Then, we analyse consistency and stability of the semi-discrete scheme

SELF-ORGANIZED DYNAMICS

FRANCIS FILBET AND CHI-WANG SHU

Abstract. This paper deals with the numerical resolution of kinetic models for systems of self-propelled particles subject to alignment interaction and attraction-repulsion. We focus on the kinetic model considered in [18, 17] where alignment is taken into account in addition of an attraction-repulsion interaction potential. We apply a discontinuous Galerkin method for the free transport and non-local drift velocity together with a spectral method for the velocity variable. Then, we analyse consistency and stability of the semi-discrete scheme. We propose several numerical experiments which provide a solid validation of the method and its underlying concepts.

Key words: Self-propelled particles, alignment dynamics, kinetic model, discontinuous Galerkin method

AMS Subject classification: 35L60, 35K55, 35Q80, 82C05, 82C22, 82C70, 92D50.

Contents

1. Introduction 1

1.1. Agent-based model of self-alignment with attraction-repulsion 3 1.2. Kinetic model of self-alignment with attraction-repulsion 3

2. Numerical Methods 4

2.1. Notations 4

2.2. The semi-discrete discontinuous Galerkin method 5

2.3. Temporal discretizations 7

3. Conservation and stability 7

4. Proof of Theorem 2.2 10

4.1. Basic results 10

4.2. Error estimates kf−f_hk_L² 15

5. Numerical simulations 16

5.1. Accuracy test 16

5.2. Taylor-Green vortex problem 17

5.3. Formation of bands problem 17

6. Conclusion and perspective 20

Acknowledgements 21

References 21

1. Introduction

Theoretical and mathematical biology communities have paid a great deal of attention to explain large scale structures in animal groups. Coherent structures appearing from seemingly direct interactions between individuals have been reported in many different species like fishes, birds, and insects [40,45,43, 5, 15,6] and many others, see also the reviews [10,8, 41]. There has been an intense literature about the modeling of interactions between individuals among

1

animal societies such as fish schools, bird flocks, herds of mammalians, etc. We refer for instance to [1,2,15,38] but an exhaustive bibliography is out of reach. Among these models, the Vicsek model [53] has received particular attention due to its simplicity and the universality of its qualitative features. This model is an individual based model or agent-based model which consists of a time-discretized set of Ordinary Differential Equations for the particle positions and velocities. A time-continuous version of this model and its kinetic formulation are available in [18]. A rigorous derivation of this kinetic model from the time-continuous Vicsek model can be found in [4] and in [17] when adding an attraction-repulsion force.

On the other hand, hydrodynamic models are attractive over particle ones due to their com- putational efficiency. For this reason, many such models have been proposed in the literature [9, 12, 23, 43, 51, 52]. However, most of them are phenomenological. For instance in [18], the authors propose one of the first rigorous derivations of a hydrodynamic version of the Vicsek model (see also [42,46, 47] for phenomenological derivations). It has been expanded in [19] to account for a model of fish behavior where particles interact through curvature control, and in [20] to include diffusive corrections. Other variants have also been investigated [16,32,33]. For instance, [32] studies the influence of a vision angle and of the dependency of the alignment frequency upon the local density, whereas in [16, 33], the authors study a modification of the model which results in phase transitions from disordered to ordered equilibria as the density increases and reaches a threshold, in a way similar to polymer models [24,44].

In this paper, we will focus on the numerical approximation of a kinetic model for self- propelled particles. The self-propulsion speed is supposed to be constant and identical for all the particles. Therefore, the velocity variable reduces to its orientation in the (d−1)-dimensional sphereS^d−1. The particle interactions consist in two parts:

• an alignment rule which tends to relax the particle velocity to the local average orien- tation;

• an attraction-repulsion rule which makes the particles move closer or farther away from each other.

This model is inspired both by the Vicsek model [53] and the Couzin model [2,15] describing interactions at the microscopic level. This approach has led to different types of models for swarming: microscopic models and macroscopic models involving macroscopic quantities (e.g.

mass, flux). Here we study an intermediate approach, called the mesoscopic scale, where we investigate the time evolution of a distribution function of particles f(t,x,v), depending on time t ≥ 0, position x∈ Ω⊂ R^d and velocity v∈ S^d−1. This distribution function is solution to a kinetic equation describing the motion of particles and their change of directions. Sev- eral works have already studied kinetic models for swarming [7, 39, 9], but few have done a numerical investigation. For macroscopic and microscopic models, we refer to [18, 21, 22] for discrete particle approximation of microscopic models and finite volume or finite difference ap- proximations [22] where complex structures may be observed as in [27,22]. Concerning kinetic model, we refer to [34] for the first work on this topic, where the authors propose a spectral discretization of the operator describing the change of direction with the flavor of what has been done for the Boltzmann equation [48, 29]. This spectral discretization is coupled with a finite volume approximation for the transport [25, 26, 28, 31]. For the study of bacteria’s motion, we also mention [30] where similar structures have been observed (band’s formation). Here we present a local discontinuous Galerkin method for computing the approximate solution to the kinetic model and to get high order approximations in time, space and velocity. Indeed, the preservation of high order accuracy allows to investigate complex structures in space as it has already been observed for macroscopic models [21, 22]. Discontinuous Galerkin methods

present a clear advantage kinetic models, which are transport dominated and non- linear but do not develop shocks. High order accuracy is mandatory to reproduce some complex phenomena involved in these models.

The paper is organized as follows: we first present precisely the kinetic model and give the main assumptions on the regularity of the unique solution to prove convergence and error estimates on the approximation to the exact smooth solution. Then, in Section 2 we develop a numerical scheme (local discontinuous Galerkin method) for the kinetic model. In Sections 3 and 4, we perform a stability and convergence analysis of the proposed numerical methods.

Numerical investigations are presented in Section 5 where the order accuracy is verified and we observe the formation of complex structures.

1.1. Agent-based model of self-alignment with attraction-repulsion. The starting point of this study is an Individual-Based Model of particles interacting through self-alignment [53]

and attraction-repulsion [2, 15]. Specifically, we consider N particles x_i ∈ R^d, with d = 2 or 3, moving at a constant speed v_i ∈ S^d−1. Each particle adjusts its velocity to align with its neighbors and to get closer or further away. Therefore, the evolution of each particle is modeled by the following dynamics: for any 1≤i≤N

(1.1)









 dx_i

dt =vi, dv_i =P_v^⊥

i v_idt + √

2ν ◦ dBⁱ_t ,

whereBⁱ_t is a Brownian motion and drepresents the noise intensity whereas P_v^⊥

i is the projec- tion matrix onto the normal plane tov_i:

P_v^⊥ = Id−v⊗v, which ensures that v_i stays of norm 1.

Both the alignment and attraction-repulsion rules are taken into account in the macroscopic velocity v_i ∈S^d−1:

v_i = 1

|J_i+R_i|(J_i + R_i),

whereJ_i counts for the alignment and R_i for the attraction-repulsion:

(1.2) J_i =

N

X

j=1

k(|x_j−x_i|)v_j, R_i = −

N

X

j=1

∇_xiφ(|x_j−x_i|),

where the kernel k is a positive function, φ⁰ can be both negative (repulsion) and positive (attraction) and for simplicity, we will assume that both k and φ are compactly supported in [0,∞).

1.2. Kinetic model of self-alignment with attraction-repulsion. When the number of particles becomes large, that is N → ∞, one can formally derive a Vlasov type equation. It describes the evolution of a system of particles under the effects of external and self-consistent fields. The unknown f(t,x,v), depending on the time t, the position x, and the velocity v, represents the distribution of particles in phase space for each species with (x,v) ∈Ω×S^d−1, d= 1, ..,3, where Ω⊂R^d. Its behaviour is given by the Vlasov equation [4,18, 50, 17],

(1.3) ∂f

∂t + v· ∇_xf =−div_v[P_v^⊥v_ff−ν∇_vf],

whereν > 0 and

(1.4)























vf = 1

|J_f +R_f|(Jf +Rf), J_f =

Z

Ω×S^d−1

k(|x−x⁰|)v⁰f(t,x⁰,v⁰)dx⁰dv⁰,

R_f =−∇_x Z

Ω×S^d−1

φ(|x−x⁰|)f(t,x⁰,v⁰)dx⁰dv⁰.

In general, the function φ is such that φ(r) → 0 when r → ∞, but here we will assume that bothk and φ are nonnegative functions which satisfy

(1.5) k, φ∈ C_c^p([0,∞)), withp≥2 and for periodic boundary conditions in space, we have















J_f(t,x) = Z

supp(k)

k(|y|)ρu(t,x+y)dy,

R_f(t,x) = −∇_x Z

supp(φ)

φ(|y|)ρ(t,x+y)dy.

Furthermore we assume that the system (1.3)-(1.4) has a smooth solution such that f ∈H^k+2([0, T]×Ω×S^d−1),

with J_f and R_f such that for any T > 0, there exists a constant ξ_T > 0 such that for all (t,x)∈[0, T]×Ω

(1.6) |Jf(t,x) +Rf(t,x)| ≥ξT.

Using the distribution f and a rescaling of (1.3), it is possible to identify the asymptotic behavior of the model in different regimes as in [17] and to recover some classical hydrodynamic model for the self-organized dynamics.

The model studied in this paper is a generalization of the model of [18] with the addition of an attraction-repulsion interaction potential [17]. We refer to [35] to the existence of global in time weak solutions for such a model.

2. Numerical Methods

In this section, we will introduce the discontinuous Galerkin algorithm for the system (1.3)- (1.4). Discontinuous Galerkin methods are particularly suited for transport type equations with several attractive properties, such as their easiness for adaptivity and parallel computation, and their nice stability properties. We refer to the survey paper [14] and the references therein for an introduction to discontinuous Galerkin methods. For discontinuous Galerkin methods solving kinetic type equations we refer to [11,3]. We consider an open set Ω⊂R^d where all boundary conditions are periodic, andf(t,x,v) is assumed to be in the unit sphere S^d−1.

2.1. Notations. Let T_h^x = {K_x} and T_h^v = {K_v} be partitions of Ω and S^d−1, respectively, with K_x and K_v being Cartesian elements ; then

T_h =

K ⊂Ω×S^d−1; K =K_x×K_v∈ T_h^x× T_h^v defines a partition of Ω×S^d−1.

LetE be the set of the edges of T_h will be E =E_x∪ E_v as







E_x = {∂K_x×K_v : K_x∈ T_h^x, K_v ∈ T_h^v}, E_v = {K_x×∂K_v : K_x ∈ T_h^x, K_v ∈ T_h^v}. Next we define the discrete spaces

(2.1) G_h^k =

g ∈L²(Ω×S^d−1) : g|_K ∈P^k(K), K ∈ T_h , and

U_h^k =

U∈[L²(Ω×S^d−1)]^d−1 : U|_K ∈[P^k(K)]^d−1, K ∈ T_h ,

where P^k(K) denotes the set of polynomials of total degree at most k on K, and k is a nonnegative integer.

Note the space G_h^k, which we use to approximate f, is called P-type, and it can be replaced by the tensor product of P-type spaces in xand v,

g ∈L²(Ω×S^d−1) : g|_K ∈P^k(K_x)×P^k(K_v), K =K_x×K_v ∈ T_h , or by the tensor product space in each variable, which is called Q-type

g ∈L²(Ω×S^d−1) : g|K ∈Q^k(Kx)×Q^k(Kv), K =Kx×Kv ∈ Th .

Here Q^k(K) denotes the set of polynomials of degree at most k in each variable on K. The numerical methods formulated in this paper, as well as the conservation, stability, and error estimates, hold when any of the spaces above is used to approximatef.

Remark 2.1. In our simulations of Section 5, we use the P-type of (2.1) as it is the smallest and therefore renders the most cost efficient algorithm.

For piecewise functions defined with respect to T_h^x or T_h^v, we further introduce the jumps and averages as follows. Forα∈ {x,v} and for any edgeσ={K_α⁺∩K_α⁻} ∈ E_α, with n^±_α as the outward unit normal to ∂K_α^±, g^± =g|_K^±

α, the jumps across σ and the averages are defined as (2.2) [g]_α =g⁺−g⁻, {g}_α = 1

2(g⁺+g⁻), α∈ {x,v}.

2.2. The semi-discrete discontinuous Galerkin method. The numerical methods pro- posed in this section are formulated for the system (1.3)-(1.4). Givenk, r≥0, the semi-discrete discontinuous Galerkin methods for the system (1.3)-(1.4) are defined by the following proce- dure: for any K = K_x×K_v ∈ T_h, we look for (f_h,q_h) ∈ G_h^k× U_h^k, v_fh ∈ U_h^r, such that for all g ∈ G_h^k,

Z

K

∂f_h

∂t g dxdv− Z

K

f_hv· ∇_xg dxdv− Z

K

(P_v^⊥v_fhf_h − νq_h)· ∇_vg dxdv (2.3)

+ Z

σx

fdhv·nxg⁻dsxdv + Z

σv

fh\P_v^⊥vf_h − νcqh

·nvg⁻dsvdx = 0,

wheren_x and n_v are outward unit normals of ∂K_x and ∂K_v, respectively, whereas q_h is given by

(2.4) Z

K

q_h·udxdv + Z

K

f_hdiv_vudxdv− Z

σv

fb_hn_v·u⁻dxds_v = 0, ∀u ∈ U_h^k. Furthermore, the velocityvf_h ∈L^∞(Ω) with kvf_hk= 1, and

(2.5) vf_h(t,x) = 1

kJ_h(t,x) +R_h(t,x)k(Jh(t,x) +Rh(t,x) ),

with J_h and R_h computed by

(2.6)











J_h(t,x) = Z

Ω

k(|x−x⁰|)ρ_hu_h(t,x⁰)dx⁰,

R_h = Z

Ω

∇_xφ(|x−x⁰|)ρ_h(t,x⁰)dx⁰, whereρ_h and u_h are defined from the distribution function f_h, by

ρ_h = Z

S^d−1

f_hdv, ρ_hu_h = Z

S^d−1

vf_hdv.

All hat functions are numerical fluxes that are determined by upwinding for convection and local DG alternating for diffusion, i.e. for the convective terms in (2.3)

(2.7)











fd_hv·n_x = v·n_x{f_h}_x − |v·n_x| 2 [f_h]_x,

f_h\P_v^⊥v_fh·n_v = P_v^⊥v_fh·n_v{f_h}_v − |P_v^⊥vf_h| 2 [f_h]_v, and for the diffusive terms in (2.4), we apply

(2.8) cqh·nv = {qh} ·nv + C11

2 [fh]v, fbhnv = {fh}nv + C22

2 [qh]v

whereC₁₁, C₂₂>0.

This completes the definition of our Discontinuous-Galerkin method, but to facilitate its study, we recast its formulation. We sum (2.3) and (2.4) over all elements and definea_h(.) and b_h(.) such that for (f_h,q_h)∈ G_h^k× U_h^k and g ∈ G_h^k

a_h(f_h,q_h,v_fh, g) :=

Z

Ω×S^d−1

∂f_h

∂t g−f_hv· ∇_xg

dxdv (2.9)

− Z

Ω×S^d−1

(P_v^⊥vf_hfh − νqh)· ∇vg dxdv

− X

σx∈Ex

Z

σx

fd_hv·n_x[g]_xds_xdv

− X

σv∈Ev

Z

σv

fh\P_v^⊥vf_h − νcqh

·nv[g]vdsvdx, and for u ∈ U_h^k

b_h(f_h,q_h, u) :=

Z

Ω×S^d−1

(q_h·u + f_hdiv_vu) dxdv (2.10)

+ X

σv∈Ev

Z

σv

fbhnv·[u]vdxdsv,

where we notice that a_h (resp. b_h) is linear with respect to (f_h,q_h) and g (resp. u).

We prove the following boundedness and error estimate results.

Theorem 2.2. Assume that the solution to (1.3)-(1.4) is such that f ∈H^k+2([0, T]×Ω×S^d−1) with the two hypothesis (1.5) and (1.6). We also assume that the initial dataf_h(0) is uniformly bounded in L²(Ω×S^d−1) and for k ≥0, we consider the numerical solution (f_h,q_h)∈ G_h^k× U_h^k

given by (2.3)-(2.8) supplemented with periodic boundary conditions. Then, for any h₀ > 0, there exists C_T >0, depending on f, T and h₀, such that for h < h₀

kf_h(t)k²_L2 + Z t

0

kq_h(s)k²_L2ds ≤ C_T, t∈[0, T] and

kf(t)−f_h(t)k_L² + Z t

0

kq(s)−q_h(s)k²_L2ds 1/2

≤ C_T h^k+1/2, t ∈[0, T].

2.3. Temporal discretizations. We use total variation diminishing (TVD) high-order Runge- Kutta methods to solve the method of lines ordinary differential equation resulting from the semi-discrete discontinuous Galerkin scheme,

df_h

dt = R(f_h).

Such time stepping methods are convex combinations of the Euler forward time discretization.

The commonly used third-order TVD Runge-Kutta method is given by

(2.11)







f_h⁽¹⁾ = f_hⁿ + 4tR(f_hⁿ), f_h⁽²⁾ = 1

4

3f_hⁿ + f_h⁽¹⁾ + 4tR(f_h⁽¹⁾)

, and

(2.12) f_hⁿ⁺¹ = 1

3

f_hⁿ + 2f_h⁽²⁾ + 24tR(f_h⁽²⁾) ,

wheref_hⁿ represents a numerical approximation of the solution at discrete time t_n.

A detailed description of the TVD Runge-Kutta method can be found in [49]; see also [36]

and [37] for strong-stability-preserving methods.

3. Conservation and stability

In this section, we will establish conservation and stability properties of the semi-discrete discontinuous Galerkin methods. In particular, we prove that for periodic boundary conditions, the total density (mass) is always conserved. We also show thatfhisL²stable, which facilitates the error analysis of Section 4.

Lemma 3.1(Mass conservation). Consider the numerical solution(f_h,q_h)∈ G_h^k×U_h^k fork ≥0 given by (2.3)-(2.8) supplemented with periodic boundary conditions. Then it satisfies

(3.1) d

dt Z

Ω×S^d−1

fhdxdv = 0.

Equivalently, for ρ_h(x, t), for any t >0, the following holds:

(3.2)

Z

Ω

ρh(t,x)dx= Z

Ω

ρh(0,x)dx.

Proof. Let g(x,v) = 1 and noticing that g ∈ G_h^k, for any k ≥ 0, is continuous ∇_xg = 0 and

∇_vg = 0. Taking this g as the test function in (2.3), one has d

dt Z

K

f_hdxdv + Z

σx

fd_hv·n_xds_xdv + Z

σv

f_h\P_v^⊥v_fh − νcq_h

·n_vds_vdx = 0.

Then summing over K ∈ T_h and thanks to the periodic boundary conditions, we get the conservation of total mass for any t≥0,

d dt

Z

Ω×S^d−1

f_h(t)dxdv = 0.

Finally from the definition ofρh and integrating in time from 0 to t, it gives (3.2).

Finally, we can obtain the L²-stability result for f_h. This result will be used in the error analysis of Section 4.

Lemma 3.2 (L²-stability of fh). Assume that the initial data fh(0) is uniformly bounded in L²(Ω×S^d−1) and for k≥0, consider the numerical solution (f_h,q_h)∈ G_h^k× U_h^k given by (2.3)- (2.8) supplemented with periodic boundary conditions. Then (f_h,q_h) satisfies for any t≥0

1 2

d dt

Z

Ω×S^d−1

|f_h|²dxdv + ν Z

Ω×S^d−1

|q_h|²dxdv

+ 1

2 X

σv∈Ev

Z

σv

(|P_v^⊥v_fh·n_v|+ν C₁₁) [f_h]²_v + ν C₂₂[q_h]²_vds_vdx

+ 1

2 X

σx∈E_x

Z

σx

|v·n_x|[f_h]²_xds_xdv

= −1 2

Z

Ω×S^d−1

f_h²div_v(P_v^⊥v_fh)dxdv.

Proof. Observing that Z

Kv

fhP_v^⊥vf_h· ∇vfhdv = 1 2

Z

Kv

P_v^⊥vf_h· ∇vf_h²dv,

= 1 2

Z

Kv

div_v P_v^⊥v_fhf_h²

− f_h²div_v(P_v^⊥v_fh) dv,

= 1 2

Z

∂Kv

P_v^⊥v_fh·n_v|f_h⁻|²ds_v − 1 2

Z

Kv

f_h²div_v(P_v^⊥v_fh) dv, and summing over all control volume K ∈ T^h and recasting the edges, it yields

Z

Ω×S^d−1

f_hP_v^⊥v_fh· ∇_vf_hdvdx = −1 2

Z

Ω×S^d−1

f_h²div_v(P_v^⊥v_fh) dvdx

− X

σv∈E_v

Z

σv

P_v^⊥v_fh·n_v{f_h}_v[f_h]_vds_vdx.

Moreover, we have after recasting Z

Ω×S^d−1

f_hv· ∇_xf_hdvdx = − X

σx∈Ex

Z

σx

v·n_x{f_h}_x[f_h]_xds_vdx.

Then we takeg =f_h in (2.9), it gives after an integration by part on each control volume

(3.3) 1

2 d dt

Z

Ω×S^d−1

|f_h|²dxdv+ 1 2

Z

Ω×S^d−1

f_h²div_v(P_v^⊥v_fh) dxdv + I₁+I₂ +I₃ = 0,

with































I₁ =− X

σx∈Ex

Z

σx

fd_hv·n_x − v·n_x{f_h}_x

[f_h]_xds_xdv,

I₂ =− X

σv∈Ev

Z

σv

f_h\P_v^⊥v_fh·n_v − P_v^⊥v_fh·n_v{f_h}_v

[f_h]_vds_vdx,

I₃ =ν Z

Ω×S^d−1

q_h· ∇_vf_hdvdx+ X

σv∈Ev

Z

σv

qc_h·n_v[f_h]_vds_vdx

! .

Let us prove that each term Ik, for 1≤k ≤ 3, is nonnegative. On the one hand using the the definition of the upwinding flux (2.7), we simply have

(3.4) I₁ = 1

2 X

σx∈Ex

Z

σx

|v·n_x|[f_h]²_xds_xdv≥0 and

(3.5) I₂ = 1

2 X

σv∈E_v

Z

σx

|P_v^⊥v_fh·n_v| [f_h]²_vds_vdx≥0.

On the other hand, to deal with the last term I₃, we choose u=q_h in (2.10), hence we get ν

Z

Ω×S^d−1

|q_h|² + f_hdiv_vq_h

dxdv + X

σv∈E_v

Z

σv

fb_hn_v·[q_h]_vdxds_v

!

= 0.

Then performing an integration by part in velocity of the second term and using the definition of I3, we have

I3 = ν Z

Ω×S^d−1

|qh|²dxdv

+ ν X

σv∈Ev

Z

σv

(f_h⁻q⁻_h − f_h⁺q⁺_h)·n_v+cq_h·n_v[f_h]_v+fb_hn_v·[q_h]_vdxds_v. Therefore from the definition of the “alternating fluxes” (2.8), we finally get that (3.6) I₃ =ν

Z

Ω×S^d−1

|q_h|²dxdv + 1 2

X

σv∈Ev

Z

σv

C₁₁[f_h]²_v + C₂₂[q_h]²_v

!

≥0.

Gathering (3.3) together with (3.4)-(3.6), we obtain the result 1

2 d dt

Z

Ω×S^d−1

|f_h|²dxdv + ν Z

Ω×S^d−1

|q_h|²dxdv +1

2 X

σv∈Ev

Z

σv

(|P_v^⊥v_fh·n_v|+ν C₁₁) [f_h]²_v + ν C₂₂[q_h]²_vds_vdx +1

2 X

σx∈Ex

Z

σx

|v·n_x|[f_h]²_xds_xdv = −1 2

Z

Ω×S^d−1

f_h²div_v(P_v^⊥v_fh) dxdv.

From this Lemma, we get L² boundedness estimates on (f_h,q_h) and on the macroscopic quantities.

Proposition 3.3. Under the assumptions of Lemma 3.2, consider the numerical solution (f_h,q_h)∈ G_h^k× U_h^k given by (2.3)-(2.8) supplemented with periodic boundary conditions. Then for any t ≥0

(3.7) kf_h(t)k²_L2 + 2ν e^t Z t

0

kq_h(s)k²_L2ds ≤ kf_h(0)k²_L2e^t.

Furthermore (ρ_h, ρ_hu_h) computed from the distribution function f_h satisfies for any t≥0 (3.8)







kρ_h(t)k_L²_(Ω) ≤ vol(S^d−1)^1/2kf_h(0)k_L²e^t, kρhuh(t)k_L²_(Ω) ≤ vol(S^d−1)^1/2kfh(0)k_L²e^t and ku_hk_L^∞_(Ω) ≤1.

Proof. Starting from Lemma 3.2 and using the fact that kv_fhk ≤1, we have d

dt Z

Ω×S^d−1

|f_h|²dxdv + 2ν Z

Ω×S^d−1

|q_h|²dxdv ≤ Z

Ω×S^d−1

f_h²dxdv, or

d dt

e^−t

Z

Ω×S^d−1

|f_h|²dxdv

+ 2ν Z

Ω×S^d−1

|q_h|²dxdv ≤ 0.

Hence after integration, it yields to theL² estimate for any t ≥0, Z

Ω×S^d−1

|f_h(t)|²dxdv + 2ν e^t Z t

0

Z

Ω×S^d−1

|q_h(s)|²dxdvds ≤ e^t Z

Ω×S^d−1

|f_h(0)|²dxdv.

The estimates on ρ_h(t) and ρ_hu_h(t) now easily follow since v∈S^d−1 and by application of the Cauchy-Schwarz inequality







kρ_h(t)k_L²_(Ω) ≤ vol(S^d−1)^1/2kf_h(t)k_L², kρ_hu_h(t)k_L²_(Ω) ≤ vol(S^d−1)^1/2kf_h(t)k_L².

Hence, we conclude the proof of (3.8) from (3.7).

4. Proof of Theorem 2.2

For any nonnegative integer m, H^m(Ω) denotes the L²-Sobolev space of order m with the standard Sobolev norm and form = 0, we use H⁰(Ω) =L²(Ω).

For any nonnegative integerk, let Πhbe theL²projection ontoG_h^k, then we have the following classical result [13].

4.1. Basic results.

Lemma 4.1 (Approximation properties). There exists a constant C > 0, such that for any g ∈H^m+1(Ω), the following hold:

(4.1) kg−Π^mgk_L²_(K) + h^1/2_K kg−Π^mgk_L²_(∂K) ≤ C h^m+1_K kgk_H^m+1_(K), ∀K ∈ T_h ,

where the constant C is independent of the mesh sizes hK but depends on m and the shape regularity parameters σx and σv of the mesh.

Moreover, we also remind the classical inverse inequality [13]

Lemma 4.2 (Inverse inequality). There exists a constant C >0, such that for any g ∈P^m(K) or P^m(K_x)×P^m(K_v) with K = (K_x×K_v)∈ T_h, the following holds:

k∇_xgk_L²_(K) ≤ C h⁻¹_K

xkgk_L²_(K), k∇_vgk_L²_(K) ≤ C h⁻¹_K

vkgk_L²_(K),

where the constant C is independent of the mesh sizes h_Kx, h_Kv, but depends on m and the shape regularity parameters σ_x and σ_v of the mesh.

Now let us start the error estimate analysis and consider (f,q=∇_vf) the exact solution to the kinetic equation (1.3) and (f_h,q_h) the approximated solution given by (2.3)-(2.8).

We introduce the consistency error function δ_h and the projected error ξ_h ∈ G_h^k × U_h^k such that

(4.2)







ξ_h = (ξ_h,1, ξ_h,2) = (f −Π_hf , q−Π_hq), δ_h = (δ_h,1, δ_h,2) = ( Π_hf−f_h, Π_hq−q_h),

where Πh represents the L² projection onto G_h^k and U_h^k. Hence, we define the total error εh =δh+ξh.

We are now ready to prove the following Lemma

Lemma 4.3 (Estimate of δ_h). Consider the numerical solution (f_h,q_h) ∈ G_h^k× U_h^k for k ≥ 0 given by (2.3)-(2.8) supplemented with periodic boundary conditions. Then for any h₀ > 0, there exists a constant C >0 depending on f and h₀, such that for h≤h₀,

(4.3) 1

2 d

dtkδ_h,1k²_L2 + νkδ_h,2k²_L2 ≤ C

kδ_h,1k²_L2 + h^2k+1 + kv_f −v_fhk_L^∞kδ_h,1k_L² .

Proof. On the one hand the numerical approximation (fh,qh,vf_h) given by (2.9)-(2.10) satisfies (4.4)







a_h(f_h,q_h,v_fh, g) = 0, ∀g ∈ G_h^k, b_h(f_h,q_h,u) = 0, ∀u ∈ U_h^k .

On the other hand since the numerical fluxes of (2.3)-(2.8) are consistent, the exact solution (f,q,v_f) satisfies

(4.5)







a_h(f,q,v_f, g) = 0, ∀g ∈ G_h^k, b_h(f,q,u) = 0, ∀u ∈ U_h^k .

Then we notice that δh ∈ G_h^k × U_h^k; by taking g = δh,1 and u = δh,2 in (4.4) and (4.5) and subtracting the two equalities, one has

(4.6)







a_h(δ_h,v_fh, δ_h,1) = −a_h(ξ_h,v_fh, δ_h,1) + R₀, b_h(δ_h, δ_h,2) + b_h(ξ_h, δ_h,2) = 0,

whereR0 contains the nonlinear terms R₀ :=

Z

Ω×S^d−1

fP_v^⊥(v_f −v_fh) · ∇_vδ_h,1dxdv

+ X

σv∈Ev

Z

σv

fP_v^⊥\(v_f −v_fh)·n_v[δ_h,1]_vds_vdx.

Following the same lines as in the proof of Lemma 3.2 and using the definition ofb_h, we get a_h(δ_h,v_fh, δ_h,1) = 1

2 d dt

Z

Ω×S^d−1

|δ_h,1|²dxdv + ν Z

Ω×S^d−1

|δ_h,2|²dxdv + 1

2 X

σv∈Ev

Z

σv

(|P_v^⊥v_fh·n_v|+ν C₁₁) [δ_h,1]²_v + ν C₂₂[δ_h,2]²_vds_vdx + 1

2 X

σx∈E_x

Z

σx

|v·n_x|[δ_h,1]²_xds_xdv + 1

2 Z

Ω×S^d−1

|δ_h,1|²div_v(P_v^⊥v_fh) dxdv + ν b_h(ξ_h, δ_h,2).

Therefore, using (4.6), it yields 1

2 d dt

Z

Ω×S^d−1

|δ_h,1|²dxdv + ν Z

Ω×S^d−1

|δ_h,2|²dxdv (4.7)

+ 1

2 X

σv∈Ev

Z

σv

(|P_v^⊥v_fh·n_v|+ν C₁₁) [δ_h,1]²_v + ν C₂₂[δ_h,2]²_vds_vdx

+ 1

2 X

σx∈E_x

Z

σx

|v·n_x|[δ_h,1]²_xds_xdv

≤ 1 2

Z

Ω×S^d−1

|δ_h,1|²|div_v(P_v^⊥v_fh)| dxdv + ν|b_h(ξ_h, δ_h,2)|+|R₀|+|a_h(ξ_h,v_fh, δ_h,1)|, where we need to evaluate the right hand side.

On the one hand, we observe that|div_v(P_v^⊥v_fh)| ≤ C, so that the first term is straightfor- ward.

Then, we evaluate the second term|b_h|in (4.7). Using thatξ_h,2 =∇_vf−Π_h∇_vf the integral on the control volume vanishes and applying the Young inequality and Lemma 4.1, we have

|b_h(ξ_h, δ_h,2)| ≤ X

σv∈Ev

Z

σv

ξc_h,1n_v·[δ_h,2]_v

dxds_v, (4.8)

≤ C₂₂ 2

X

σv∈E_v

Z

σv

[δ_h,2]²_vdxds_v + C

2C₂₂kfk²_Hk+1h^2k+1.

Now we treat the third term |R₀| in (4.7) and use the fact that f and P^⊥_v are continuous and v_f −v_fh does not depend on v, hence after an integration by part and by consistency of the flux, all the integrals over σ_v ∈ E_v vanish and there exists a constant C >0, only depending onk∇vfk_W^1,∞, such that

(4.9) |R₀| ≤ Ckv_f −v_fhk_L^∞kδ_h,2k_L².

Finally the last term in (4.7) is a_h(ξ_h,v_fh, δ_h,1), we split it in two parts (4.10) a_h(ξ_h,v_fh, δ_h,1) = R₁+R₂,

whereR₁ and R₂ are given by



























R₁ = Z

Ω×S^d−1

∂ξ_h,1

∂t δ_h,1−ξ_h,1v· ∇_xδ_h,1 − (P_v^⊥v_fhξ_h,1−ν ξ_h,2)· ∇_vδ_h,1

dxdv,

R2 = − X

σx∈Ex

Z

σx

ξ[h,1v·nx[δh,1]xdsxdv

− X

σv∈Ev

Z

σv

ξ_h,1\P_v^⊥v_fh − νξc_h,2

·n_v[δ_h,1]_vds_vdx.

Let us first evaluate the term R1 and decompose it as R1 =R11+R12+R13, with

|R11|:=

Z

Ω×S^d−1

∂ξ_h,1

∂t δh,1dxdv

≤ C h^k+1k∂tfk_H^k+1kδh,1k_L² and R₁₂ is

R₁₂ :=

Z

Ω×S^d−1

ξ_h,1(v−v₀)· ∇_xδ_h,1dxdv+ Z

Ω×S^d−1

ξ_h,1v₀· ∇_xδ_h,1dxdv,

wherev₀ be theL² projection of the functionv onto the piecewise constant space with respect toT_h. Hence , by definition of the L² projectionξ_h,1 =f −Π_hf the last term vanishes and we have from Lemma 4.1 and since kv−v₀k_L^∞ ≤C h,

|R₁₂| ≤ Ckv−v₀k_L^∞h^kkfk_H^k+1kδ_h,1k_L² ≤ C h^k+1kfk_H^k+1kδ_h,1k_L². Finally, we evaluate R₁₃ defined as

R₁₃ := − Z

Ω×S^d−1

P_v^⊥(v_fh−v_f)ξ_h,1· ∇_vδ_h,1dxdv

− Z

Ω×S^d−1

P_v^⊥v_f ξ_h,1· ∇_vδ_h,1dxdv

+ ν

Z

Ω×S^d−1

ξ_h,2· ∇_vδ_h,1dxdv.

Using the definition of the L² projection ξ_h,2 = q−Π_hq, we first observe that the last term vanishes

Z

Ω×S^d−1

ξ_h,2· ∇_vδ_h,1dxdv= 0,

then we proceed as on the estimate of R₁₂ by introducing the L² projection of the function P_v^⊥v_f onto the piecewise constant space with respect to T^h, whereas we apply the Cauchy- Schwarz inequality to treat the first term. It yields that there exists a constant C >0,

|R₁₃| ≤ C kv_f −v_fhk_L^∞kξ_h,1k_L²k∇_vδ_h,1k_L² + h^k+1kfk_H^k+1kδ_h,1k_L² . Again we apply the two Lemmas 4.2 and 4.1, which gives that

|R13| ≤ C h^k (kvf −vf_hkL^∞ + h)kfk_H^k+1kδh,1k_L².

Gathering these results on R11,R12 and R13, we get the following estimate on R1, (4.11) |R₁| ≤ C h^k (kv_f −v_fhk_L^∞ + h)kfk_Hk+1kδ_h,1k_L².