In the specific example given in there, the 3rd order derivative occurs as a result of a discretization scheme but I reasoned the general theory also applies to any 3rd order term.
Yeah, that's generally true -- in the modified equation analysis they are doing, they are asking the question "My discretization is only approximately equal to the original PDE, but what is another PDE that my discretization is exactly equal to?"
Also in the example of the square pulse it so turns out that its Fourier decomposition contains only odd harmonics, but I presume the wiggles would still appear for - say - a sawtooth signal.
Most of the "old textbooks" I have (Crawford and others) deal with dispersion from the perspective of a wave equation, which would be double-derivative in time. Does the interpretation of the third order derivative in space depend on the degree of the time derivative?
It might depend on the time derivative -- in fact, I was pondering this exact thing just before you got here because I thought it might come up. When I said odd terms are dispersive, that was coming from the position of a first-order time derivative.
This is some analysis we did for a particular purpose (we're in the process of writing the journal version now), but we dig into the dispersion and dissipation of various methods
Anyway, in that paper we were able to show that a purely dissipative operator applied to the solution gives you dissipation on the solution, but a dissipative operator applied to du/dt and then integrated forward in time gives you purely dispersive behavior in u
I'm struggling with such terms because I'm working with vector fields in a non-standard coordinate system - a choice had to be made of either simple functions and complicated coordinates or simple coordinates and complicated functions...
and we think these terms are an artefact of the coordinate system.
nevertheless they might have some interpretation in terms of the "easy" set of functions.
As for whether it happens to mixed derivatives, that's also an interesting question. I think it will because if we think of it like d/dy (d^2/dx^2) then it's going to look like a convection of dissipation, which behaves like dispersion I think
I'm not positive on mixed terms though, those don't show up in Navier-Stokes so I haven't given it a ton of consideration
Hey everyone! I've got a question but I don't really think it fits in the site. If you had the metaphysical power to change one and only one (mainstream physics')formula (in any way you want, changing powers or order of derivation, adding extra terms) what would it be and why?
@MauroGiliberti the problem with changes in dimensionful constants is that they're meaningless if you don't say what else you're changing/keeping constant
@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
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!
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.
