A Computational Investigation of a Continuum Model for Flocking Dynamics
This study introduces a novel method for computationally investigating 2D nonlocal nonlinear continuum models. Leveraging the fourier transform and using a singular kernel for the nonlocal operator, the computations involved efficiently compute the convolution operator to preserve the time and power needed to investigate this system in 2D and higher.. In 2D and higher, the lack of a conserved quantity prevents a proof for well-posedness. However, guided by analytical investigation, candidates for conserved quantities are able to be numerically tested as a result of this computational investigation.
icon search Searchable Transcript
Toggle between list and paragraph view.
[00:00:01.590]Hi everyone today,
[00:00:02.640]I will be talking about a competitional
investigation of a continuum model for
[00:00:07.590]This project has been advised by professor
Adam logos and the department of math
[00:00:16.040]we would like to motivate the problem
by saying what is flocking. Exactly.
[00:00:19.580]And there are many examples of
flocking in nature as pictured above.
[00:00:23.930]We have birds flocking together in
a nice cohesive group with aligning
[00:00:29.570]We also have fish swarming throughout
the sea in this self-organizing kind of
[00:00:34.400]pattern on this on a smaller
scale, yet larger amount of agents.
[00:00:38.930]We have white blood cells swarming
towards a source of infection, perhaps,
[00:00:43.130]and in the bottom, right?
[00:00:43.970]We have robots flying throughout the
sky with perhaps a flocking algorithm
[00:00:49.010]coded into the description.
[00:00:50.990]There are three key notions that
determine what exactly flocking
[00:00:55.670]is. And this is cohesion of positions,
[00:00:59.150]alignment of velocities in
separation of positions as well.
[00:01:03.230]So separation is important in that
you don't constantly see birds, uh,
[00:01:07.250]crashing into each other as
they're trying to flock. And again,
[00:01:11.390]the examples that we outlined the
methods for this study involved
[00:01:16.250]using a suitable spectral algorithm
with two-thirds D aliasing a
[00:01:20.960]rung a cutter for, for the time stepping,
[00:01:23.780]integrating factors to handle the linear
diffusion fusion terms and spectral
[00:01:27.980]viscosity, um, which is an
essence high-frequency diffusion.
[00:01:32.750]And to make the simulation
[00:01:36.230]this introduced a novel
[00:01:39.140]which could be easily adapted to
different PDE models of similar
[00:01:46.790]The goals of this project
were to computationally
investigate a recent flocking
[00:01:50.630]model, identify and
monitor key quantities,
[00:01:53.660]which characterize flocking and
adapt the model to include wind
[00:01:58.640]and lastly track conserved quantities,
[00:02:01.160]which is useful for a theoretical
investigation. As we will explain
[00:02:07.220]to begin, we will talk
about the continuum model,
[00:02:11.420]but Rhodes denote.
[00:02:12.350]The density here is the description
of the non-local applause Sheehan.
[00:02:17.540]And here is the partial
[00:02:21.470]Now we have density transport
in the first equation.
[00:02:24.380]So some density of say a fluid is
being pushed around by its velocity.
[00:02:29.390]we have a oil Larian type description for
the velocity term where the right-hand
[00:02:34.130]side is a commutator structure
of non-local applause aliens.
[00:02:37.190]And this constitutes flocking behavior
as, um, derived in the straight coy,
[00:02:42.110]tad more a paper
[00:02:45.440]note in 2017, should courts had more,
[00:02:48.590]they used a singular influence function.
[00:02:52.520]This influence function
dictates how strong, um,
[00:02:57.680]agents communicate with another.
[00:03:00.370]In that too agents are birds who are
further away from one another will have
[00:03:04.680]less strength on their aligning velocities
than say two birds whose position
[00:03:09.630]are close together.
[00:03:11.100]It should be noted that theoretically
this model was only shown in one D to
[00:03:15.870]exhibit flocking solutions.
And as we will see, uh,
[00:03:20.490]we were able to obtain evidence pointing
towards two dimensional flocking.
[00:03:24.810]So here we have a graphic
of a simulation and note,
[00:03:28.860]we start off with random initial
velocities and densities and allow the
[00:03:32.370]equations to evolve in time.
And by about time equals to 20,
[00:03:36.900]we can visibly see cohesion in the
densities where the density becomes
[00:03:41.370]uniformly, distributed and
alignment in velocities.
[00:03:45.510]It should be noted that both of these
occur at an exponential rate in time.
[00:03:51.690]And this is great news as before one day,
[00:03:55.500]flocking was only known theoretically,
[00:03:57.600]and that this could perhaps pave
the way for two dimensional,
[00:04:00.720]theoretical investigations moving forward.
[00:04:04.020]We also tracked some conserved quantities
now in one D it is well known that
[00:04:08.520]this conserve quantity exists. Um,
[00:04:11.160]here's a short proof indicating,
or really verifying, um,
[00:04:15.900]how it works. And yeah,
[00:04:19.380]the next goal was to computationally
investigate some perhaps two-dimensional
[00:04:24.150]conserved quantities, which are not, um,
[00:04:26.970]known to exist at the moment and
could further aid a theoretical
dimensions higher than one.
[00:04:38.130]Therefore we computationally
[00:04:41.310]to obtain experimental evidence of
conserved quantities. So here we have, um,
[00:04:46.140]one graph indicating
some of the values of the
[00:04:51.120]conserved coin and used throughout time.
[00:04:52.710]And it should be noted that if these
quantities are to be conserved,
[00:04:56.640]they should be time invariant,
[00:04:58.440]or we should only see horizontal
lines for their value.
[00:05:03.210]Although, as the graph suggests,
[00:05:05.790]these quantities are actually dynamic
throughout time and therefore are not
[00:05:12.180]The next investigation we looked into
was disordered and alignment in this
[00:05:16.770]recent paper by Leslie and
shitcoin they investigate how far
[00:05:21.480]Rowe deviates from uniform
distribution in one dimension.
[00:05:24.900]Our goal is different in that
we are investigating this,
[00:05:29.370]these quantities in two,
[00:05:32.490]the key quantity we use
is this, uh, L one norm,
[00:05:36.810]which is really the difference
from the average density
[00:05:41.170]quantity for alignment was,
[00:05:42.690]which was identified in shitcoin
tad Moore's 2017 paper is this,
[00:05:48.360]um, L infinity difference of the, uh,
[00:05:52.590]V minus V-Bar were Vivar is the average.
[00:06:00.380]We ran some investigations and note in
the bottom plot representing alignment.
[00:06:05.600]We received stronger alignment for
higher and higher values of alpha,
[00:06:13.930]which indicates the stronger the
kernel is the better alignment we
[00:06:18.520]receive. Now, if we
fix alpha equal to one,
[00:06:23.320]and again, computationally investigate
this 10, the D the disorder.
[00:06:27.610]Now we notice, um,
[00:06:31.150]this order occurs in that the density
approaches uniform distribution
[00:06:36.040]at an exponential rate. And
no, this is a log linear plot,
exponential decay. However,
[00:06:43.030]one caveat of this is that alignment
is not nearly as strong and does not
[00:06:47.170]exhibit the same, uh,
[00:06:48.640]decreasing behavior for
higher values of the Colonel.
[00:06:54.460]Finally, we looked into flocking
with wind. This is very practical,
[00:06:58.360]as one could imagine, birds flying
throughout the sky and a nice flock,
[00:07:02.020]and eventually, uh, being
disturbed by some say,
[00:07:04.600]wind farm or external wind
force. So in doing this,
[00:07:08.470]we slightly adapted the model that we
had and followed some particle, uh,
[00:07:13.630]derivations from a recent paper.
[00:07:17.740]And then the idea was to start from a
flock state and throw in different amounts
[00:07:21.370]of wind notice that the disorder or namely
[00:07:26.320]the cohesion of the flock behaved,
how one should expect in that.
[00:07:30.550]When we threw wind into the
mix, the birds became disturbed,
[00:07:36.190]however, starting from a
random state and again,
[00:07:38.470]throwing in constant sources
of wind the model, again,
[00:07:41.890]behaved how we would expect
alignment was seen less often,
[00:07:46.600]and the spectrum became highly, uh, uh,
[00:07:51.640]contaminated as the simulation
became under result.
[00:07:57.130]Finally asked for future work. We aim
to investigate in two dimensions, well,
[00:08:01.240]posing this using producer
and type criteria.
[00:08:05.200]It should be noted that global existence
of well posingness was recently
[00:08:11.020]in multidimensions in a paper
two months ago, by a tad more,
[00:08:16.150]we aim to adapt some of the ideas from
this paper in a two dimensional setting
[00:08:21.070]with focus on less restrictions.
[00:08:25.360]And the final goal is to prove well posed
in this perhaps under integratability
[00:08:29.170]assumptions or even less, if possible,
[00:08:32.140]I would like to acknowledge you cares
program for funding this project,
[00:08:37.180]along with the national science
foundation grants for providing travel
[00:08:41.920]funds and support.
Thank you for your time.
Log in to post comments