A Computational Investigation of a Continuum Model for Flocking Dynamics

Diego Galvan Author

04/02/2021 Added

26 Plays

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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.

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[00:00:01.590]Hi everyone today,
[00:00:02.640]I will be talking about a competitional investigation of a continuum model for
[00:00:06.260]flocking dynamics.
[00:00:07.590]This project has been advised by professor Adam logos and the department of math
[00:00:11.700]medics. Again,
[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
[00:00:28.580]velocities.
[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 less artificial,
[00:01:36.230]this introduced a novel numerical algorithm,
[00:01:39.140]which could be easily adapted to different PDE models of similar
[00:01:44.120]structure.
[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 differential equation.
[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.
[00:02:29.090]Well,
[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
[00:04:31.740]investigation in dimensions higher than one.
[00:04:38.130]Therefore we computationally investigate, uh,
[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
[00:05:10.170]conserved.
[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.
[00:05:58.760]So computationally.
[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,
[00:06:40.000]which indicates 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
[00:08:08.740]showing, um,
[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.

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A Computational Investigation of a Continuum Model for Flocking Dynamics

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