Semi-linear Poisson-mediated Flocking in a Cucker-Smale Model
Christos N. Mavridis, Amoolya Tirumalai, John S. Baras, Ion Matei
aa r X i v : . [ c s . C E ] F e b Semi-linear Poisson-mediated Flocking in aCucker-Smale Model ⋆ Christos N. Mavridis ∗ Amoolya Tirumalai ∗ John S. Baras ∗ Ion Matei ∗∗∗
Electrical and Computer Engineering Department and the Institute forSystems Research, University of Maryland, College Park, MD 20742 USA,(emails: { mavridis, ast256, baras } @umd.edu ) ∗∗ Palo Alto Research Center (PARC), Palo Alto, CA 94304 USA(email: [email protected] ) Abstract:
We propose a family of compactly supported parametric interaction functions in the generalCucker-Smale flocking dynamics such that the mean-field macroscopic system of mass and momentumbalance equations with non-local damping terms can be converted from a system of partial integro-differential equations to an augmented system of partial differential equations in a compact set. We treatthe interaction functions as Green’s functions for an operator corresponding to a semi-linear Poissonequation and compute the density and momentum in a translating reference frame, i.e. one that is takenin reference to the flock’s centroid. This allows us to consider the dynamics in a fixed, flock-centeredcompact set without loss of generality. We approach the computation of the non-local damping using thestandard finite difference treatment of the chosen differential operator, resulting in a tridiagonal systemwhich can be solved quickly.
Keywords:
Control of Distributed Parameter Systems, Networked Control Systems, Large ScaleSystems1. INTRODUCTIONCollective motion of autonomous agents is a widespread phe-nomenon appearing in numerous applications ranging from ani-mal herding to complex networks and social dynamics (Okubo,1986; Cucker and Smale, 2007; Giardina, 2008).In general, there are two broad approaches when investigatingthe underlying dynamics for flocks or swarms: the microscopic,particle models described by ordinary differential equations(ODEs) or stochastic differential equations, and the macro-scopic continuum models, described by partial differential equa-tions (PDEs). Agent-based models assume behavioral rules atthe individual level, such as velocity alignment, attraction, andrepulsion (Cucker and Smale, 2007; Giardina, 2008; Balleriniet al., 2008) and are often used in numerical simulations andin learning schemes where the interaction rules are inferred(Matei et al., 2019). As the number of interacting agents getslarge, the agent-based models become computationally expen-sive (Carrillo et al., 2010). Considering pairwise interactions,the growth is O ( N ) , where N is the number of agents. Aswe approach the mean-field limit, it is useful to consider theprobability density of the agents. Using Vlasov-like arguments(Carrillo et al., 2010), we can construct an equation analogousto the Fokker-Planck-Kolmogorov equation. We can then definemomentum and density and construct a system of compressiblehydrodynamic PDEs (Carrillo et al., 2010; Shvydkoy and Tad-mor, 2017). ⋆ This material is based upon work supported by the Defense Advanced Re-search Projects Agency (DARPA) under Agreement No. HR00111990027. Thispaper has been accepted for publication in the 24th International Symposiumon Mathematical Theory of Networks and Systems (MTNS 2020), Cambridge,UK.
In flocking dynamics (Cucker and Smale, 2007; Carrillo et al.,2010), the velocity alignment term is not only nonlocal butcan also be nonlinear (Shvydkoy and Tadmor, 2017; Maoet al., 2018). The computation of the corresponding hydrody-namic equations with nonlocal forces becomes quite costly dueto the approximation of the convolution integrals or integraltransforms using the various quadrature methods. The simplest‘quadrature’ method is the Riemann sum, whose complexity is O ( n ) , where n is the number of grid points, when estimating aconvolution integral as a convolution sum in one dimension. Onthe other hand, an equivalent solution may be obtained usingfinite differences if the interaction kernel is associated with adifferential operator. If that operator can be put into a sparseform, ideally a tridiagonal form, a solution can be obtainedefficiently.In this work, we modify the classical Cucker-Smale model ofnonlocal particle interaction for velocity consensus (Cuckerand Smale, 2007; Ha et al., 2009). We propose a family ofparametric interaction functions in R d , d ∈ { , , } , that areGreen’s functions for appropriately defined linear partial differ-ential operators, which allow us to speed-up computation of thenonlocal interaction terms. We investigate the conditions underwhich time-asymptotic flocking is achieved in the microscopicformulation in a centroid-fixed frame. We solve the macro-scopic formulation using the Kurganov-Tadmor MUSCL finitevolume method (Kurganov and Tadmor, 2000) and a second-order finite difference discretization of our chosen differentialoperator. The method is compared to bulk variables computedfrom the microscopic formulation for validation.The rest of the manuscript is organized as follows: Section 2introduces the agent-based Cucker-Smale flocking dynamicsand the macroscopic Euler equations. Section 3 describes theonversion of the Euler equations to an augmented system ofPDEs, and the formulation of the boundary value problem. InSection 4 a family of interaction functions is proposed and thecomputation process is explained. Finally, Section 5 comparesthe numerical results and Section 6 concludes the paper.2. MATHEMATICAL MODELSIn this section we introduce the Cucker-Smale dynamics undergeneral interaction functions, define time-asymptotic flocking,and present the mean-field macroscopic equations. Consider an interacting system of N identical autonomousagents with unit mass in R d , d ∈ { , , } . Let x i ( t ) , v i ( t ) ∈ R d represent the position and velocity of the i th -particle at eachtime t ≥
0, respectively, for 1 ≤ i ≤ N . Then the general Cucker-Smale dynamical system (Cucker and Smale, 2007) of ( N ) ODEs reads as: dx i dt = v i dv i dt = N N ∑ j = ψ ( x j , x i )( v j − v i ) (1)where x i ( ) , are v i ( ) are given for all i = , . . . , N , and ψ : R d × R d → R represents the interaction function betweeneach pair of particles.The center of mass system ( x c , v c ) of { ( x i , v i ) } Ni = is defined as x c = N N ∑ i = x i , v c = N N ∑ i = v i (2)When ψ is symmetric, i.e., ψ ( x , s ) = ψ ( s , x ) , system (1) implies dx c dt = v c , dv c dt = x c ( t ) = x c ( ) + tv c ( ) , t ≥ We investigate the additional assumptions on the initial con-ditions and the interaction function ψ , such that system (1)converges to a velocity consensus, a phenomenon known in theliterature as time-asymptotic flocking , defined in terms of thecenter of mass system as Definition 1 (Asymptotic Flocking) . An N − body interactingsystem G = { ( x i , v i ) } Ni = exhibits time-asymptotic flocking if andonly if the following two relations hold: • (Velocity alignment): lim t → ∞ ∑ Ni = k v i ( t ) − v c ( t ) k = , • (Spatial coherence): sup ≤ t ≤ ∞ ∑ Ni = k x i ( t ) − x c ( t ) k < ∞ . We consider the new variables ( ˆ x i , ˆ v i ) : = ( x i − x c , v i − v c ) (5)which correspond to the fluctuations around the center of masssystem, and define ˆ x : = ( ˆ x , . . . , ˆ x N ) , ˆ v : = ( ˆ v , . . . , ˆ v N ) , | ˆ x | = (cid:0) ∑ Ni = k ˆ x i k (cid:1) / , and | ˆ v | = (cid:0) ∑ Ni = k ˆ v i k (cid:1) / , where k · k repre-sents the standard l -norm in R d . Based on Definition 1, asymp-totic flocking is achieved if | ˆ x ( t ) | < ∞ , t ≥ , and lim t → ∞ | ˆ v ( t ) | = d | ˆ x | dt = (cid:28) d ˆ xdt , ˆ x (cid:29) ≤ | ˆ x || ˆ v | (7)which implies d | ˆ x | dt ≤ | ˆ v | (8)Suppose the interaction function ψ is chosen such that ψ ( x , s ) = ˜ ψ ( k x − s k ) , with ˜ ψ : R + → R + being a non-negative and non-increasing function. Then ( ˆ x i , ˆ v i ) are governed by the dynamicalsystem (1), and d | ˆ v | dt = − N ∑ ≤ i , j ≤ N ˜ ψ ( k ˆ x j − ˆ x i k ) k ˆ v j − ˆ v i k ≤ − N ˜ ψ ( | ˆ x | ) ∑ ≤ i , j ≤ N k ˆ v j − ˆ v i k = − N ˜ ψ ( | ˆ x | ) | ˆ v | (9)which implies d | ˆ v | dt ≤ − N ˜ ψ ( | ˆ x | ) | ˆ v | : = − φ ( | ˆ x | ) | ˆ v | (10)where we have used the fact that ∑ Ni = ˆ v i ( t ) = t ≥
0, andmax ≤ i , j ≤ N k ˆ x i − ˆ x j k ≤ | ˆ x | (11)The following Theorem by (Ha et al., 2009) provides sufficientconditions for time-asymptotic flocking. Theorem 1.
Suppose ( | x | , | v | ) satisfy the system of dissipativedifferential inequalities (8), (10) with φ ≥ . Then if | v ( ) | < R ∞ | x ( ) | φ ( s ) ds, there is a x M ≥ such that | v ( ) | = R x M | x ( ) | φ ( s ) ds,and for every t ≥ , | x ( t ) | ≤ x M , and | v ( t ) | ≤ | v ( ) | e − φ ( x M ) t . The following is an immediate consequence of Theorem 1.
Proposition 1.
Let G = { ( x i , v i ) } Ni = be an N − body interact-ing system with dynamics given by (1). Suppose ψ ( x , s ) = ˜ ψ ( k x − s k ) , with ˜ ψ : R + → R + being a non-negative and non-increasing function. Then if | v ( ) − v c ( ) | < R ∞ | x ( ) − x c ( ) | N ˜ ψ ( s ) ds, G exhibits time-asymptotic flocking.2.3 The Mean-Field Limit Consider the empirical joint probability distribution of theparticle positions and velocities { x i , v i } Ni = F Nxv ( t , x , v ) : = N N ∑ i = δ ( x i , v i ) (12)where δ ( · , · ) is the Dirac measure on R d . As the numberof particles N → ∞ , we can use McKean-Vlasov argumentsto show that the empirical distribution converges weakly to adistribution whose density f xv evolves according to the forwardKolmogorov equation (Carrillo et al., 2010) ∂ t f xv + ∇ x · ( v f xv ) + ∇ v · ( A f xv ) = A : = Z R d ψ ( x , s )( w − v ) f xv ( t , s , w ) dsdw . (13)We define ρ ( t , x ) : = Z R d f xv ( t , x , v ) dvm ( t , x ) : = ρ ( t , x ) u ( t , x ) : = Z R d v f xv ( t , x , v ) dv . (14)hich are the marginal probability and momentum densityfunctions. Substituting these into (13) yields the following ( d + ) compressible Euler equations with non-local forcing: (cid:26) ∂ t ρ + ∇ x · ( ρ u ) = ∂ t ( ρ u ) + ∇ x · ( ρ u ⊗ u ) = ρ L ψ ( ρ u ) − ρ u L ψ ρ (15)where u is the mean velocity, ρ ( , x ) and u ( , x ) are given and L ψ f ( t , x ) : = Z R d ψ ( x , s ) f ( t , s ) ds . (16)3. SEMI-LINEAR POISSON MEDIATED FLOCKING We think of the function ψ as a Green’s function, i.e., as theimpulse response of a linear differential equation, representedby the operator L x , such that L x y ( t , x ) = g ( t , x ) (17)implies y ( t , x ) = Z R d ψ ( x , s ) g ( t , s ) ds (18)which results in L − ψ = L x (19)for all t ≥
0, where L x ψ ( x , s ) = δ ( x − s ) , x , s ∈ R d . (20)Then the following proposition holds: Proposition 2.
Suppose ψ is a Green’s function with respectto a linear differential operator L x . Then system (15) is equiv-alent to the augmented system of ( d + ) partial differentialequations: ∂ t ρ + ∇ x · ( ρ u ) = L x y = (cid:2) ρ u ρ (cid:3) T ∂ t ( ρ u ) + ∇ x · ( ρ u ⊗ u ) = d ∑ i = ( ρ y i − ρ u i y d + ) · ˆ e i (21) where { ˆ e i } di = is the standard basis in R d .3.2 The Boundary Value Problem Due to the time-dependence of the center of mass (4), x i , i = , . . . , N , will escape any fixed and open bounded domain Ω ⊂ R d , unless in the trivial case where v c ( ) =
0. Because ofthe flocking behavior (Definition 1), the position fluctuationswith respect to the center of mass are uniformly bounded, i.e.,sup ≤ t ≤ ∞ N ∑ i = k x i ( t ) − x c ( t ) k < ∞ (22)and, therefore we can define a Boundary Value Problem (BVP)in the moving domain Ω c ( t ) = { x + x c ( t ) : x ∈ Ω } (23)where it is assumed that 0 d ∈ Ω , 0 d being the origin of R d .We notice that solving system (21) for ( x , u ) , x ∈ Ω c is equiv-alent to solving it for the fluctuation variables ( ˆ x , ˆ u ) (5), withˆ x ∈ Ω .We note that the boundedness of the domain has an effecton both the Green’s function and the flocking behavior of thesystem of interacting particles, which should satisfy x i ( t ) − x c ( t ) ∈ Ω , i = , . . . , N , t ≥ . (24) 4. ONE-DIMENSIONAL CASEThe BVP of the augmented system of PDEs (21) for d =
1, on Ω = { ˆ x ∈ [ − L , L ] } reads as: ∂ t ρ + ∂ ˆ x ( ρ u ) = L ˆ x y = (cid:2) ρ u ρ (cid:3) T ∂ t ( ρ u ) + ∂ ˆ x ( ρ u ) = ρ y − ρ uy (25)with homogeneous Dirichlet boundary conditions and initialconditions ρ ( , ˆ x ) = ρ ( ˆ x ) , u ( , ˆ x ) = u ( ˆ x ) (26)which are smooth functions.We select the linear partial differential operator L x = − k ( ∂ ∂ x − λ ) (27)with k = λ =
0, for which the associated parametric fam-ily of Green’s functions with homogeneous Dirichlet boundaryconditions on [ , L ] reads as:ˆ ψ ( x , s ) = ( c ( s )( e λ x − e − λ x ) s ≤ xc ( s )( e λ ( x − L ) − e − λ x ) s > x (28) c ( s ) = k λ ( e − L λ − ) ( e λ ( s − L ) − e − λ s ) c ( s ) = k λ ( e − L λ − ) ( e λ s − e − λ s ) (29)The solution over any interval of length L can be obtained by asimple translation of coordinates.The profile of the Green’s function ˆ ψ and the effect of thebounded domain on on it is illustrated in Fig. 1, where, fordifferent fixed values of x , ˆ ψ ( x , s ) is compared to the function ψ ( x , s ) = k λ e − λ k x − s k (30)which is the Green’s function corresponding to L x in an infinitedomain. We note that the parameters ( k , λ , L ) generate a familyof interaction functions (see also (Mavridis et al., 2020)) thatcan simulate widely used interaction functions as the one foundin the original Cucker-Smale model (Cucker and Smale, 2007): G ( x , s ) = K ( + k x − s k ) γ (31)for given parameters ( K , γ ) . − − − s (arb. units) ψ ( x , s ) x = − π x = − π x = 0 x = π x = π Fig. 1. Illustration of ˆ ψ ( x , · ) (28) for different values of x ,and for λ = k = [ − π , π ] . The function ψ ( x , s ) = k λ e − λ k x − s k , which is the Green’s function for L x in infinitedomain, is depicted in the dashed-dotted lines. .1 Asymptotic Flocking Next we provide sufficient conditions such that the solution { ( x i ( t ) , v i ( t )) } Ni = , t ≥
0, of system (1) with interaction functionˆ ψ as defined in (28), (29), satisfy the flocking conditions inDefinition 1, with ˆ x i ( t ) ∈ Ω , for all t ≥ d | ˆ x | dt ≤ | ˆ v | (32)From (11) and the fact that k ˆ x i k ≤ max ≤ i , j ≤ N k ˆ x i − ˆ x j k , we get | ˆ x | ≤ ˆ x M = ⇒ k ˆ x i k ≤ ˆ x M , i = , . . . , N (33)Therefore, we are interested in showing asymptotic flockingwith | ˆ x ( t ) | ∈ [ , ˆ x M ] , for all t ≥ | ˆ x i ( ) | , there is a large enoughvalue of L such that there exist an ˆ x M ∈ [ , L ) for whichˆ x M > | ˆ x ( ) | (34)From (28), (29) it follows that for | ˆ x | ≤ ˆ x M ,ˆ ψ ( x j , x i ) ≥ ˆ ψ ( − ˆ x M , k ˆ x j − ˆ x i k ) ≥ ˆ ψ ( − ˆ x M , | ˆ x | ) (35)which implies that d | ˆ v | dt = − N ∑ ≤ i , j ≤ N ˆ ψ ( ˆ x j , ˆ x i ) k ˆ v j − ˆ v i k ≤ − N ˆ ψ ( − ˆ x M , | ˆ x | ) | ˆ v | (36)and d | ˆ v | dt ≤ − N ˆ ψ ( − ˆ x M , | ˆ x | ) | ˆ v | : = − φ ( | ˆ x | ) | ˆ v | (37)Next we notice that the Lyapunov function V ( | x | , | v | ) : = | ˆ v | + Z | ˆ x | α φ ( s ) ds , α ≥ ( | ˆ x ( t ) | , | ˆ v ( t ) | ) of thesystem of dissipative differential inequalities (8) and (10), for | ˆ x ( t ) | ≤ ˆ x M , since ddt V ( | ˆ x | , | ˆ v | ) = d | ˆ v | dt + φ ( | ˆ x | ) d | ˆ x | dt ≤ φ ( | ˆ x | ) (cid:18) −| v | + d | ˆ x | dt (cid:19) ≤ | ˆ v ( t ) | + Z | ˆ x || ˆ x | φ ( s ) ds ≤ | ˆ v ( ) | , | ˆ x | ≤ ˆ x M | ˆ v ( ) | such that | ˆ v ( ) | < R ˆ x M / | ˆ x ( ) | φ ( s ) ds ,and, since φ is non-negative for | ˆ x ( t ) | ≤ ˆ x M , there exists a¯ x ∈ [ | ˆ x ( ) | , ˆ x M ] for which | ˆ v ( ) | = Z ¯ x | ˆ x ( ) | φ ( s ) ds (41)Suppose there exists a t ∗ ≥
0, such that ˆ x ∗ : = | ˆ x ( t ∗ ) | ∈ ( ¯ x , ˆ x M ] .Then Z ˆ x ∗ | ˆ x ( ) | φ ( s ) ds > | v ( ) | (42) which contradicts (40). Therefore | ˆ x ( t ) | ≤ ¯ x ≤ ˆ x M , t ≥ | ˆ v ( t ) | ≤ | ˆ v ( ) | e − φ ( ¯ x ) t , t ≥ . (44) Lemma 1.
The operator (27) L x on C ∞ R , C ( Ω ) , the space of com-pactly supported test functions, is self-adjoint and invertible,and therefore has a self-adjoint inverse L − x on C ∞ R , C ( Ω ) .Proof. Self-adjointness of the inverse follows immediatelyfrom self-adjointness of L x and the existence of the inverse(Taylor, 2010). It is clear that L x has an inverse since theGreen’s function is nontrivial.We shall now show that the operator L x is self-adjoint on C ∞ R , C ( Ω ) . Consider two functions u , w ∈ C ∞ R , C ( Ω ) , u = w , thespace of test functions, and associated f u , f w ∈ C ∞ R , C , f u : = L x u , f w : = L x w . Let Ω : = [ − L , L ] . We have Z Ω ( w L x u − u L x w ) dx = − k Z Ω ( w ∂ x u − u ∂ x w ) dx . (45)since the semi-linear term drops out. Using Green’s secondidentity, and the compact support of u , w , we have that Z Ω ( w ∂ x u − u ∂ x w ) dx = Z ∂Ω ( w ∂ n u − u ∂ n w ) dx = . (46)Thus, L x is self-adjoint and has a self-adjoint inverse, i.e. Z Ω ( f w L − x f u − f u L − x f w ) dx = Z Ω ( f w u − f u w ) dx = . (47) Proposition 3.
If y is compactly supported, and psi is as given,then mass and momentum are conserved, i.e.ddt Z Ω [ ρ ρ u ] T d ˆ x = Z Ω [ ρ y − ρ uy ] T d ˆ x = . (48) Proof.
We obtain (48) by simply integrating the conservationlaws in (25) over the entire space and apply the Leibniz rule.The conclusion follows directly from the self adjointness ofthe inverse in (47). The proposition holds for any self-adjointalignment operator.
For compactness, we re-write the PDEs (25) as (cid:26) ∂ t U + ∂ ˆ x F ( U ) = S ( U , Y ) L ˆ x Y = U (49)with U = [ ρ , ρ u ] T , Y = [ y , y ] T , F = (cid:2) ρ u , ρ u (cid:3) T , and S =[ , ρ y − ρ uy ] T . Recall the transformation m = ρ u . From this,the flux Jacobian is given by D U F : = (cid:20) − u u (cid:21) (50)which is not diagonalizable, and thus the system is only weaklyhyperbolic. Its eigenvalues are ± u . With these notations estab-lished, we now detail the numerical solution of the PDEs. yperbolic Solver. To solve the hyperbolic system, we applythe finite volume method (LeVeque, 2002). To begin, we definethe sequence of points { ˆ x , ..., ˆ x i , ..., ˆ x N } which are the centersof the cells I i : = [ ˆ x i − , ˆ x i + ) . Then, we average the PDE overthese cells, which gives1 λ ( I i ) ddt Z I i Ud ˆ x = − λ ( I i ) Z I i ∂ ˆ x Fd ˆ x + λ ( I i ) Z I i Sd ˆ x (51)where λ ( · ) denotes the length of an interval. Suppose these areidentical, so ∆ ˆ x : = λ ( I i ) ∀ i . Then, using the divergence theorem,and replacing the integrals of U , F , S with their cell-averages,i.e. their midpoint values ¯ U , ¯ F , ¯ S , we obtain ddt ¯ U i = − ∆ ˆ x ( ¯ F i + − ¯ F i − ) + ¯ S i (52)where ¯ U i : = ¯ Y ( ˆ x i ) , ¯ F i : = ¯ F ( ˆ x i ) , ¯ S : = ¯ S ( ˆ x i ) . In this work, we em-ploy the second-order strong stability preserving Runge-Kuttascheme (Kurganov and Tadmor, 2000) for time integration. Forthe fluxes, we assume piecewise linearity and use the Kurganov-Tadmor flux (Kurganov and Tadmor, 2000). The fluxes aregiven by¯ F i + : = [ F ∗ i + F ∗ i + − max {| u ∗ i | , | u ∗ i + |} ( U ∗ i + − U ∗ i )] U ∗ i + : = U i + − ∆ ˆ x minmod ( U i + − U i + ∆ ˆ x , U i + − U i ∆ ˆ x ) U ∗ i : = U i + ∆ ˆ x minmod ( U i + − U i ∆ ˆ x , U i − U i − ∆ ˆ x ) (53)where minmod ( a , b ) : = ( sign ( a ) + sign ( b )) min ( | a | , | b | ) . Elliptic Solver.
To solve the elliptic equations, we apply theclassical second-order finite difference method, which is y ji + − y ji + y ji − ∆ ˆ x − λ y ji = − kU ji (54)Over the interior points, this yields linear equations ( ∆ ˆ x A − λ I ) y jint = − kU jint − ∆ ˆ x (cid:2) y j . . . y jN (cid:3) T , (55) A = − . . . . . . − . . .
00 1 − . . . − (56)The matrix in (55) is tridiagonal, so banded matrix algorithms(Golub and Van Loan, 2013) can be used to solve the corre-sponding system of equations. As shown in Fig. 2, using finitedifferences is much faster than a convolution (Riemann) sum,even when the embarrassing parallelism of the sum is exploited. Particle Solver.
We solve the system of particle equationsusing the velocity Verlet algorithm (Mao et al., 2018). Givena system of ODEs of the form dxdt = vdvdt = a ( x , v , t ) , (57)with appropriate initial conditions and a time-discretization atsteps { , , ..., i , ... } with increment ∆ t , the discretization is v i + = v i + a ( x i , v i , t i ) ∆ tx i + = x i + ∆ tv i + v i + = v i + ∆ t [ a ( x i , v i , t i ) + a ( x i + , v i + , t i + )] . (58) n (Array Elements) − − − − − − − C o m pu t a t i o n T i m e ( s ec s ) Finite Differences (Tridiagonal Algorithm)Riemann Sum (6 cores) n · − Fig. 2. Computation Times for Nonlocal Terms using FiniteDifferences and Riemann Sum.5. NUMERICAL RESULTS AND HIGHER DIMENSIONSIn this section we present numerical simulations of one-dimensional nonlocal flocking dynamics, by solving ( a ) theagent-based Cucker-Smale model using the velocity Verletmethod, and ( b ) the macroscopic model with initial conditionswhose support is the interval [ − π , π ] . Our aim is to verify thatthe agent based and continuum based approaches to the flockingproblem produce similar results.In the following, the initial density and velocity are given by ρ ( ˆ x ) = π L cos ( π ˆ xL ) , (59) u ( ˆ x ) = − c sin ( π ˆ x ) L ) , ˆ x ∈ [ − L , L ] , (60)i.e. it is assumed that ρ ( ˆ x ) = u ( ˆ x ) = , ∀ ˆ x / ∈ [ − L , L ] , wherewe have used L = π . In all simulations, we take λ = k =
4. For the particlesimulation, we use N = particles. For the macro-scalesimulation, we use ∆ ˆ x = π as the spatial increment. In bothsimulations, we take ∆ t = .
001 as the time increment.In both cases, the support of the initial profile shrinks as the bulkcomes together. The semi-linear Poisson-forced Euler systemis highly dissipative, and the momentum profile is dampeduntil it flattens (although it is conserved over the domain), andthe system attains an equilibrium distribution. Fig. 3 showsthe agreement between the particle model and the macro-scalemodel.
In higher dimensions, the radial symmetry of the interactionfunction ψ suggests the use of a singular kernel . Singularkernels have been extensively studied in the literature and,under mild assumptions in the initial conditions, have beenshown to result in flocking behavior while, at the same time,avoiding collisions (Ahn et al., 2012).In the BVP of the augmented system of PDEs (21) with theinitial and boundary conditions (26), we select the linear differ-ential operator (see also (Mavridis et al., 2020)): L x = − k − d / ( ∇ x − λ ) (61) − − x (arb. units) . . . . . . . . ρ Macro, t = 0 t = . t = 1 t = 1 . t = 10 Particle, t = 0 t = . t = 1 t = 1 . t = 10 − − − x (arb. units) − . − . . . . ρ u Fig. 3. Evolution of the Probability Densities ρ ( t , ˆ x ) and Mo-mentum Densities m ( t , ˆ x ) as computed by solving themacro-scale model and the particle model (dashed-line).and Ω = B d ( , r ) : = (cid:8) x ∈ R d : k x k < r (cid:9) , which results in aGreen’s function of the formˆ ψ ( x , s ) = ψ ( x − s ) + φ ( x , s ) (62)where ψ is given by ψ ( x , s ) = ˜ ψ ( k x − s k )= (cid:18) k π (cid:19) d / (cid:18) λ k x − s k (cid:19) d / − K d / − ( λ k x − s k ) (63)with K α ( · ) being the modified Bessel function of the secondkind of order α , and φ is a function such that L s φ ( x , s ) = , s ∈ B d ( , r ) φ ( x , s ) = − ψ ( x , s ) , s ∈ ∂ B d ( , r ) (64)For s ∈ ∂ B d ( , r ) we have k x − s k = k x k − h x , s i + k s k = k x k k sr − rx k x k k (65)and it can be shown that φ ( x , s ) = − ˜ ψ ( r k x kk s − r x k x k k ) . (66)The interaction function ˆ ψ is affected by the bounded domainin the same way as in the one-dimensional case, and dependson the parameter values k and λ as illustrated in Fig.4 for the2-dimensional case.Fig. 4. The effect of the parameters k , λ on the profile ofthe interaction function ˆ ψ (( , . ) , s ) , s ∈ B ( , ) . Left: ( k , λ ) = ( , . ) . Right: ( k , λ ) = ( , ) . 6. CONCLUSIONA family of compactly supported parametric interaction func-tions in the general Cucker-Smale flocking dynamics was pro-posed such that the macroscopic system of mass and mo-mentum balance equations with non-local damping terms canbe converted to an augmented system of coupled PDEs in acompact set. We approached the computation of the non-localdamping using the standard finite difference treatment of thechosen differential operator, which was solved using bandedmatrix algorithms. The expressiveness of the proposed inter-action functions may be utilized for parametric learning fromtrajectory data. REFERENCESAhn, S., Choi, H., Ha, S.Y., and Lee, H. (2012). On collision-avoiding initial configurations to cucker-smale type flockingmodels. Communications in Mathematical Sciences , 10. doi:10.4310/CMS.2012.v10.n2.a10.Ballerini, M., Cabibbo, N., Candelier, R., Cavagna, A., Cisbani,E., Giardina, I., Lecomte, V., Orlandi, A., Parisi, G., Procac-cini, A., et al. (2008). Interaction ruling animal collectivebehavior depends on topological rather than metric distance:Evidence from a field study.
Proceedings of the nationalacademy of sciences , 105(4), 1232–1237.Carrillo, J.A., Fornasier, M., Toscani, G., and Vecil, F. (2010).Particle, kinetic, and hydrodynamic models of swarming.In
Mathematical modeling of collective behavior in socio-economic and life sciences , 297–336. Springer.Cucker, F. and Smale, S. (2007). Emergent behavior in flocks.
IEEE Transactions on automatic control , 52(5), 852–862.Giardina, I. (2008). Collective behavior in animal groups:theoretical models and empirical studies.
HFSP journal .Golub, G.H. and Van Loan, C.F. (2013).
Matrix Computations .The Johns Hopkins University Press, fourth edition.Ha, S.Y., Liu, J.G., et al. (2009). A simple proof of the cucker-smale flocking dynamics and mean-field limit.
Communica-tions in Mathematical Sciences , 7(2), 297–325.Kurganov, A. and Tadmor, E. (2000). New high-resolutioncentral schemes for nonlinear conservation laws and convec-tion–diffusion equations.
Journal of Computational Physics ,160(1), 241 – 282.LeVeque, R.J. (2002).
Finite Volume Methods for HyperbolicProblems . Cambridge Texts in Applied Mathematics. Cam-bridge University Press. doi:10.1017/CBO9780511791253.Mao, Z., Li, Z., and Karniadakis, G. (2018). Nonlocal flockingdynamics: Learning the fractional order of pdes from particlesimulations. arXiv preprint arXiv:1810.11596 .Matei, I., Mavridis, C., Baras, J.S., and Zhenirovskyy, M.(2019). Inferring particle interaction physical models andtheir dynamical properties. In , 4615–4621. IEEE.Mavridis, C.N., Tirumalai, A., and Baras, J.S. (2020). Learninginteraction dynamics from particle trajectories and densityevolution. In . IEEE.Okubo, A. (1986). Dynamical aspects of animal grouping:swarms, schools, flocks, and herds.
Advances in biophysics .Shvydkoy, R. and Tadmor, E. (2017). Eulerian dynamics witha commutator forcing ii: Flocking. arXiv .Taylor, M. (2010).