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arXiv:2101.03950v1 [cond-mat.stat-mech] 8 Jan 2021

Density dependent diﬀusion models for the interaction of

particle ensembles with boundaries

Jennifer Weissen∗

, Simone G¨ottlich∗

, Dieter Armbruster†

January 12, 2021

Abstract

The transition from a microscopic model for the movement of many particles to a

macroscopic continuum model for a density ﬂow is studied. The microscopic model for

the free ﬂow is completely deterministic, described by an interaction potential that leads

to a coherent motion where all particles move in the same direction with the same speed

known as a ﬂock. Interaction of the ﬂock with boundaries, obstacles and other ﬂocks

leads to a temporary destruction of the coherent motion that macroscopically can be

modeled through density dependent diﬀusion. The resulting macroscopic model is an

advection-diﬀusion equation for the particle density whose diﬀusion coeﬃcient is density

dependent. Examples describing i) the interaction of material ﬂow on a conveyor belt with

an obstacle that redirects or restricts the material ﬂow and ii) the interaction of ﬂocks (of

ﬁsh or birds) with boundaries and iii) the scattering of two ﬂocks as they bounce oﬀ each

other are discussed. In each case, the advection-diﬀusion equation is strictly hyperbolic

before and after the interaction while the interaction phase is described by a parabolic

equation. A numerical algorithm to solve the advection-diﬀusion equation through the

transition is presented.

AMS Classiﬁcation. 35M10, 35K65, 35L65

Keywords. Interacting particle systems, mean ﬁeld limit, advection-diﬀusion equation, nu-

merical simulations, boundary interactions, material ﬂow, swarming.

1 Introduction

We study the transition from microscopic models of interacting particles to the macroscopic

limit describing their motion as coherent ensembles. Such problems naturally arise in the

description of biological swarms such as ﬂocks of birds [4,13,16], schools of ﬁsh [3], ant [7]

or bacterial colonies [24], the movement of pedestrian crowds [19,21] or transport of material

[17,30]. In some production facilities, e.g. bottling plants, work in process (i.e. bottles) is

transported on conveyor belts from one processing station to the next. Bottles are positioned

in an initial conﬁguration on a conveyor belt. This initial conﬁguration remains intact and

is transported with constant speed as long as individual objects do not collide. On the

biological level, social animals gather together and move collectively, often in synchronized

∗University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany (jennifer.weissen@uni-

mannheim.de, goettlich@uni-mannheim.de)

†Arizona State University, School of Mathematical and Statistical Sciences, Tempe, AZ 85287-1804, USA

(dieter@asu.edu)

1

and coherent patterns [10]. Thus individuals organize into swarms and build formations that

remain stable over time. When such swarms move with constant velocity and form a well-

deﬁned translational steady state where all individuals head in a common direction they are

called a ﬂock. While each individual has their own initial velocity the interactions between

individuals lead to a stable formation where, in the absence of external perturbations, all

individuals have exactly the same velocity. In that sense, the movement of a ﬂock acts like a

conveyor belt. In this paper, we model situations where the stable formation is perturbed or

destroyed by external interactions with boundaries and the movement of the particle ensemble

is reorganized.

On the microscopic level we consider the motion of Nindividuals moving according to New-

ton’s law. On a conveyor belt the relative motion of single objects is determined by frictional

forces which emerge due to contact with the conveyor belt, its geometric restrictions and

collisions with other objects [17]. The resulting system of ordinary diﬀerential equations

describes transport of all parts on the moving belt. For biological swarms, the individual

motion is typically governed by a velocity selection mechanism and attractive and repulsive

interactions between individuals leading to similar systems of coupled ordinary diﬀerential

equations. Among the microscopic models for swarming, the Vicsek model [33] and the

attraction-repulsion model [16] have received considerable attention in the literature.

There have been a few studies at the microscopic level to describe the phenomenology of

the interaction of ﬂocks with geometrical boundaries. Studies in ﬁnite domains of the Vicsek

model, see [5] and references therein, and the attraction-repulsion model [4] illustrate how the

geometry of the domain inﬂuences steady state ﬂocking solutions. In particular, boundaries

generate internal excitations in the swarm which causes the ﬂocking solution to break apart.

Depending on the geometry of the domain the ﬂock reorganizes with a diﬀerent direction,

similar to particle scattering. In the Vicsek model, the noise level and the inﬂuence horizon

impact the formation of the collective. In the attraction-repulsion model the relative strength

of attraction and repulsion in comparison to the self-propelling forces determine whether ﬂocks

scatter elastically or inelastically from a boundary. Similarly when two or more ﬂocks [4]

collide they may cross almost without interacting or their formation may be temporarily

destroyed and they re-emerge as two ﬂocks with diﬀerent directions of motions or they may

merge into one ﬂock.

The prototype for micro-macro transitions is kinetic gas theory developed by Boltzmann in

the 19th century which linked the macroscopic measurable quantities of heat and temperature

to microscopic particle motion. One principal advantage of macroscopic models is that they

are computationally independent from the number of individuals. Microscopic models are

computationally expensive for large numbers of individuals that often appear in nature where

animal societies might contain thousands or millions of individuals. Another reason to use

macroscopic models is that macroscopic solutions like ﬂocks can be observed within micro-

scopic simulations and the emergent properties of their motion can be studied and described

well at a macroscopic level.

Hence continuum models are developed for the limit as the number of particles goes to inﬁnity.

Being aware of the fact that this limit is a much better approximation for gases than it is

for birds or bottles we derive macroscopic models for self-organized ﬂow via a mean ﬁeld

limit from the underlying microscopic models. We are especially interested in the description

of stable motions which are disturbed when interacting with boundaries or obstacles. The

boundary interaction leads to internal perturbation which is modeled by diﬀusion on the

macroscopic scale.

2

The general type of the macroscopic equation that we will derive is the (strongly degenerate)

advection-diﬀusion equation for the particle density ρ=ρ(x, t)

∂tρ+∇x·(f(ρ, x)−k(ρ)∇ρ) = 0 (x, t)∈R2×(0, T ),(1)

where diﬀusion is generally coming from random motion in the underlying microscopic model.

Under several assumptions, among them the requirements that k(ρ) is suﬃciently smooth,

Volpert and Hudjaev [34] showed existence of a BV entropy solution for (1) in several space

dimensions and unbounded domains. Uniqueness of weak solutions in the class of bounded

integrable functions for the purely parabolic case and nondecreasing k(ρ) has been proven by

Brezis and Crandall [8]. Yin [23] showed uniqueness of weak solutions in L∞∩BV for the

Cauchy Problem of (1) and strictly increasing k(ρ). Carillo [11] showed uniqueness of entropy

solutions for particular boundary value problems with Dirichlet boundary conditions.

In our case, the hyperbolic part of equation (1) allows for the correct modeling of material

transport and ﬂocks. In the undisturbed situation our model is diﬀusion free, i.e. k(ρ) = 0.

Whenever the bulk of material or the ﬂock is disturbed, diﬀusion becomes active and is

generated from the ergodic properties of the large number of particle interactions. These

disturbances are not constant in time but instead depend on time and space.

The paper is organized as follows: In Section 2, we present the microscopic model and give

a short overview on existing macroscopic limits before we formally derive our macroscopic

equation. Starting from the macroscopic equation for material transport in Section 2.3.1,

we generalize the concept to model biological swarms in Section 2.3.3. Section 3introduces

the operator splitting method to compute approximate solutions to the advection-diﬀusion

equation. We study the properties of the material ﬂow model numerically in Section 3.2

and compare our results to a non-local macroscopic model as well as to experimental data.

Section 3.3 discusses numerical results for the movement of ﬂocks in bounded domains and

scattering interactions of two swarms. Section 4summarizes our results.

2 Derivation of the macroscopic limit

2.1 Discrete model and macroscopic treatment

We consider the deterministic, second order microscopic model describing the movement of

particles driven by a velocity selection mechanism and a pairwise interaction force [13]. The

equations of motion for Nparticles are

dxi

dt =vi

mdvi

dt =G(vi) + X

j6=i

F(xi−xj),i= 1,...,N (2)

where xi, vi∈R2are the position and velocity of the particle iand mis the mass. G(v) is

the velocity selection mechanism and F(x) is the interaction force depending on the positions

and distance of particles.

To describe material ﬂow on conveyor belts, the interaction of individual particles in the force

term Fis reduced to short-range repulsion when two particles collide. First order equations

derived from (2) have been considered in [18] where a non-local second order model is derived

3

from the microscopic model (2) via mean ﬁeld limit and a second order macroscopic model

which couples the continuity equation

∂tρ+∇x·(ρv) = 0,(3)

with a momentum equation. The continuity equation is closed with a closure velocity de-

rived from the momentum equation. The resulting ﬁrst order model is the continuity equa-

tion with non-local velocity v(ρ, x, t). Alternatively, the (non-degenerate) advection-diﬀusion

equation (1) is proposed where the density is conserved while particles travel with the con-

stant average speed of the conveyor belt f(ρ, x) = vTρand their movements are subject to

diﬀusion with strength k(ρ, x, t) = Cρ with C > 0, see [18].

In connection with animal swarming, the focus has been mainly on models in which the density

of the population ρsatisﬁes the advection-diﬀusion equation (1) where f(ρ, x) = vρ. The

velocity v=v(ρ, x, t) is a non-local velocity and k(ρ, x, t) is the diﬀusion coeﬃcient [25,31].

The non-locality models spatially decaying social forces including attraction and repulsion

between the individuals and is based on the fact that interactions between individuals via

sight, sound or smell often take place at a larger distance [32]. Diﬀusion in the continuum

limit leads to disordering and dispersal within the swarm. Density independent diﬀusion

leads to disintegration of swarms on large time scales, while density dependent diﬀusion can

stabilize swarms [25]. In comparison to non-locality in the advection term, non-local eﬀects in

the diﬀusion term do not lead to qualitatively new patterns in the movement of animals [25].

The usual approach to derive ﬁrst order continuum equations from ﬁrst order microscopic

models involves a Fokker-Planck approximation [27]. The seminal derivation of the advection-

diﬀusion-terms from stochastic microscopic models was considered in [2,20,28]. The micro-

scopic models describe motion of individual cells or organisms subject to random jumps or

turns modeled by (biased) random walks [2,28]. In addition, cells sense and are inﬂuenced by

the number of neighbours which is assumed to be distributed with Poisson probability [20].

The main assumption is that individual movements in the microscopic model include suﬃ-

ciently large random motion. Then, individual-based stochastic simulations agree well with

the behaviour described by the limit equation.

Microscopic second order models for swarming are connected to their macroscopic counterpart

using kinetic theory as a middle stage. The kinetic equation for the single particle probability

distribution function is derived from the particle scale and then related to the macroscopic

limit equation with additional assumptions [1,12,13,15]. The book [26] (Part III, Section 2-4)

contains a comprehensive summary on microscopic swarming models and their second order

continuum limits. The momentum equations are non-local equations which depict interactions

over a broader range of space. For a general overview on microscopic and macroscopic models

for swarming, we especially refer the reader to the papers [13,32] and references therein.

To our knowledge, ﬁrst order models derived from second order microscopic models do not

exist in the literature in the context of swarming. Thus, in the next section, we follow the

approach of [18] to motivate a ﬁrst order macroscopic limit from the second order macroscopic

model (2). Speciﬁcally, the full macroscopic equations which couple the continuity equation

(3) to a non-local momentum equation are reduced to a local ﬁrst order limit equation through

the identiﬁcation of a local closure velocity.

4

2.2 Mean ﬁeld limit and macroscopic equation

Starting from the microscopic model (2), we derive a macroscopic limit equation via mean

ﬁeld limit as middle stage. Initially, we consider N0ball shaped particles with radius R0and

mass m0. We let N→ ∞, R →0, where Nis the number of individuals and Ris the radius.

We rescale and keep the total mass and the total surface covered by the particles constant

Nm =N0m0, N πR2=N0πR2

0.

Let f(N)(x, v, t) = m0N0

NPN

i=1 δxi(t)×vi(t), then

Zρ(x, t)dx =Z Z f(N)(x, v, t)dvdx =m0N0.

The corresponding mean ﬁeld equation is

∂tf+v· ∇xf+Sf= 0,

Sf=∇v·1

mG(v) + Z Z F(x−y)f(y, w, t)dwdyf(x, v, t),

where F(0) = 0. Using a mono-kinetic closure, we arrive at the macroscopic limit equation

∂tρ+∇ · (ρu) = 0 (4)

G(u) + ZF(x−y)ρ(y)dy = 0,(5)

where we have left out the time and space dependency of ρ, u whenever the meaning is clear,

see [18] for further details. Apparently, the equations (4) and (5) are coupled via the velocity

u. To derive a closed model consisting of a single equation only for the density in (4), an

explicit closure relation for the velocity uis needed. However, the velocity is only implicitly

given by G(u) in (5). So the key idea in the following section is to determine the velocity u

depending on the force term F, i.e. u=G−1(RF(x−y)ρ(y)dy). As we will see, the choice

of the interaction potential leads to diﬀerent types of advection-diﬀusion equations for (4).

2.2.1 Macroscopic limit for interaction potentials with compact support

We are especially interested in interaction forces of the form

F(x) = H(dR− kxk)FR(x),(6)

where His the Heaviside function and FRis the gradient ﬁeld of a potential UR, i.e., FR=

∇UR. Note that FRis odd, i.e. FR(−x) = −FR(x). The interaction force (6) between

two particles is only active up to the distance dR. We assume that the distance dRcan be

expressed depending on the radius of the particle, i.e., a constant ratio R/dR. The distance

dRis the horizon up to which a single particle can sense others.

Consider the force F(xi−xj) acting on particle iinduced by the particle j. On the microscopic

level, particles are described by their center of mass. If their centers of mass are in close

proximity they have a repulsive impact on each other such that the particles experience a

force which pushes them apart. Usually particle iis pushed in the direction −(xi−xj)

opposite to particle j. On the macroscopic level, the density distribution describes the spatial

5

concentration of the mass. Thus there are no particles as a density distribution represents

the collection of inﬁnitely many inﬁnitely small particles with zero distance and hence the

distance between particles is not deﬁned.

Since macroscopically the integral over the density is the mass, repulsion should only be

active, if microscopic particles overlap, indicated by a cumulated density that is too high.

Therefore, we switch from scaling the force in terms of the distance microscopically to scaling

it macroscopically in terms of the mass.

Without any changes, we rewrite the microscopic force term (6) as follows

F(xi−xj) = H(dR− kxi−xjk)FR(xi−xj)(H(dR−kxi−xjk)=1)

=H(dR− kxi−xjk)FR(xi−xj)(PjmH(dR−kxi−xjk)>m).

We interpret the expressions mand PjmH(dR−kxi−xjk) in the additional indicator function

macroscopically and use a Taylor expansion to reformulate

m∼m0N0

N=m0R2

R2

0

,

X

j

mH(dR− kxi−xjk)∼ZBdR(x)

ρ(y)dy ≈ZBdR(x)

ρ(x) + ∇ρ(x)·(x−y)dy

=π(dR)2ρ(x) + ∇ρ(x)·ZBdR(0)

˜z d˜z

|{z }

=0

.

This way, we derive an expression for the density threshold ρcrit above which diﬀusion is

observed

X

j

mH(dR− kxi−xjk)> m

∼ρ(x)>m0R2

π(dR)2R2

0

=: ρcrit,(7)

which is meaningful even for R→0, as the ratio R/dRis ﬁxed by assumption. Then, we plug

in the expression for the force term into equation (5), exploit that FRis odd and use again

Taylor expansion to obtain

ZF(x−y)ρ(y)dy =ZBdR(x)

FR(x−y)ρ(y)RBdR(x)ρ(y)dy> m0N0

Ndy

=ZBdR(x)

FR(x−y)ρ(y)(ρ(x)>ρcrit)dy

=ZBdR(0)

FR(−z)ρ(x+z)dz H(ρ(x)−ρcrit)

≈ − ZBdR(0)

FR(z)h∇ρ(x), zidz H(ρ(x)−ρcrit ).

We are interested in the limit of the force term for R→0. Therefore, we assume that the

interaction force is chosen such that the limit can be reformulated as follows

lim

R→0ZBdR(0)

FR(z)h∇ρ(x), zidz =C∇ρ(x),where C < ∞.(8)

6

Then, the local approximation of the force is denoted by

ΨdR(ρ, ∇ρ) := lim

R→0ZBdR(0)

FR(z)h∇ρ(x), zidz H(ρ(x)−ρcrit )

=C∇ρ(x)H(ρ(x)−ρcrit).

If the self-propelling force Gis invertible, we can solve (5) for the velocity u=G−1(ΨdR(ρ, ∇ρ))

and substitute uin (4), such that we obtain the ﬁrst order macroscopic limit equation

∂tρ+∇x·ρG−1(ΨdR(ρ, ∇ρ))= 0.(9)

Remark 1. In the case of interaction potentials with unlimited support we achieve a similar

result. Let us consider interaction forces F(x) = F∞(x)with support supp(F) = R2and

instead of ﬁxing the ratio R/dR, we assume dR=∞in equation (6). The resulting critical

density where diﬀusion becomes active is ρcrit = 0. Deﬁning

Ψ∞(ρ, ∇ρ) = lim

R→0ZR2

F∞(z)h∇ρ(x), zidz =¯

C∇ρ(x)<∞,(10)

leads to the limit equation

∂tρ+∇x·ρG−1(Ψ∞(ρ, ∇ρ))= 0.(11)

In the following section, we present some exemplary particle systems and interaction forces to

illustrate the two types of limit equations, i.e. the degenerate advection-diﬀusion equation (9)

or the non-degenerate advection-diﬀusion equation (11).

2.3 Applications

2.3.1 Material Flow

The microscopic model for material ﬂow on a conveyor belt Ω ⊂R2describes the transport of

identical and homogeneous parts with mass m0and radius R0with velocity vT∈R2, see [17].

The regularized bottom friction

G(v) = −γb(v−vT),(12)

corrects deviations of the parts’ velocities from the conveyor belt velocity where γbis the

bottom viscous damping.

The interaction force Fis given by a spring-damper model of the form

F(x) = H(2R0− kxk)F2R0(x).(13)

Proposition 1. For the material ﬂow model (2)with bottom friction (12)and interaction

force (13)obeying (8), the macroscopic limit given by equation (9)is the degenerate advection-

diﬀusion equation

∂tρ+∇x·(ρvT−k(ρ)∇ρ) = 0,(14)

with threshold density ρcrit =m0

π4R2

0and k(ρ) = ¯

C

γbρH(ρ−ρcrit).

7

Proof. The interaction force Fhas to satisfy (8). Without loss of generality, we carry out the

analysis for FR(z) = kmz

kzk

(2R−kzk)

R4, km>0. It holds that

lim

R→0ZB2R(0)

FR(z)h∇ρ(x), zidz

= lim

R→0ZB2R(0)

km

z

kzk

(2R− kzk)

R4(∂x(1) ρz(1) +∂x(2) ρz(2) )dz

= lim

R→0kmπ∇ρZ2R

0

r2(2R−r)

R4dr =8

3kmπ∇ρ=¯

C∇ρ,

where we set the macroscopic diﬀusion constant ¯

Cequal to the microscopic term 8

3kmπ. Thus

the scale of the interaction force kmdetermines the strength of the diﬀusion coeﬃcient ¯

C.

We obtain

Ψ2R(ρ, ∇ρ) = ¯

C∇ρH(ρ−ρcrit).

Using G−1(y) = vT−y

γb, the velocity uis then given by

u=vT−¯

C∇ρH(ρ−ρcrit)

γb

.

The Kirchhoﬀ transformation of k(ρ) given by b(ρ) = Rρ

0k(y)dy, ∇b=k(ρ)∇ρallows to

recast (14)

∂tρ+∇ · (f(ρ, x)− ∇b(ρ)) = 0 x∈Ω,(15)

which we consider in the following in a bounded domain Ω ⊂R2equipped with boundary

and initial conditions

(f(ρ, x)− ∇b(ρ)) ·~n = 0 x∈∂Ω,(16)

ρ(x, 0) = ρ0(x),(17)

where ~n is the outer normal vector at the boundary ∂Ω. The limit equation is the de-

generate advection-diﬀusion equation with convective ﬂux f(ρ, x) = ρvTand diﬀusive ﬂux

∇b=k(ρ)∇ρ. Equation (14) is formally parabolic, but is purely hyperbolic when k(ρ)≥0

vanishes. Since k′(ρ)≥0 is zero on a set of positive measures, the equation is strongly de-

generate. In the purely hyperbolic case, i.e. k(ρ) = 0 the equation reduces to the transport

equation describing transport at the velocity vTof the conveyor belt

∂tρ+∇x·(ρvT) = 0.

If the conveyor belt is stopped (i.e. vT= 0), we obtain the purely parabolic case

∂tρ= ∆b(ρ),

where material spreads out from regions which are densely packed, i.e. regions with density

values above the density value ρcrit.

8

The density ρcrit matches the microscopic situation. The density in (14) corresponds to the

concentration of mass of a single part with total mass m0in the circle with radius R0, i.e,

a maximum density ρcrit at which parts are packed as closely as possible without overlaps.

This deﬁnition of the maximum density is derived from ball shaped particles. With initial

data kρ0kL∞< ρcrit in unbounded domains, the equation describes a trivial transport process

without diﬀusive inﬂuence.

The most interesting situation occurs when the compactly supported interaction force leads to

the distinction between ρ < ρcrit and ρ≥ρcrit . In particular the mixed hyperbolic-parabolic

equation (14) is purely hyperbolic for ρ < ρcrit and contains a parabolic region if ρ > ρcrit .

The diﬀusion in equation (14) is only activated when the transport of the material is dis-

turbed and the material density increases above the maximum density. The diﬀusion models

dispersal, is spatially local and acts in the opposite direction of the gradient. Once diﬀusion

is active, it is linear in both ρand ∇ρ. The ratio between vTand C

γbdetermines whether the

problem is advection or diﬀusion dominated. As the viscous damping γbincreases, diﬀusion

decreases and the transport process becomes advection dominated.

A typical experiment involves the placement of bottles on a conveyor belt. Without obstacles,

the conveyor belt moves them with a constant velocity and no relative velocity between them

hence the initial conﬁguration is preserved. The spatial translation ρ(x, t) = ρ0(x−vTt)

is accurately represented by the hyperbolic part of (14) and sharp material gradients are

maintained. As soon as an obstacle is placed on the belt, the transportation process is

disturbed. As a consequence, the initial conﬁguration of the bottles is destroyed. They are

pressed closer together and when the maximum density in the macroscopic model is exceeded,

as a result of the diﬀusion, the velocity of the bottles will deviate from the transport velocity

vT.

Generally in advection-diﬀusion equations the P´eclet number measures the relative strength

of advection and diﬀusion. The P´eclet number vanishes for pure diﬀusion and is inﬁnite in

case of pure advection. We remark that the P´eclet number for the degenerate advection-

diﬀusion equation (14) is deﬁned locally and strongly varies with the density. For regions

where the undisturbed transport process takes place, the P´eclet number is inﬁnite, while for

regions with high density values and strong diﬀusion i.e. near disturbances, very small P´eclet

numbers can be observed.

The following Lemma shows that for interaction potentials with unlimited support and in-

teraction potentials chosen such that (10) holds true, the limit equation is given by the

advection-diﬀusion equation where the diﬀusion term is always active for ρ > 0.

Lemma 1. For the material ﬂow model (2)with bottom friction (12)and interaction force (6)

with dR=∞obeying (10), the macroscopic limit given by equation (11)is the (non-degenerate)

advection-diﬀusion equation

∂tρ+∇x·(ρvT−k(ρ)∇ρ) = 0,(18)

where k(ρ) = Cρ, C =¯

C

γb.

Proof. For the exemplary interaction potential F∞(x) = kmx

kxkexp(−3kxk3), we obtain

Ψ∞(ρ, ∇ρ) = 1

9kmπ

|{z}

C

∇ρH(ρ).

The rest of the proof is analogous to the proof of Proposition 1.

9

Qualitatively, this limit equation is also derived in [18]. Again, the diﬀusion is density de-

pendent and scales linearly with the density. The existence and uniqueness results, compare

Section 1, are applicable for (18) in an unbounded domain, but not for (14), since k(ρ) is not

continuous and does not meet the regularity requirements to obtain mentioned results. More-

over, existence and uniqueness of the solution is also not proven for (14), (18) on a bounded

domain Ω ⊂R2equipped with the boundary conditions (16). However, we see in Section 3

that numerical examples provide good results in bounded domains.

2.3.2 Pedestrian Crowds

The dynamical behaviour of pedestrian crowds has been modeled by the deterministic mi-

croscopic social force model [19,21]. Our microscopic model (2) can be written in this way

by considering the velocity selection mechanism G(v) as the destination force Gdest(x, v) =

1

τ(vCD(x)−v). Here, vC>0 is a comfort speed which is achieved within the relaxation time

τ > 0 and D:R2→R2describes the direction to a destination. The interaction force F

in this context describes attraction and repulsion between pedestrians. We assume that Fis

chosen such that (8) is satisﬁed. Exploiting the inverse function G−1(y) = vCD(x)−yτ , the

limit equation is the degenerate advection-diﬀusion equation with space dependent advection

term

∂tρ+∇x·ρvcD(x)−τ¯

C∇ρH(ρ−ρcrit)= 0.(19)

2.3.3 Swarming

We consider the microscopic attraction-repulsion model for biological swarms

dxi

dt =vi

mdvi

dt = (α−βkvk2)v+λ∇xiX

j6=i

U(xi−xj),(20)

where Uis a potential. The potential strength λ > 0 scales the impact of the potential forces

relative to the self-propelling force [1]. The velocity selection mechanism is given by

G(v) = (α−βkvk2)v, (21)

with α, β ≥0. For velocity independent forces, i.e., α, β = 0, equations (20) form a Hamilto-

nian system with energy conservation. Acceleration αv and deceleration dynamics −βkvk2v

are added to the system with α, β > 0. In the case α, β ≥0, after a transition phase, the

Hamiltonian system is recovered when particles travel at equilibrium speed kvk=pα/β.

An important property to describe the large particle limit of the system is H-stability of the

potential U. In particular, H-stability of the potential ensures that particles do not collapse

for N→ ∞. For non-H-stable (catastrophic) systems, increasing particle numbers Nreduce

the particle spacing and particles collapse. In either case, for H-stable as well as catastrophic

systems, stationary solutions can emerge and are characterized by α, β and the potential

characteristic, see [16].

For α, β > 0 and a Morse potential where the potential minimum exists, stationary states of

the large particle systems are observed in which the particles form a coherent structure and

travel at speed with kvk=pα/β for the H-stable as well as the catastrophic situation [16].

10

For N→ ∞, a well-deﬁned spacing is maintained for the H-stable potential, which is also

true for ﬁnite N for the catastrophic potential.

To model the collective behavior macroscopically for α, β > 01, we consider (19) with average

velocity v, kvk=vc=pα/β. The macroscopic limit is then given by

∂tρ+∇x·(ρv−ρC∇ρH(ρ−ρcrit)) ,(22)

where vas the average speed of the swarm. The density threshold ρcrit above which diﬀusion

in a swarm occurs is not to be understood as a maximum density, when individuals are as

closely together as possible, but instead as the preferred density of the undisturbed ﬂock above

which diﬀusion is active. Collisions of the swarm with obstacles lead to internal diﬀusion of

the swarm and can change the direction v. The collisional behavior will be validated by

numerical comparisons of (22) to corresponding microscopic behavior in Section 3.

For α= 0 and β > 0, the macroscopic equation (9) is given by the following Lemma.

Lemma 2. For the attraction-repulsion model (20)with velocity selection mechanism (21),

α= 0, β > 0and interaction force F=∇Uobeying (8), the macroscopic limit given by

equation (9)is the diﬀusion equation

∂tρ− ∇x·(k(ρ)∇ρ) = 0,(23)

where

k(ρ) = ρ¯

C∇ρH(ρ−ρcrit)

3

qk¯

C∇ρH(ρ−ρcrit)kβ

,

and threshold density ρcrit =m0

πR2

0z2if R/dR=z > 0. For an interaction force (6)with dR=

∞obeying (10), the macroscopic limit given by equation (11)is the diﬀusion equation (23)

with ρcrit = 0.

Proof. We restrict ourselves to the case with R/dR=z > 0. Since the force term satisﬁes (8),

we have ΨdR(ρ, ∇ρ) = ¯

C∇ρH(ρ−ρcrit).

We have to determine G−1(y) for y∈R2. Note that we are able to uniquely determine G−1(0)

because G(v) = −βkvkv= 0 has only one solution. Therefore, we have

G−1(y) =

y

−3

pkykβif kyk 6= 0,

0 if kyk= 0,

and the limit is the diﬀusion equation

∂tρ+∇x·

−ρ¯

C∇ρH(ρ−ρcrit)

3

qk¯

C∇ρH(ρ−ρcrit)kβ

= 0.

1We remark that for α, β > 0, Gis not invertible and we cannot directly apply (9).

11

3 Numerical evaluation in bounded domains

For the numerical evaluation of the diﬀerent types of (advection-)diﬀusion equations, we

present a tailored discretization scheme with operator splitting. In our numerical examples,

we particularly focus on the inﬂuence of boundaries on the dynamics of the macroscopic

models.

3.1 Operator splitting method

We discretize a rectangular spatial domain (Ω∪∂Ω) ⊂R2with grid points xij = (i∆x(1), j∆x(2)),

(i, j)∈A={1,...Nx(1) }× ∈ {1,...Nx(2) }. The boundary is described by the set of indices

B⊂A. The time discretization is given by ts=s∆t. We compute the approximate solution

ρ(x, t) = ρs

ij for (x∈Cij

t∈[ts, ts+1),

to (15)-(17) where Cij =(i−1

2)∆x(1),(i+1

2)∆x(1)×(j−1

2)∆x(2),(j+1

2)∆x(2). We use

an operator splitting method to separate the advective and diﬀusive terms.

Since the hyperbolic part of the material ﬂow limit (14) and the swarming limit (22) reduces

to linear transport, we solve the advective part with ﬂux f(l)(ρ, x) = v(l)(x)ρ, l = 1,2 using

the Upwind scheme combined with dimensional splitting

˜ρs+1

ij =ρs

ij −∆t

∆x(1) F(1)

up (ρs

ij , ρs

i+1j)−F(1)

up (ρs

i−1j, ρs

ij )

ρs+1

ij = ˜ρs

ij −∆t

∆x(2) F(2)

up (˜ρs

ij ,˜ρs

ij+1 )−F(2)

up (˜ρs

ij−1,˜ρs

ij ),

(24)

where

F(1)

up (ρs

ij , ρs

i+1j) = (ρs

ij v(1)

ij if v(1)

ij ≥0

ρs

i+1jv(1)

ij otherwise,

with v(1)

ij =v(1)(xij ) and F(2)

up is deﬁned analogously. The diﬀusive part is solved with the

implicit ﬁnite diﬀerence method

ρs+1

ij =ρs

ij +∆t

∆x(1)∆x(2) bs+1

i−1j+bs+1

i+1j−4bs+1

ij +bs+1

ij−1+bs+1

ij+1 ,(25)

where bs+1

ij =b(ρs+1

ij ). This combination allows to use the time step restriction given by the

CFL condition of the hyperbolic part

∆t≤min

(i,j)

1

|v(1)

ij |

∆x(1) +|v(2)

ij |

∆x(2)

,(26)

without any additional restriction from the diﬀusive part, see also [22]. As a result, the

splitting method enables for large time steps even for relatively high diﬀusion.

12

Numerically, we approximate the Heaviside function H(ρ−ρcrit), k(ρ) = CρH(ρ−ρcrit) and

b(ρ) = Rρ

0k(y)dy with smooth approximations

Hξ,ρcrit (ρ) = Zρ

ρcrit

2

ξmax 0,1−

2(y−ρcrit)

ξ−1dy

kξ,ρcrit (ρ) = C ρHξ ,ρcrit (ρ)

bξ,ρcrit (ρ) = Zρ

0

CyHξ,ρcrit (y)dy,

(27)

for which

kξ,ρcrit (ρcrit) = 0, kξ,ρcrit (ρcrit +ξ) = C(ρcrit +ξ),

k′

ξ,ρcrit (ρcrit) = 0, k′

ξ,ρcrit (ρcrit +ξ) = C.

Figure 1a-1b show the approximations for ρcrit = 1 and varying diﬀusion constants C. In the

following, we choose consistently ξ= 10−2in all simulations.

0 1 1+ξ

0

0.5

1

ρ

Hξ,1

(a) Heaviside function Hξ,1for ρcrit = 1.

0 1 1+ξ

0

2

4

6

ρ

kξ,1

C= 1

C= 2

C= 3

(b) Strength of diﬀusion kξ,1ρcrit = 1.

Figure 1: Numerical approximations of the Heaviside function (27).

To satisfy the boundary condition (16), we apply zero ﬂux boundary conditions to the advec-

tive and to the diﬀusive ﬂux

f·~n = 0, x ∈∂Ω

(k(ρ)∇ρ)·~n = 0, x ∈∂Ω,

where ~n = (n(1), n(2))Tis the outer normal vector at the boundary.

3.2 Numerical results for the material ﬂow model

3.2.1 Comparisons of local models: degenerate vs. non-degenerate

To compare the limit equations (14) and (18), we perform the following experiment: A bulk

of material with sharp edges is placed with uniform spacing on a conveyor belt. The initial

13

Figure 2: Experimental setup.

density inside the bulk is set to ρ0= 0.8. We send the bulk of material with the conveyor

belt velocity vT= (1,0)Tagainst a boundary which blocks the transportation, as shown in

Figure 2.

We expect that the initial conﬁguration of the bulk governs a linear transport until it reaches

the boundary at time t= 0.1. Since the density is below the maximum density ρcrit = 1, the

material might be compressed up to the density ρcrit when interacting with the boundary.

We use the operator splitting method with step sizes ∆x(1) = ∆x(2) = 10−2and the CFL

condition (26).

For the degenerate advection-diﬀusion equation (14) with diﬀusion coeﬃcients k1= 10ρHξ,1(ρ)

and k2=ρHξ,1(ρ), the bulk is transported until it reaches the boundary at t= 0.1. The

maximum material density maxxρ(x, t) = ρ0is constant for t < 0.1 and increases when the

bulk hits the boundary. For k2(ρ), the numerical density exceeds the maximum density, while

for k1(ρ) the numerical density stays at the level of the critical density ρcrit = 1 suggesting a

maximum principle (Fig. 3).

0 0.05 0.1 0.15

1

1.5

2

t

maxxρ(x, t)

Figure 3: Maximum density as a function of time for Eq.(14) with diﬀusion coeﬃcient k1

(solid line) and k2(dashed line), and for Eq.(18) with diﬀusion coeﬃcient k3(dashed dotted)

and k4(dotted line).

For the non-degenerate advection-diﬀusion equation (18), small diﬀusion coeﬃcients have to

14

be considered to portray the free ﬂow properly. With the diﬀusion coeﬃcient k3(ρ) = 0.05ρ,

the inﬂuence of the diﬀusion is too high in the free ﬂow phase where the density of the bulk

is reduced before the interaction with the boundary happens. For k4(ρ) = 0.02ρ, the density

formation is better captured, but diﬀusion is not strong enough to portray the boundary

interaction correctly. In both cases, when the boundary is reached, diﬀusion is not strong

enough to properly capture the dynamics. Moreover, the impact of the diﬀusion in the free

ﬂow phase misrepresents the time at which the material reaches the boundary.

Figure 4shows the density plots of the solutions at t= 0.15. Figures 4a-4b show sharp edges

at the rear edge of the bulk and the spread out of material at the boundary. Diﬀusion smears

out the edges of the bulk in Figures 4c-4d and the material is compressed at the boundary.

Summarizing, Figs. 3and 4show that the model (14) is a better approach, compared to (18),

to describe the evolution of material ﬂow on a conveyor belt.

(a) k1(ρ).(b) k2(ρ).

(c) k3(ρ).(d) k4(ρ).

Figure 4: Density plots of the solutions at t= 0.15 for diﬀerent diﬀusion coeﬃcients k(ρ).

3.2.2 Comparison to a non-local model and experimental data

In contrast to the local models we have presented so far, non-local models have received an

increasing interest over the past decades to mimic transport phenomena in bounded domains

such as crowd motion [14] or material ﬂow [17]. Therefore, we recall a non-local version of

the degenerate equation (14), see [17]. We consider a kernel function ηand deﬁne a non-local

15

model for the density ρin the following way:

∂tρ+∇x·ρv(x)−C(η∗ ∇ρ)H(ρ−ρcrit)

γb= 0.(28)

The asterisk ∗denotes the spatial convolution. At an arbitrary point xin space, we have to

consider the non-local gradient ∇(η∗ρ) (x). If we use a molliﬁer ηfor the non-local gradient,

deﬁne ǫ=C

γband normalize the non-local gradient, we obtain the non-local macroscopic

model [17] for material ﬂow

∂tρ+∇x· ρ v(x)−ǫ∇(η∗ρ)

p1 + k∇ (η∗ρ)k2

2

H(ρ−ρcrit)!!= 0,(29)

as a special case of equation (28).The non-local model (29) has been proven to accurately

describe the dynamics in good agreement with experimental data [17,29,30]. For a compari-

son of the (local) advection-diﬀusion equation (14) to the non-local model (29), we study the

experiment of material ﬂow on a conveyor belt Ω ∈R2, cf. [17,30]. The experiment consists

of N= 192 cylindrical parts which are transported on a conveyor belt and redirected by a

deﬂector with angle ν, see Figure 5a. The conveyor belt is modeled with the time-independent

velocity ﬁeld v(x) representing the transport along the belt and the deﬂector, see [17] for a

full description. Details for the simulation of (29) are discussed in the Appendix A.

The diﬀerent modeling approaches are compared in Figure 5. Obviously, a good agreement

of the approximate density for the advection-diﬀusion equation with both, the experimental

data and the solution to the non-local model, is achieved. Figure 6depicts the maximum value

of the density maxx∈Ωρ(x, t). The critical density ρcrit = 1 is achieved with the advection-

diﬀusion equation (Hξ,1) when congestion at the deﬂector occurs, while the non-local equation

and the advection-diﬀusion equation with Htan smear out the density proﬁle and the critical

density is not reached at any time.

3.3 Numerical results for the swarming model

3.3.1 Scattering a single swarm at a boundary

In [4] the collision of ﬂocks with walls has been analyzed for the microscopic attraction-

repulsion model (20). Setting F(xi−xj) = λ∇xiU(xi−xj) and varying λ > 0 changes

the relative strength of the damping and the swarming potential. Reﬂecting the individual’s

velocity specularly at the wall, diﬀerent reﬂection patterns for the swarm have been obtained.

Speciﬁcally, reﬂection laws that show the outgoing angle of the ﬂock as a function of the

incoming angle θ0for diﬀerent scalings of the interaction potential λhave been determined.

In the following, we study the boundary behaviour for swarms and for varying diﬀusion

coeﬃcients Cto validate the macroscopic model (22).

We write the initial velocity with initial heading θ0as v0=cos(θ0),sin(θ0)such that

k¯v0k= 1 and set the initial density to ρ0=ρcrit (kxk≤r), r > 0,representing a ﬂock centered

at x= (0,0)T. The boundary is placed at x(1) = 0.2 with ~n = (1,0)T(see Fig. 7a). The

critical density is set to ρcrit = 1. More details of the simulation can be found in Appendix

B.

Figure 7b shows the initial conﬁguration of a ﬂock which is directed towards the boundary

with θ0= 45. We introduce δ=C/v(1),0to evaluate the boundary interaction of the swarm

16

(a) Experimental data. (b) Advection-Diﬀusion (Hξ,1) .

(c) Advection-Diﬀusion (Htan) . (d) Non-Local (Htan ).

Figure 5: Real data and density plots of the solutions t= 1.5s.

for diﬀerent values of δ≥1. Thus diﬀusion in regions where the critical density is exceeded

increases with δ. The contour of the swarm density after the boundary interaction (at t= 0.3)

is obtained with Algorithm 1(Appendix B) and shown in Figure 7c2. The arrows depict the

direction of the ﬂock after the interaction with the wall.

The reﬂection angle θrfor increasing values of δand angles θ0∈ {30,45,60}are displayed

in Figure 8. We ﬁnd that for increasing θ0, the reﬂection angle θrincreases, matching the

microscopic observations in the attraction-repulsion model [4]. For small diﬀusion, the swarm

aligns with the wall. Increasing diﬀusion, the outgoing angles decrease and the swarms are

reﬂected away from the wall. The asterisk in Figure 8marks the diﬀusion constant for which

the ﬂock as a whole reﬂects specularly, i.e. θ0=θr. The results of Table 1are marked with

a circle in Figure 8.

Table 1: Reﬂection angle of the experiments in Figure 7c.

δ123

θr52.81 24.93 16.15

2The swarm proﬁle slightly enlarges over time due to numerical diﬀusion.

17

Advection-Diﬀusion (Hξ,1) Advection-Diﬀusion (Htan) Non-Local (Htan)

0123

0.2

0.4

0.6

0.8

1

Time

maxxρ(x, t)

Figure 6: Maximum density over time.

3.3.2 Collision of several swarms

We consider two swarms described by the density vector ρ= (ρ1, ρ2), where the density of

swarm iis given by ρi(x, t), i = 1,2. Assuming that each swarm ihas an identity and adjusts

its velocity according to the mean velocity of its members, while diﬀusion is driven by the

total density taking into consideration crowding from the other swarm, we obtain the system

of non-linear advection-diﬀusion equations

∂tρi+∇ · ρivi−ρiC H 2

X

i=1

ρi> ρcrit!∇ 2

X

i=1

ρi!!= 0,

in two space dimensions. Let

fi(ρ) = ρ¯vi, gi(ρ) = ρiC(Φ(ρ)>ρcr it),Φ(ρ) =

2

X

z=i

ρi.

We can reformulate this equation in the more general form as

∂tρi+∇ · (fi(ρi)−gi(ρ)∇Φ(ρ)) = 0,(30)

frequently used in batch settling and sedimentation processes [6,9].

We consider the collision of two swarms heading in opposite directions discussed in [4] (see

Figure 9a). The initial velocities are vL0= (cos(45),sin(45))Tand vR0= (−cos(45),sin(45))T

for the left and the right swarm, respectively. We set ρcrit = 1. Initially, the swarm densities

are set to ρ0

L=ρ0

R= 0.8ρcrit, such that diﬀusion is activated when the swarms collide. Note

that for ρ0<0.5, the two swarms just pass each other since their cumulated density is always

below ρcrit in agreement with [4].

Using Algorithm 1(Appendix B) with ∆x(1) = ∆x(2) = 5 ·10−2, we compute the approximate

solution to equation (30). Initial positioning and simulation results are displayed in Figure 9.

18

Wall

θ0

θr

(a) Incoming angle and reﬂection angle. -0.8 -0.6 -0.4 -0.2 0 0.2

-0.2

0

0.2

0.4

0.6

0.8

(b) Initial conﬁguration.

-0.8 -0.6 -0.4 -0.2 0 0.2

-0.2

0

0.2

0.4

0.6

0.8 =1

=2

=3

(c) Swarm contours after collision for diﬀerent

δ.

Figure 7: Collision of a swarm with a boundary.

12345

0

20

40

60

80

reflection angle r

30 deg

45 deg

60 deg

Figure 8: Reﬂection angle for θ0= 30, 45 and 60 deg.

19

For the lower diﬀusion coeﬃcient C= 0.1, we observe that the two swarms merge, interact

with each other and after the interaction continue on separate paths. After the interaction,

the density inside of the swarms is 1

2ρcrit. In contrast, for the diﬀusion coeﬃcient C= 2, the

two swarms are still merged3. The density inside the new swarm is approximately ρcrit.

(a) Initial positions. (b) C= 0.1 at t= 1.2. (c) C= 2 at t= 1.2.

Figure 9: Collision of swarms.

4 Conclusion

We have shown how locally repelling forces modeling geometric exclusion principles at the

microscopic level lead to a discontinuous advection-diﬀusion model at the macroscopic limit for

very generic setups of interacting particle ﬂows. This closes an important gap in the theoretical

program to develop macroscopic models that describe emergent phenomena of large ensembles

of interacting particles as limits of their microscopic behavior. Speciﬁcally, such geometric

exclusion applies to boundaries and obstacles whose macroscopic representation have typically

been done by ad-hoc boundary conditions.

The resulting transport equation is a generalization of [18]. It is strictly hyperbolic as long

as the density of individuals is below a critical density and becomes parabolic when the

density exceeds that threshold. The parabolic part is then described by a density dependent

diﬀusion coeﬃcient. We introduced the operator splitting algorithm to solve the hybrid partial

diﬀerential equation splitting the advective from the diﬀusive part of the equation and thus

being able to handle the discontinuous change in type.

Applications are far reaching: Material ﬂows on conveyor belts ﬁt that model as an undis-

turbed conveyor belt will transport all parts on it with the same velocity without any relative

motion. Interacting with deﬂectors or obstacles will push the parts together such that they

collide and create relative movement that can best be described by diﬀusion [17]. We show

that numerical solutions of the degenerate advection-diﬀusion model for conveyor belt ﬂows

around a deﬂector reproduce quantitatively the experimental results for similar situations.

In another application, the attraction repulsion model [13] of self-propelling particles generates

ﬂock solutions where the forcing potential reaches a minimum and all particles move with the

same velocity and a crystal-like ﬁxed relative position. Again, relative motion is introduced

when the particles become close enough for collisions which macroscopically is depicted by

exceedance of the critical density. The macroscopic model successfully replicated microscopic

3We observe a small drift of the density in Figure 9c to the left because we compute ﬁrst the left swarm

and afterwards the right swarm in each time step.

20

reﬂection laws and high-impact vs. low-impact scattering depending on the relative strength

of the potential forcing vs. the damping forces, reported in [4].

Acknowledgments

The authors are grateful for the support of their joint research by the DAAD (Project-ID

57444394). J. Weissen and S. G¨ottlich are supported by the DFG project GO 1920/7-1.

Further, the authors would like to thank Stephan Knapp for valuable discussions and helpful

suggestions which contributed to the emergence of the present work.

A Simulation of the non-local model

The space step for our simulations discussed in section 3.2.2 is ∆x(1) = ∆x(2) = 10−2. We

consider numerical results for equation (28) computed by the ﬁnite volume Roe scheme,

see [17]. We set ρcrit = 1, ǫ= 2v(1)

Tand the molliﬁer ηis

η(x) = σ

2πe−1/2σkxk2

2,

with σ= 104. The CFL time step for the Roe scheme is ∆t= 4.7328 ·10−4[30]. The CFL

step is relatively small due the approximation of the Heaviside function

Htan(u) = arctan(50(u−ρcrit))

π+1

2,(31)

in the numerical ﬂux function of the Roe scheme. Sharper approximations would further

strengthen the time step size.

For comparability, we compute the advection-diﬀusion equation once with the Heaviside ap-

proximation (31) and once with the sharper approximation (27). We ﬁx C:= ¯

C/γbin the

diﬀusion to C= 2v(1)

T. We treat the deﬂector as internal boundary ∂Ω⊂Ω in the advection-

diﬀusion equation. We use the operator splitting method (24)-(25) with the boundary condi-

tions at the deﬂector from Section 3.1. The CFL time step (26) is ∆t= 7.32 ·10−2. Note that

the application of the operator splitting allows to compute with larger time steps because

the implicit method is used to compute the parabolic part. However, in comparison to the

explicit Roe scheme, a system of nonlinear equations has to be solved in each iteration.

B Simulation of swarms

To compute an approximate solution to (22), we discretize with step sizes ∆x(1),∆x(2),∆t

and have to iteratively determine the velocity ¯vs, s = 1,...Nt. For a given location xij ∈Ω

and ﬁxed time ts=s∆t, we determine the velocity

vs

ij =vs−1−C∇ρs−1

ij H(ρs−1

ij −ρcrit),(32)

and compute the new mean velocity as the weighted average

¯vs=X

(i,j)

ρs−1

ij

P(i,j)ρs−1

ij

vs

ij,¯vs= ¯vsk¯v0k

k¯vsk,(33)

21

Algorithm 1 Numerical simulation of a swarm in a bounded domain

Require: Domain with boundary Ω ∪∂Ω, initial conditions ρ0, v0for x∈Ω∪∂Ω, diﬀusion

coeﬃcient Cand critical density ρcrit, step sizes ∆x(1) ,∆x(2),∆t

Ensure: Densities vectors ρs= (ρs

ij )i∈{1,...,Nx(1)},j ∈{1,...,Nx(2) }

1: Set s= 0, ts= 0

2: while ts< T do

3: Set ∆taccording to the CFL condition (26) and set s=s+ 1, ts=ts−1+ ∆t

4: for i= 1,...,Nx(1) do

5: for j= 1,...,Nx(2) do

6: Compute the velocity vs

ij (32).

7: If xi+1j, xij+1, xi−1jor xij−1∈∂Ω and hvs

ij, ~ni>0, apply specular reﬂection

vij,new =vs

ij −2hvs

ij , ~ni~n,

where ~n is the outer normal vector at the boundary and update vs

ij =vij,new.

8: end for

9: end for

10: Compute the new average velocity ¯vsof the swarm (33) and compute the solution

ρ(x, ts) to

∂tρ+∇x·(ρvs−ρC∇ρH(ρ−ρcrit))) = 0 x∈Ω

(ρvs−ρC∇ρH(ρ−ρcrit)) ·~n = 0 x∈∂Ω

ρ(x, ts−1) = ρs−1

ij x∈Cij

with the operator splitting method (24)-(25).

11: end while

which is normalized such that k¯vsk=kv0k. For inner grid cells, we determine ∇ρs−1

ij with cen-

tral diﬀerences. To determine the velocity vs

ij in a cell (i, j) at the boundary, we approximate

the gradient ∇ρij using neighbouring cells in transport direction as follows

∇ρs−1

ij ≈

ρs−1

i+1j−ρs−1

ij

∆x(1) if (i+ 1, j)∈Band ¯v(1),s−1≥0,

ρs−1

ij −ρs−1

i−1j

∆x(1) if (i+ 1, j)∈Band ¯v(1),s−1<0.

The gradient is approximated using forward diﬀerences if the ﬂock is moving towards the

boundary, i.e., ¯v(1),s−1≥0, to reﬂect the swarm from the boundary. If instead of approaching

the wall, the ﬂock moves away from the boundary, i.e., v(1),s−1<0, the gradient is approx-

imated with the backward diﬀerence. When the ﬂock approaches the boundary, the ﬂock

solution breaks apart due to the boundary inﬂuence. If hvs

ij , ~ni>0, we assume that the

velocity is reﬂected specularly for an individual cell. This is in line with the microscopic

treatment [4] where single individuals are reﬂected specularly at the boundary. In particular,

the macroscopic velocity (32) is updated as follows

vs

ij,new =vs

ij −2hvs

ij , ~ni~n,

before calculating the new average velocity.

22

To evaluate whether the density is at the level of the critical density, we use Hξ,ρcrit−ξ(ρs−1

ij ),

such that Hξ,ρcrit−ξ(ρcr it) = 1. Below the maximum density, the transport velocity vsis

reﬂected specularly. When the critical density ρs−1

ij =ρcrit is reached in the cell (i, j ), the

velocity in x(1)-direction is reﬂected. If v(1),s

ij,new <0, the velocity for this particular cell changes

its sign. If the sign of the velocity changes in a suﬃciently large number of cells, the new

mean velocity ¯v(1),s is smaller than zero and the entire swarm will have negative mean velocity

v(1),s in x(1)-direction and move away from the boundary in the next time step. For each

experiment, we choose the end of the time horizon Tsuch that the change in the reﬂection

angle after the wall collision is small, i.e., we choose Ntsuch that θr(tNt)−θr(tN t−1)<10−3.

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