepens, the back side wiggles, and because it's species mass fractions it gets renormalized to get rid of negative mass fractions... which funny enough makes this case turn into a "flame" eventually even though it's non-reacting. Pesky numerical errors!
@ZeroTheHero There's a picture showing oscillations in a Gaussian pulse. But, looking back through my notes, it's because it's a nonlinear set of equations and a pure gaussian pulse in the species (methane pulse in air background) is nonlinear
