Numerical simulation of a kinetic model for chemotaxis

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Numerical simulation of a kinetic model for chemotaxis

Nicolas Vauchelet

To cite this version:

Nicolas Vauchelet. Numerical simulation of a kinetic model for chemotaxis. Kinetic and Related

Models , AIMS, 2010, 3 (3), pp.501-528. �10.3934/krm.2010.3.501�. �hal-00844174�

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N. Vau helet

UPMC,UnivParis06,UMR7598LJLL,ParisF-75005Fran e;

CNRS, UMR7598LJLL,Paris,F-75005Fran e,

andINRIAProjetBANG,

Tel.: (+33)(0)144273772 Fax: (+33)(0)144277200

E-mailaddresses: vau heletann.jussieu.fr

Abstra t

This paperis devoted to numeri al simulations of a kineti model des ribing

hemo-taxis. This kineti framework has been investigated sin e the 80's when experimental

observationshaveshownthatthemotion ofba teriaisdueto thealternan eof'runs and

tumbles'. Sin eparaboli andhyperboli modelsdonottakeintoa ountthemi ros opi

movement of individual ells, kineti models have be omeof a great interest. Dolak and

S hmeiser (2005) have then proposed a kineti model des ribing themotion of ba teria

responding to temporal gradients of hemoattra tants along their paths. An existen e

result for this system is provided and a numeri al s heme relying on a semi-Lagrangian

method is presented and analyzed. An implementation of this s heme allows to obtain

numeri al simulationsofthe modelandobserve blow-uppatternsthatdiergreatly from

the aseof Keller-Segeltype of models.

Keywords. Chemotaxis; Kineti equations;semi-Lagrangianmethod; onvergen e analysis.

AMS subje t lassi ations: 92C17;92B05;65M12; 82C80.

1 Introdu tion and modelling

Chemotaxis is the phenomenon in whi h ells dire t their movements a ording to ertain

hemi als in their environment. A possible issue of a positive hemota ti al movement is

the aggregation of organisms involved to form a more omplex organism or body. Many

attempts for des ribing hemotaxis from a Partial Dierential Equations viewpoint, i.e. for

a large population, have been onsidered. At the ma ros opi level the most famous is the

Patlak, Keller and Segel model[28, 34℄. Although this models have been su essfully used to

des ribeaggregation ofthe population(see [25, 26, 41℄ forsurveys), these ma ros opi models

have several short omings, for instan e they do not take into a ount the detailed individual

movement of ells.

Therefore another approa h involving kineti equations to des ribe hemotaxis has been

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distribution fun tion

f

of ba teria at time

t

, position

x ∈ ω

and velo ity

v ∈ V

and of the

on entration of hemoattra tant

S

. The system writes inthe following way :











∂

t

f + v · ∇

x

f =

Z

v

′

∈V

(T [S](v

′

→ v)f(v

′

) − T [S](v → v

′

)f (v)) dv

′

,

−∆S + S = ρ(t, x) :=

Z

v∈V

f (t, x, v) dv,

(1.1)

ompleted with the initial ondition

f (0, x, v) = f

0 (x, v).

(1.2)

The turningkernel

T [S](v

′

→ v)

denotes the rate of ells hanging their velo ity from

v

′

to

v

.

This system models the evolution of agellated ba teria su h as E. Coli. It has been

observed that aba teriummovesalong straightlines, suddenlystop to hoose a newdire tion

andthen ontinuemovinginthenewdire tionuntilthe ellsre eptorssaturate. Themovement

of the ba teriumis thendue to the alternan eof these 'run'and 'tumble' phases[1,40℄. Cells

areableto omparethepresent hemi al on entrationstopreviousonesandthustorespondto

temporalgradientsalongtheirpaths. The de isionto hangedire tionand turnorto ontinue

moving depends then on the on entration prole of the hemi al

S

along the traje tories of

ells and detailed models have been proposed in [18, 19, 16℄. In [16℄ the authors propose to

onsider simply aturning kernel of the form :

T [S](v

′

→ v) = φ(∂

t

S + v

′

· ∇

x

S).

(1.3)

Therateof turningisgreaterif thegradient on entrationalongthetraje tory

∂

t

S + v · ∇

x

S

is negativethan whenitispositive. Experimentally,intheabsen e ofgradientsof on entration,

an individual ell of E. Coli performs a random walk with a mean duration of run times of

1s

(see [30℄). Due to the inuen e of the hemoattra tant a ellsensing a positive gradientof

on entration has a run 4 times longer. Then, in this simpliedmodel, we onsider that

φ

is

a positive nonin reasingsmooth fun tion; morepre isely,

φ ∈ C

∞

(R),

φ

′

< 0,

φ(z) =

1

if

z < −α,

1/4

if

z > α,

(1.4)

for agiven positive

α

small.

Inthis work,weare interestedintheevolutionoftheba teria on entrationinaPetri box,

whi his approximated by a bounded domain

ω ⊂ R

2

. The velo ity of ba teriahas a onstant

modulus

V

,thereforewetake

V = S

V

:= {v ∈ R

2

with

kvk = V }

. We denote

Ω = ω × V

. The

system an bethen rewritten in the followingway :











∂

t

f + v · ∇

x

f =

Z

v

′

∈V

φ(∂

t

S + v

′

∇

x

S)f (v

′

) dv

′

− 2πφ(∂

t

S + v∇

x

S)f (v),

f (0, x, v) = f

0 _{(x, v),}

−∆S + S = ρ(t, x) :=

Z

v∈V

f (t, x, v) dv,

(1.5)

This system is ompleted with the spe ular ree tion onditions

(4)

where

ν(x)

istheoutwardunitve toratthepoint

x

oftheboundary

∂ω

. Andforthe

hemoat-tra tant on entration,we set Neumannboundary onditions :

∂

ν

S(t, x) = 0,

∀ x ∈ ∂ω.

(1.7) Thissystem is omposedofakineti equation oupledtoanellipti equation. Therefore we

proposeinthisworktouse te hniques whi hhaveproven theire ien y forthenumeri al

res-olutionofthe Vlasov-Poissonsystem inplasmaphysi stodealwiththe numeri alapproa hof

system(1.5). LagrangianmethodslikeParti le-In-Cellmethodswhi h onsistofapproximating

theplasmabyanitenumberofma ro-parti lesareusuallyperformedforthe Vlasovequation

(see [4℄). However, thesemethodsareknown tobevery useful forlarges ale problemsbut are

very noisy and do poor job on the tailof the distribution fun tion. To remedy this problem,

Eulerian methods have been proposed. They onsist of dis retizing the Vlasov equation on a

mesh of phase spa e. Among them, nite volume s hemes are known to be robust and

om-putationally heap [9, 12, 14, 22℄ but very onstrained by a CFL ondition. Semi-Lagrangian

methodsare otherkindsof Eulerianmethodallowingtoobtaina uratedes riptionof the

dis-tributionfun tion [39, 2, 3, 15℄. They onsist of dire tly omputing the distribution fun tion

atea htimestep onaxed Cartesianmesh ofthe phasespa e byfollowingthe hara teristi s

urvesba kward andinterpolatingthevalueatthe baseofthe hara teristi s. Wereferto[23℄

for a review on Eulerianmethods. Then, we will use in this work a semi-Lagrangian method

for the numeri alresolution of model(1.5).

The paper is organized as follows. In the next se tion we state and prove an existen e

and uniqueness result for the system (1.5)(1.7). For the sake of simpli ity this study is

onsidered in the whole domain

R

2

, whi h allows to have an expli it expression of

S

thanks

to the Bessel potential. Se tion 3 is devoted to the numeri al resolution of this system. We

rst re all the semi-Lagrangian method used for the dis retization of the kineti transport

equation. Then we present the algorithm of resolution of the whole system (1.5). Finally, an

analysis of this s heme under additional assumptions furnishes a onvergen e result in

L

2

of

the dis reteapproximationtowards the solution ofmodel(1.5). Numeri alsimulations,whi h

shows the aggregation phenomenon observed forba teria E.Coli, are presented and dis ussed

in omparison to those for Keller-Segelin se tion4.

2 Existen e result

Forthe sake of simpli ity,we onsider in this se tion that

ω = R

2

and

Ω = R

2 _{× V}

.

The existen e of solutionstokineti models of hemotaxishas been investigated inseveral

papers. In[11,27℄,globalexisten efortheinitialvalueproblem(1.1)in

R

3

andin

R

2

hasbeen

obtainedundertheassumptionthattheturningkernelis ontrolledby termsinvolving

S(t, x+

v)

and

S(t, x − v

′

)

. Dispersive methods are used to obtain a priori estimates. These results

has been extended in [8℄formore generalassumptions onthe turningkernel. All thesepapers

took intoa ount the ee t ofthe gradient ofthe hemi alsignaland showed global existen e

of solutions. However, these rigorous global existen e results have not in luded the temporal

derivative of the signal in the growth ondition of the turning frequen y. In ref. [17℄, the

authorsinvestigate global existen e of solutions(not ne essary unique) for general hyperboli

hemotaxis models where the turning kernel takes into a ount the temporal derivative of

the hemoattra tant through the evolution of internal states but only in the one-dimensional

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(1.5) onsidered in the whole domain

Ω = R

2 _{× V}

and to establish existen e and uniqueness

of global-in-time solution. The main novelty is due to the dependan e of the ross-se tion of

the turningoperatoron the timederivative ofthe hemoattra tant on entration. Weuse the

followingexpression for the hemoattra tant on entration :

S(t, x) = (G ∗ ρ(t))(x),

where

G(x) =

1 4π

Z

∞

0 e

−

π

|x|

_4s

2 −

s

4π

ds

s

.

The fun tion

G

isknown asthe Besselpotential. The idea toover ome thedi ulty raisedby

the term

∂

t

S

inthe turning kernel expression (1.3) isto use the onservation of the density :

∂

t

ρ + ∇

x

· J = 0,

J(t, x) =

Z

V

vf (t, x, v) dv.

(2.1) Wehave then

∂

t

S = G ∗ ∂

t

ρ = −G ∗ ∇

x

· J = −∇

x

G ∗ J,

(2.2)

where the onvolutionbetween two ve tors is dened by

∇

x

G ∗ J = ∂

x

1 G ∗ J

1 + ∂

x

2 G ∗ J

2

. It

implies a ontrol onthe partial derivatives of

S

with respe t to time thanks to ontrols on

J

(see [7℄). We an then rewritethe problem (1.5) as







∂

t

f + v · ∇

x

f =

Z

v

′

∈V

φ(∇

x

G ∗ (v

′

ρ − J))f(v

′

) dv

′

− 2πφ(∇

x

G ∗ (vρ − J))f(v),

f (0, x, v) = f

0 _{(x, v),}

(2.3)

Werst denethe notionof weak solutions for(2.3) on

(0, T ) × Ω

.

Denition 2.1 We say that

f

is a weak solution of (2.3) on

L

q

_{(0, t}

0 ; L

p

(Ω))

for

t

0 > 0

and

p, q ≥ 1

, if forany test fun tion

ψ ∈ D([0, t

0 ) × Ω)

, we have

Z

(0,t

0 )×Ω

(∂

t

ψ + v · ∇

x

ψ)f dxdvdt = −

Z

(0,t

0 )×Ω

Z

V

φ(∇

x

G ∗ (v

′

ρ − J))f(v

′

)ψ dtdxdvdv

′

+2π

Z

(0,t

0 )×Ω

φ(∇

x

G ∗ (vρ − J))fψ dtdxdv +

Z

Ω

f

0 _{(x, v)ψ(0, x, v) dxdv,}

where the ells density

ρ

and the urrent density

J

are dened by

ρ(t, x) =

Z

V

f (t, x, v) dv,

J(t, x) =

Z

V

vf (t, x, v) dv.

(2.4)

Westate the following existen e and uniqueness result :

Theorem 2.2 Assume

f

0 _{∈ L}

1 +

∩ L

∞

(Ω)

and that the turning kernel

T

is dened by (1.3)

(1.4). Then the initial value problem (2.3) admits a unique global weak solution

f

satisfying

f ∈ L

∞

((0, ∞); L

1 +

∩ L

∞

(Ω))

. Moreover, if

f

0 _{∈ W}

2,2

_{(Ω) ∩ W}

1,∞

_(Ω)

then for all

t

0 > 0

, there exists a onstant

C

0

depending on

t

0

and on the data su h thatthe weak solution of (2.3) satises

kfk

L

∞

_((0,t

₀

_);W

2,2

_(Ω)∩W

1,∞

_(Ω))

≤ C

₀

;

and

k∂

t

f k

L

∞

((0,t

(6)

of a unique solution. The smoothness assumption (1.4) on

φ

provides thanks to a xed point

pro edure the uniqueness of solution. Then, thanks to a priori estimates, we extend this

solution up to

t

0

for all

t

0 > 0

and then re over global existen e. The main tool in the proof

of Theorem 2.2is ana priori estimategiven in the following Lemma:

Lemma 2.3 Let

t

0 > 0

and

f

0 _{∈ L}

∞

(Ω)

. Let

f

be a weak solution of (2.3), su h that

f ∈

L

1 _{((0, t}

0 ), L

1 +

∩ L

∞

(Ω))

. Then we have for a.e.

t ∈ (0, t

0 )

,

kf(t, ·, ·)k

L

1 (Ω)

= kf

0 _k

L

1 _(Ω)

and

kfk

L

∞

_((0,t

₀

_);L

∞

_(Ω))

≤ C(kf

0 k

_L

∞

_(Ω)

) e

2πt

,

where the onstant

C(kf

0 _k

L

∞

_(Ω)

)

depends only on the initial data.

Proof. First the onservation of the mass shows that

kf(t, ·, ·)k

L

1 _(Ω)

= kf

0 k

_L

1 _(Ω)

.

From the bound

1/4 ≤ φ ≤ 1

of

φ

(1.4) we dedu e

∂

t

f (t, x, v) + v · ∇

x

f (t, x, v) ≤

Z

V

f (t, x, v

′

) dv

′

.

Integrating alongthe traje tories, we nd

f (t, x, v) ≤ f

0 _{(x − tv, v) +}

Z

t

0 ρ(s, x + (s − t)v) ds.

(2.5)

We an boundtherighthandside termbyits

L

∞

x

norm. Integratingwithrespe tto

v

provides

kρ(t, ·)k

L

∞

_(R

2 ₎

≤ 2πkf

0 k

_L

∞

_(Ω)

+ 2π

Z

t

0 kρ(s, ·)k

L

∞

_(R

2 ₎

ds.

We obtain a bound on the

L

∞

norm on

ρ

thanks to Gronwall's inequality and on lude the

proof with (2.5).

Proofof Theorem2.2. Thelo al-in-timeexisten eisobtainedbyaxedpointargument.

Let

t

0 > 0

, the map

F

on

L

1 _{((0, t}

0 ); L

∞

(Ω))

is dened for all

f

by :

F(f)

is a weak solution

of the problem







∂

t

F + v · ∇

x

F =

Z

v

′

∈V

φ(∇

x

G ∗ (v

′

ρ − J))F(v

′

) dv

′

− 2πφ(∇

x

G ∗ (vρ − J))F(v),

F(0, ·, ·) = f

0 ,

where

ρ =

R

V

f (t, x, v) dv

and

J =

R

V

vf (t, x, v) dv

. We willshow that this map denes a on-tra tionon

L

1 _{((0, τ ), L}

∞

(Ω))

for

τ

smallenough. Let

f

1

and

f

2

begiven in

L

1 _{((0, t}

0 ), L

∞

(Ω))

and denoting

F

12 = F(f

1 ) − F(f

2 )

we have

∂

t

F

12 + v · ∇

x

F

12 =

Z

V

φ(∇

x

· G ∗ (vρ

1 − J

1 ))F

12 (v

′

) dv

′

− 2πφ(∇

x

· G ∗ (vρ

1 − J

1 ))F

12 −2πF(f

2 )(φ(∇

x

· G ∗ (vρ

1 − J

1 )) − φ(∇

x

· G ∗ (vρ

2 − J

2 ))

+

Z

V

F(f

2 )(v

′

)(φ(∇

x

· G ∗ (v

′

ρ

1 − J

1 )) − φ(∇

x

· G ∗ (v

′

ρ

2 − J

2 )) dv

′

,

(7)

with thenotations

ρ

i

=

R

V

f

i

(t, x, v) dv

and

J

i

=

R

V

vf

i

(t, x, v) dv

. We an rewritethis identity

inthe following way

∂

t

F

12 + v · ∇

x

F

12 + 2πφ(∇

x

· G ∗ (vρ

1 − J

1 ))F

12 = G,

(2.6) where

G(t, x, v) =

Z

V

φ(∇

x

· G ∗ (v

′

ρ

1 − J

1 ))F

12 (v

′

) dv

′

_{− 2πF(f}

2 )(φ(∇

x

· G ∗ (vρ

1 − J

1 ))−

φ(∇

x

· G ∗ (vρ

2 − J

2 )) +

Z

V

F(f

2 )(v

′

)(φ(∇

x

· G ∗ (v

′

ρ

1 − J

1 )) − φ(∇

x

· G ∗ (v

′

ρ

2 − J

2 )) dv

′

.

(2.7)

Usingthe hara teristi sof the system, we an rewriteequation (2.6) as

d

ds

e2πφ(∇

x

· G ∗ (vρ

1 − J

1 ))(τ, x + v(τ − t))F

12 (s, x + v(s − t), v)

=

e2πφ(∇

x

· G ∗ (vρ

1 − J

1 ))(τ, x + v(τ − t))G(s, x + v(s − t), v).

Atthe initialtime,

F

12 (0, ·, ·) = 0

. Integrating the latterequality between

0

and

t

, we have

F

12 (t, x, v) =

Z

t

0 exp

2π

Z

s

t

φ(∇

x

· G ∗ (vρ

1 − J

1 ))(τ, x + v(τ − t)) dτ

G(s, x+v(s−t), v) ds.

Sin e

φ

(1.4) is bounded from below by

1/4

we dedu e that for all

0 < t < t

0

,

|F

12 (t, x, v)| ≤

Z

t

0 |G(t − s, x − vs, v)| ds.

(2.8)

Moreover, from(2.7) and the assumptionson

φ

(1.4), we dedu e

|G| ≤

Z

V

|F

12 (v

′

)| dv

′

+ kφ

′

k

∞

2πF(f

2 ) +

Z

V

F(f

2 )(v

′

) dv

′

×

× (|∇

x

G ∗ (J

1 − J

2 )| + V |∇

x

G ∗ (ρ

1 − ρ

2 )|) ,

where

V = max

v∈V

kvk

. Noti ingthat

|J

1 − J

2 | ≤ V |ρ

1 − ρ

2 |

, we have moreover

k|∇

x

G ∗ (J

1 − J

2 )(t, ·)| + V |∇

x

G ∗ (ρ

1 − ρ

2 )(t, ·)|k

L

∞

_(R

2 ₎

≤ V k∇

_x

Gk

_L

1 _(R

2 ₎

k(ρ

₁

− ρ

₂

)(t, ·)k

_L

∞

_(R

2 ₎

.

Finally,from (2.8)wededu e the bound

kF

12 (t, ·, ·)k

L

∞

_(Ω)

≤ C

₁

Z

t

0 kF

12 (t − s, ·, ·)k

L

∞

_(Ω)

ds+

C

2 Z

t

0 kF(f

2 )(t − s, ·, ·)k

L

∞

_(Ω)

k∇

_x

Gk

_L

1 _(R

2 ₎

k(ρ

₁

− ρ

₂

)(t − s, ·)k

_L

∞

_(R

2 ₎

ds.

Therefore, using a Gronwall Lemma, we on lude that for

τ > 0

small enough,

F

denes a

ontra tion on

L

1 _{((0, τ ), L}

∞

(Ω))

. It allows to onstru t a unique solution as the xed point

of the map

F

on the interval

(0, τ )

. Using the a priori estimates established in Lemma 2.3,

we an extend this solution on

(0, t

0 )

for all

t

0 > 0

and we have a bound on this solution in

L

∞

(8)

For the proof of the se ond point of Theorem 2.2, let us assume that

f

0 _{∈ W}

2,∞

_(Ω)

and

that

t

0 > 0

is xed. By dierentiating with respe t to

x

1

the kineti equation (2.3) satised by

f

, we obtain

∂

t

∂

x

1 f + v · ∇

x

∂

x

1 f + 2πφ(∇

x

· G ∗ (vρ − J))∂

x

1 f =

R

V

φ(∇

x

· G ∗ (v

′

ρ − J))∂

x

1 f (v

′

) dv

′

+

R

V

∇

x

G ∗ (v

′

∂

x

1 ρ − ∂

x

1 J)φ

′

(∇

x

G ∗ (v

′

ρ − J))f(v

′

) dv

′

−2π∇

x

G ∗ (v∂

x

1 ρ − ∂

x

1 J)φ

′

(∇

x

G ∗ (vρ − J))f(v).

The righthand side isbounded by

Z

V

|∂

x

1 f (v

′

)| dv

′

+ 8πV kφ

′

k

L

∞

kfk

_L

∞

_(Ω)

k∇

_x

Gk

_L

1 _(Ω)

Z

V

|∂

x

1 f (v

′

) dv

′

|.

Integrating alongthe hara teristi sandpro eedingasabove,weobtainthatforall

0 ≤ t ≤ t

0

,

|∂

x

1 f (t, x, v)| ≤ C

1 |∂

x

1 f

0 (x − tv, v)| + C

2 Z

t

0 Z

V

|∂

x

1 f (t − s, x − sv, v)| ds,

where

C

1

and

C

2

stand for nonnegative onstants depending only on

t

0

and on the data. By

the same token as in proof of Lemma 2.3 using Gronwall's inequality, we obtain a bound on

∂

x

1 f

in

L

∞

((0, t

0 ); L

∞

(Ω))

. Dierentiating (2.3) with respe t to

x

2

and

v

, we dedu e by the same token that for all

t ∈ (0, t

0 )

, we have

f (t, ·, ·) ∈ W

1,∞

_(Ω)

. With a similar argument we

dedu e after straightforward al ulationsthat

f

isbounded in

L

∞

((0, t

0 ); W

2,2

(Ω))

. Moreover from (2.3) we have an expression of

∂

t

f

with respe t to

f

,

ρ

,

J

,

∇

x

G

and

∇

x

f

allowing to obtain aboundon

∂

t

f

in

L

∞

((0, t

0 ); L

∞

(Ω))

.

Remark 2.4 Forthesakeof simpli ity,thisexisten eresulthasbeenestablishedinthedomain

ω = R

2

. However, this existen e and uniqueness result is still available in a bounded domain

ω ⊂ R

2

providedthattheboundary onditionsallows tousetheellipti regularityfortheellipti

equationsatised by

S

. For numeri al analysis,we will onstraint the domain

ω

to be bounded and make use of this existen e result in this framework.

3 Numeri al approa h

Inthisse tion,wepresentthenumeri alapproa hforsolving(1.5). The omputationaldomain

isdenedby

(x, v) ∈ Ω = ω×S

V

where

ω

isare tangulardomainof

R

2

,

ω = [0, L

x

]×[0, ℓ

y

]

,and

S

_V

is thesphere

S

V

= {v ∈ R

2 _,

su hthat

kvk

2 = V }

, foragiven onstant velo ity

V > 0

. We

redu e this 4-dimensional problem to a 3-dimensional problem by onsidering the ylindri al

oordinate

θ

.

At the boundary of the domain, we assume to have spe ular ree tion at the boundaries

(9)

on entration

S

we set Neumann onditions onthe entire boundary :

f (x, 0, θ) = f (x, 0, 2π − θ),

for

θ ∈ [0, π], x ∈ [0, L

x

],

f (x, ℓ

y

, θ) = f (x, ℓ

y

, 2π − θ),

for

θ ∈ [π, 2π], x ∈ [0, L

x

],

f (0, y, θ) = f (0, y, π − θ),

for

θ ∈ [0, π/2] ∪ [3π/2, 2π], y ∈ [0, ℓ

y

],

f (L

x

, y, θ) = f (L

x

, y, π − θ),

for

θ ∈ [π/2, 3π/2], y ∈ [0, ℓ

y

],

∂

x

S(0, y) = ∂

x

S(L

x

, y) = 0,

for

y ∈ [0, ℓ

y

],

∂

y

S(x, 0) = ∂

y

S(x, ℓ

y

) = 0,

for

x ∈ [0, L

x

].

(3.1)

Obviously, inthe

θ

dire tionf is

2π

-periodi . Weintrodu ethenodes

(x

i

= i h

x

)

i=0,··· ,N

x

−

1

,

(y

j

= j h

y

)

j=0,··· ,N

y

−

1

and

(θ

k

= k h

θ

)

θ=0,··· ,N

θ

−

1

where

h

x

= L

x

/(N

x

− 1)

,

h

y

= ℓ

y

/(N

y

− 1)

and

h

θ

= 2π/N

θ

. We denote

x

i

= (x

i

1 , y

i

2 )

with

i = i

1 + i

2 N

x

and wemeshthe domain

ω

withre tangulartrianglesusingthenodes

x

i

. There-forethetriangulationisregularandallniteelementsareaneequivalenttoasinglereferen e

element. We denote the time step

∆t

and set

t

n

_{= n ∆t}

for

n = 0, . . . , N

t

.

3.1 Semi-Lagrangian methods

In this se tion we des ribe the semi-Lagrangian method used for the numeri al resolution in

ω ⊂ R

2

of the kineti equation:

∂

t

f + v · ∇

x

f =

Z

S

_V

φ(∂

t

S + v

′

· ∇

x

S)f (v

′

) dv

′

− 2πφ(∂

t

S + v · ∇

x

S)f (v),

(3.2)

ompleted with the initial datum (1.2). We assume in this se tion that the hemoattra tant

on entration is known and we will therefore onsider the turningkernel as a fun tion of

t

,

x

and

v

:

T (t, x, v) = φ(∂

t

S(t, x) + v · ∇

x

S(t, x))

. Its numeri al approximation will be denoted

T

h

.

Semi-Lagrangian methods onsist in al ulating the distribution fun tion at time

t

n+1

₌

t

n

_{+ ∆t}

thanks to the one whi h has been obtained at the time

t

n

by using the onservation

relationalongthe hara teristi s urves. Werstdenethe hara teristi s

(X, Θ)

ofthesystem for all

0 ≤ s ≤ t

by :











d

ds

X(s; x, θ, t) = v

Θ

,

with

v

Θ

= (V cos Θ, V sin Θ) ;

X(t; x, θ, t) = x,

d

ds

Θ(s; x, θ, t) = 0 ;

Θ(t; x, θ, t) = θ,

(3.3)

if

X(s; x, θ, t) ∈ (0, L

x

) × (0, ℓ

y

)

. Therefore the velo ity remains onstant, ex ept when the traje torymeetsaboundaryofthedomain: ifthereexistsatime

s > 0

su hthat

X(s; x, θ, t) ∈

(0, L

x

)×{0, ℓ

y

}

thentheangle

Θ(s; x, θ, t)

is hangedinto

2π−Θ(s; x, θ, t)

;ifthereexistsatime

s > 0

su hthat

X(s; x, θ, t) ∈ {0, L

x

}×(0, ℓ

y

)

then

Θ(s; x, θ, t)

issubstitutedby

π−Θ(s; x, θ, t)

.

More pre isely, wehave

X(s; x, θ, t) = x + v

θ

(s − t),

with

v

θ

= (V cos θ, V sin θ),

Θ(s; x, θ, t) = θ,

for all

s ∈ R

+

for whi h the traje tory does not ross the boundaries. By ree tion, if the

(10)

t < t

0 < s

,

X(s; x, θ, t) = x + v

θ

(t

0 − t) + v

2π−θ

(s − t

0 )

and

Θ(s; x, θ, t) = 2π − θ.

Ifthe traje tory rosses the boundary

x = 0

or

x = L

x

attime

t

1

, thenfor

t < t

1 < s

, wehave

X(s; x, θ, t) = x + v

θ

(t

1 − t) + v

π−θ

(s − t

1 )

and

Θ(s; x, θ, t) = π − θ.

Obviously, if the traje tory meets several times the boundaries, we make others ree tions.

Usingthe hara teristi s,we an rewrite the kineti equation (3.2) inthe following way :

d

ds

f (s, X(s; x, θ, t), Θ(s; x, θ, t)) =

Z

2π

0 (T (s, X(s; x, θ, t), θ

′

)f (s, X(s; x, θ, t), θ

′

) dθ

′

−2π T (s, X(s; x, θ, t), Θ(s; x, θ, t)) f(s, X(s; x, θ, t), Θ(s; x, θ, t))).

(3.4)

The semi-Lagrangian method relies on a dis rete approximation of relation (3.4). We

assumetoknow thedistribution fun tionattime

t

n

. Weuse anexpli itintime Eulers heme

to ompute this quantity attime

t

n+1

_{= t}

n

_{+ ∆t}

on ea h point

(x

i

, θ

j

)

of the grid. It leads to the followingsystem :

f

h

(t

n+1

, x

i

, θ

j

) = f

h

(t

n+1

, X(t

n

; x

i

, θ

j

, t

n+1

), Θ(t

n

; x

i

, θ

j

, t

n+1

))

+∆t

Z

2π

0 (T

h

(t

n

, X(t

n

; x

i

, θ

j

, t

n+1

), θ

′

)f

h

(X(t

n

; x

i

, θ

j

, t

n+1

), θ

′

) dθ

′

−2π∆t T

h

(t

n

, X(t

n

; x

i

, θ

j

, t

n+1

), Θ(t

n

; x

i

, θ

j

, t

n+1

)) f

h

(X(t

n

; x

i

, θ

j

, t

n+1

), Θ(t

n

; x

i

, θ

j

, t

n+1

))),

(3.5)

where

f

h

and

T

h

stand forapproximationsof

f

and

T

. Forthe sakeof larity,we willuse from

now on the notations

X

n

i,j

(s)

instead of

X(s; x

i

, θ

j

, t

n

₎

and

Θ

n

i,j

(s)

instead of

Θ(s; x

i

, θ

j

, t

n

₎₎

.

The dis retizationrelies on two main steps :

1. Findthepoint

(X(t

n

_{; x}

i

, θ

j

, t

n

+∆t), Θ(t

n

; x

i

, θ

j

, t

n

+∆t)) := (X

i,j

n+1

(t

n

), Θ

n+1

i,j

(t

n

))

. Start-ingfrom

(x

i

, θ

j

)

,itsu es tofollowba kward the hara teristi s urvesduring the time step

∆t

. Tothis end, wehave tosolve (3.3). Sin e the resolution ofthe se ond equation in(3.3) is lear, this step is simple (ina generalframeworksee e.g. [39, 2,15℄)

X

_i,j

n+1

(t

n

) = x

i

− ∆t v

Θ

n+1

_i,j

,

Θ

n+1

i,j

(t

n

) = θ

j

.

Obviously, if the traje tory rea hes a boundary of the domain, then we use spe ular

ree tion :

X

n+1

i,j

(t

n

)

is repla ed by its symmetri with respe t to this boundary and

Θ

n+1

_i,j

(t

n

₎

by

2π − Θ

n+1

i,j

(t

n

)

at the verti al boundary

y = 0

or

y = ℓ

y

, orby

π − Θ

n+1

i,j

(t

n

)

atthe boundary

x = 0

or

x = L

x

.

Therefore the omputationof the footof the hara teristi sis exa t.

2. Sin e the fun tions

f

and

T

at time

t

n

are only known on the nodes of the mesh, we

interpolate these fun tions at the points

(X

n+1

i,j

(t

n

), Θ

n+1

i,j

(t

n

))

. A tually, sin e we have

Θ

n+1

_i,j

(t

n

_{) = θ}

j

or

π − θ

j

or

2π − θ

j

, by taking

N

θ

odd we only have a 2D interpolation

method to implement. In fa t, we have that

2π − θ

j

= θ

N

θ

−

j

and

π − θ

j

= θ

N

θ

/2−j

if

(11)

an use a linear interpolation : we dene the linear interpolation operator

Π

onto the

spa e of Lagrangian polynomialsof degree lesser than orequal to1 by :

Πf (x, y) =

N

x

N

y

−

1 X

i=0

f (x

i

)l

i

(x, y),

with

l

i

∈ P

1 _(ω)

su h that

l

i

(x

j

) = δ

ij

,

(3.6) where

P

1 _(ω)

is the set of pie ewise polynomialswhose restri tion onea htriangleof the

meshholdsinpolynomialsfun tionof degreelesserthanorequalto1. Theinterpolation

error estimations inSobolev spa esgive that for all

f ∈ W

2,2

_(ω)

(see [13℄ Theorem 16.2

p. 128)

kf − Πfk

L

2 _(ω)

≤ C max{h

2 x

, h

2 y

} kfk

W

2,2

_(ω)

.

(3.7)

Moreover, this linear interpolationmethodpreserves the

L

∞

bound :

kΠfk

L

∞

_(ω)

≤ kfk

_L

∞

_(ω)

.

(3.8)

For more pre ision on solutions, a Hermite spline interpolation method whi h is a well

established high order interpolation method an be used. We refer to [39, 15℄ for more

details about this interpolation step. However, spurious os illations (e.g. Runge

phe-nomena) an appear with high order interpolationmethods.

Finally,usingadis retizationofthe integralin(3.5),theapproximationofthe distribution

fun tion attime

t

n+1

is obtained by

f

h

(t

n+1

, x

i

, θ

j

) = Πf

h

(t

n

, X

i,j

n+1

(t

n

), Θ

i,j

n+1

(t

n

)) − 2π∆t Π(T

h

f

h

)(t

n

, X

i,j

n+1

(t

n

), Θ

n+1

i,j

(t

n

))+

+∆t

N

X

θ

−

1 k=0

h

θ

Π(T

h

f

h

)(t

n

, X

i,j

n+1

(t

n

), θ

k

).

(3.9)

3.2 Numeri al resolution of the oupled system

We re all the following notations :

X

n+1

i,j

(t

n

)

instead of

X(t

n

_{; x}

i

, θ

j

, t

n+1

)

,

Θ

n+1

i,j

(t

n

)

instead of

Θ(t

n

_{; x}

i

, θ

j

, t

n+1

))

and

Π

istheinterpolationoperator. Thenumeri alresolutionofthe oupled

problem (1.5) is then ta kled in the following way. We assume that the approximation of

the distribution fun tion

f

h

and of the hemoattra tant on entration

S

h

are known at time

t

n

at ea h nodes of the mesh. We des ribe the pro ess to ompute

f

h

and

S

h

at time

t

n+1

.

As des ribed previously, the distribution fun tion at time

t

n+1

is approximated thanks to the

relation:

f

h

(t

n+1

, x

i

, θ

j

) = Πf

h

(t

n

, X

i,j

n+1

(t

n

), Θ

i,j

n+1

(t

n

)) − 2π∆t Π(T

h

f

h

)(t

n

, X

i,j

n+1

(t

n

), Θ

n+1

i,j

(t

n

))+

+∆t

N

X

θ

−

1 k=0

h

θ

Π(T

h

f

h

)(t

n

, X

i,j

n+1

(t

n

), θ

k

),

(3.10)

where the turning kernel

T

h

is omputed by adis retization of (1.3). As we have seen for the

(12)

to get an estimation of the term involving temporal derivativein the denition of

T

. In fa t, the quantity

∂

t

S

satises

−∆∂

t

S + ∂

t

S = −∇ · J,

(3.11)

ompletedwith Neumannboundary onditionsdedu ed from(3.1). We denethe

approxima-tion of the urrent by

J

h

(t

n

, x

i

) =

N

X

θ

−

1 k=0

h

θ

v

_θ

k

f

h

(t

n

_{, x}

i

, θ

k

).

(3.12)

Then the approximation of

∂

t

S

, denoted

S

th

, is omputed by solving (3.11) with

J

h

using

onforming

P

1

nite elements :

∀ V

h

∈ X

h

,

Z

ω

(∇S

th

· ∇V

h

+ S

th

V

h

) dx =

Z

ω

J

h

· ∇V

h

dx,

(3.13)

where

X

h

isthe set of fun tion of

C

0 _(ω)

whose restri tion toea h triangleon the mesh holds

in

P

1

. Wedene the approximation

T

h

of the turningkernel by

T

h

(t

n

, x

i

, θ

j

) = φ

S

th

(t, x) +

1 ∆t

(S

h

(t

n

_{, x}

i

) − ΠS

h

(t

n

, X

i,j

n

(t

n−1

))

.

(3.14)

On e

f

h

is known at time

t

n+1

, the hemoattra tant on entration is updated by solving

the followingellipti equation with onforming

P

1

nite elements:

−∆S

h

(t

n+1

, x

i

) + S

h

(t

n+1

, x

i

) = ρ

h

(t

n+1

, x

i

) :=

N

X

θ

−

1 k=0

f

h

(t

n+1

, x

i

, θ

k

) h

θ

.

(3.15)

This system is ompleted with boundary onditions (3.1).

From basi error estimates on ellipti problem,wehave (see [13℄ Theorem 18.1 p. 138) :

Proposition 3.1 Let

u

be a solution of the variational problem

a(u, v) = l(v)

where

a

is

bilinear ontinuous symmetri oer ive on

H

1 _(ω)

and

l

is linear ontinuous on

H

1 _(ω)

. Then,

if

u

h

is the dis rete approximation omputed by onforming

P

1

nite elements, there exists a

nonnegative onstant

C

su h that

ku − u

h

k

H

1 _(ω)

≤ C max{h

_x

, h

_y

}kuk

_H

2 _(ω)

.

3.3 Convergen e analysis

We are interested in this se tion in the onvergen e of the s heme (3.10)(3.15) towards

so-lutions of model (1.5). Convergen e analysis of semi-Lagrangianmethodin the framework of

Vlasov-Poissonsystem havebeenobtained in[3℄;thoseresultsarepresented in

L

∞

. Duetothe

la k ofregularity inour ase ausedby the presen e ofthe turningoperator,we presentinthe

following theorem a onvergen e result in

L

2

. The main result of this se tion is presented in

Theorem3.4 belowunderthefollowingadditionalassumptionallowingtosimplifyallintegrals

(13)

allparti lesareinitially onnedinthe enter ofthedevi eandthatthetime

t

0

issmallenough to avoid parti les to meet the boundary of the domain. Thus all traje tories are straight lines

and

X(s; x, θ, t) = x + v

θ

(s − t);

Θ(s; x, θ, t) = θ,

for

0 ≤ s ≤ t.

Moreover, sin e

f

h

vanishes near the boundary of the domain

ω

, we have

Z

ω

f

h

(t, X(s; x, θ, t), θ) dx =

Z

ω

f

h

(t, x, θ) dx,

for

0 ≤ s ≤ t

small enough.

From now on, we x

t

0 > 0

small enough and assume

f

0 _{∈ W}

2,2

_{∩ W}

1,∞

_(Ω)

is hosen

su h that Assumption 3.2holds. Therefore, in this onvergen e analysis ells donot 'see' the

boundary. Thes heme (3.10)(3.15)allowstodenetheapproximatedfun tion

f

h

onlyonthe

nodes on the mesh. We extend this denition on all

(x, θ) ∈ ω × [0, 2π]

thanks to the linear

interpolation operator

Π

(3.6). We rst establish the positivity and a priori estimates on the

dis reteapproximation

f

h

.

Lemma 3.3 Let

t

0 > 0

and assume that

f

0 _{∈ W}

2,2

_{∩ W}

1,∞

_{(ω × [0, 2π])}

is a nonnegative

fun tion su h that Assumption 3.2 holds. If

∆t ≤ 1/(2π)

, then the s heme dened by (3.10)

(3.15) givesa nonnegative approximation

f

h

of thedistribution fun tion. Moreover

f

h

satises the followingestimate

∀ t ∈ (0, t

0 ),

kΠf

h

(t, ·, ·)k

L

∞

_{(ω×[0,2π])}

≤ e

2πt

0 kf

0 k

_L

∞

_{(ω×[0,2π])}

,

kΠf

_h

(t, ·, ·)k

_H

1 _{(ω×[0,2π])}

≤ C

₀

,

where

C

0

is a nonnegative onstant depending on

t

0

and

kf

0 _k

H

1 _{(ω×[0,2π])}

.

Proof. From the denition of (3.6) and the assumptions on

φ

(1.4) we have that for all

nonnegative fun tion

f

1

4 Πf ≤ Π(T f) ≤ Πf.

Therefore assuming

f (t

n

_{, ·, ·)}

nonnegative, we dedu e from(3.10) that

f

h

(t

n+1

, x

i

, θ

j

) ≥ (1 − 2π∆t) Πf

h

(t

n

, x

i

− v

θ

j

∆t, θ

j

) ≥ 0.

Moreover, from(3.10) and (3.8) we have

kΠf

h

(t

n+1

, ·, ·)k

L

∞

_{(ω×[0,2π])}

≤ (1 + 2π∆t) kΠf

_h

(t

n

, ·, ·)k

_L

∞

_{(ω×[0,2π])}

.

Applying adis reteGronwallinequalityallowsto on lude theproof ofthe

L

∞

bound.

Dier-entiating (3.10)with respe t to

x

1

gives

∂

x

1 f

h

(t

n+1

_{, x}

i

, θ

j

) = ∂

x

1 Πf

h

(t

n

_{, x}

i

− v

θ

j

∆t, θ

j

)

−2π∆t ∂

x

1 Π(T

h

f

h

)(t

n

_{, x}

i

− v

θ

j

∆t, θ

j

) + ∆t

P

N

θ

−

1 k=0

h

θ

∂

x

1 Π(T

h

f

h

)(t

n

_{, x}

i

− v

θ

j

∆t, θ

k

).

(3.16)

By linearity of the interpolationoperator,we have that

(14)

From the denition of

T

h

(3.14),we dedu e that

k∂

x

1 (ΠT

h

)(t

n

)k

L

2 _(ω)

≤ k∂

_x

₁

S

_th

(t

n

)k

_L

2 _(ω)

+

1 ∆t

k∂

x

1 S

h

(t

n

, ·) − ∂

x

1 ΠS

h

(t

n

_{, X(t}

n−1

; ·, θ, t

n

))k

L

2 _(ω)

.

From the ellipti regularity onthe equation(3.13) and the bound of

f

h

in

L

∞

(ω × [0, 2π])

, we

havethat

S

th

isbounded in

H

1 _(ω)

. The ellipti regularity on (3.15)gives

kS

h

(t

n

, ·) − ΠS

h

(t

n

, X(t

n−1

; ·, θ, t

n−1

))k

H

1 _(ω)

≤ kρ

_h

(t

n

, ·) − Πρ

_h

(t

n

, X(t

n−1

; ·, θ, t

n

))k

_L

2 _(ω)

.

Usinga Taylor expansion gives

ρ

h

(t

n

, x) − Πρ

h

(t

n

, x − v

θ

∆t) =

Z

0 −

_∆t

v

_θ

· ∇Πρ

_h

(t, x + v

_θ

s) ds.

From Assumption 3.2wededu e that

kρ

h

(t

n

, x) − Πρ

h

(t

n

, x − v

θ

∆t)k

L

2 _(ω)

≤ ∆tV k∇(Πρ

_h

)(t

n

, ·)k

_L

2 _(ω)

.

Thus,

k∂

x

1 (ΠT

h

)k

L

2 (ω×[0,2π])

≤ C(1 + k∇(Πρ

h

)k

L

2 (ω×[0,2π])

).

Doingthe same with

x

2

,(3.16) leads to the following estimate

k∇Πf

h

(t

n+1

)k

L

2 _{(ω×[0,2π])}

= k∇Πf

_h

(t

n

)k

_L

2 _{(ω×[0,2π])}

+ C∆t(1 + k∇Πf

_h

(t

n

)k

_L

2 _{(ω×[0,2π])}

).

A Gronwallinequality allows to on lude the proof.

Theorem 3.4 Let

t

0 > 0

and assume that

f

0 _{∈ W}

2,2

_{∩ W}

1,∞

_{(ω × [0, 2π])}

is a nonnegative

fun tion su h that Assumption 3.2 holds. Let

f

be the global weak solution of (1.5) on

(0, t

0 )

and

f

h

be its approximation omputed atthe nodes of themesh thanksto the algorithm(3.10)

(3.15) where

Π

is dened in (3.6). Then there exists a nonnegative onstant

C

depending on

t

0

,

f

0

and the data su h that

kf − Πf

h

k

L

∞

_(0,t

₀

_;L

2 _{(ω×[0,2π]))}

≤ C(∆t + h

2 +

h

∆t

+ h),

where

h = max{h

x

, h

y

, h

θ

}

.

Proof. From Theorem 2.2wehave

f (t, ·, ·) ∈ W

1,∞

_{∩ W}

2,2

_{(ω × [0, 2π])}

We denethe global

error attime

t

n+1

by

ǫ

n+1

_{= kf(t}

n+1

_{, x, θ) − Πf}

h

(t

n+1

, x, θ)k

L

2 _{(ω×[0,2π])}

.

(3.17) A rst remark isthat sin e hara teristi s are straight lines the numeri al omputationof the

hara teristi s

(X, Θ)

isexa t. From (1.5), wededu e that for

0 ≤ s ≤ t

d

ds

f (s, x − v

θ

(t − s), θ) =

Z

2π

0 T (s, x − v

θ

(t − s), θ

′

)f (s, x − v

θ

(t − s), θ

′

) dθ

′

−2π T (s, x − v

θ

(t − s), Θ

n

(s))f (s, x − v

θ

(t − s), θ),

(15)

T (t, x, θ) = φ(∂

t

S + v

θ

· ∇

x

S),

and

S

being the solution of the ellipti problem

−∆S + S =

Z

2π

0 f (t, x, θ) dθ.

We dedu e from the regularity of the fun tion

f

proved in Theorem 2.2 that the fun tion

s 7→ f(s, x − v

θ

(t − s), θ)

is bounded in

W

2,2

_{(0, t}

0 )

. Hen e a Taylor expansion gives, under

Assumption 3.2:

f (t

n+1

, x, θ) = f (t

n

_{, x − v}

θ

∆t, θ) + ∆t

Z

2π

0 T (t

n

_{, x − v}

θ

∆t, θ

′

)f (t

n

, x − v

θ

∆t, θ

′

) dθ

′

−2π∆t T (t

n

, x − v

θ

∆t, θ)f (t

n

, x − v

θ

∆t, θ) + O

L

2 (∆t

2 ),

where

O

L

2 (∆t)

means that there exists

C > 0

su h that

kO

L

2 (∆t

2 )k

_L

2 ≤ C∆t

2

. Therefore, using the denition (3.10), we rewritethe dieren e

f (t

n+1

_{, x, θ) − Πf}

h

(t

n+1

, x, θ)

as

f (t

n+1

_{, x, θ) − Πf}

h

(t

n+1

, x, θ) = f (t

n

, x − v

θ

∆t, θ) − Πf

h

(t

n

, x − v

θ

∆t, θ)

+∆t

Z

2π

0 (T f )(t

n

_{, x − v}

θ

∆t, θ

′

) dθ

′

₋

N

X

θ

−

1 k=0

h

θ

Π(T

h

f

h

)(t

n

, x − v

θ

∆t, θ

k

)

!

−2π∆t ((T f)(t

n

, x − v

θ

∆t, θ) − Π(T

h

f

h

)(t

n

, x − v

θ

∆t, θ)) + O

L

2 (∆t

2 ).

To evaluate the global error

ǫ

n+1

, we de ompose

f (t

n+1

_{, x, θ) − f}

h

(t

n+1

, x, θ)

as

f (t

n+1

_{, x, θ) − Πf}

h

(t

n+1

, x, θ) = f (t

n

, x − v

θ

∆t, θ) − Πf

h

(t

n

, x − v

θ

∆t, θ)

+∆t((1 − Π)I

1 + I

2 + I

3 ) + O

L

2 (∆t

2 ),

(3.18) where

I

1 =

N

X

θ

−

1 k=0

h

θ

(T f )(t

n

, x − v

θ

∆t, θ

k

) − 2π(T f)(t

n

, x − v

θ

∆t, θ),

(3.19)

I

2 =

N

X

θ

−

1 k=0

h

θ

(T f − T

h

f

h

)(t

n

, x − v

θ

∆t, θ

k

) − 2π(T f − T

h

f

h

)(t

n

, x − v

θ

∆t, θ),

(3.20)

I

3 =

Z

2π

0 (T f )(t

n

_{, x − v}

θ

∆t, θ

′

) dθ

′

−

N

X

θ

−

1 k=0

h

θ

(T f )(t

n

, x − v

θ

∆t, θ

k

).

(3.21) Takingthe

L

2

normof (3.18) implies with Assumption 3.2

ǫ

n+1

_{≤ ǫ}

n

_{+ ∆t(k(1 − Π)I}

1 k

L

2 _{(ω×[0,2π])}

+ kI

₂

k

_L

2 _{(ω×[0,2π])}

+ kI

₃

k

_L

2 _{(ω×[0,2π])}

) + C∆t

2 .

(3.22)

Wewillestimate ea h term separately thanks tothe following Lemmata.

Lemma 3.5 Let

(f, S)

being solution of (1.5) for

f

0 _{∈ W}

1,∞

_{∩ W}

2,2

_{(ω × [0, 2π])}

. If

I

1

is dened by (3.19), then there exists a nonnegative onstant

C

su h that for all

t ∈ (0, t

0 )

,

k(1 − Π)I

1 k

L

2 _{(ω×[0,2π])}

≤ C max{h

2

(16)

Proof. Let

0 ≤ t ≤ t

0

. From Theorem 2.2, we have that

f (t, ·, ·) ∈ W

1,∞

_{∩ W}

2,2

_{(ω × [0, 2π])}

.

Therefore, by ellipti regularity, we dedu e that

∇S(t, ·) ∈ W

3,2

_(ω)

. Moreover, (3.11) implies

−∆∂

t

S(t, ·) + ∂

t

S(t, ·) = −∇ · J(t, ·) ∈ W

1,2

(ω).

Then theellipti regularityfurnishes

∂

t

S(t, ·) ∈ W

3,2

_(ω)

and sin ewith our denition on

φ

we

get

φ

′

∈ C

∞

c

(R)

,wededu e that

T (t, ·, ·) = φ(∂

t

S + v

θ

· ∇S) ∈ W

3,2

_{(ω × [0, 2π])}

. Then

∂

_x

2 _i

_x

_j

(T f ) = (∂

x

i

x

j

T )f + ∂

x

i

T ∂

x

j

f + ∂

x

j

T ∂

x

i

f + T (∂

x

i

x

j

f )

∈ L

2 (ω × [0, 2π]).

Hen e

(T f )(t, ·, ·) ∈ W

2,2

_{(ω × [0, 2π])}

. Finally, the result of the Lemma is a straightforward

onsequen e of the interpolation error (3.7).

Lemma 3.6 Let

f

and

T

beingdened previously,there existsa nonnegative onstant

C

su h

that for all

0 ≤ t ≤ t

0

and

x

∈ ω

,

Z

2π

0 (T f )(t, x, θ

′

) dθ

′

−

N

X

θ

−

1 k=0

h

θ

(T f )(t, x, θ

k

)

L

2 _(ω)

≤ Ch

2 θ

.

Proof. Let

0 ≤ t ≤ t

0

and

x ∈ ω

. As noti ed in the proof of Lemma 3.5 we have that

(T f )

belongsto

W

2,2

_{(ω ×[0, 2π])}

. Hen etheresultofLemma3.6isa onsequen eofthewell-known

error estimatefor the trapezoidalrule : if

g ∈ W

2,2

_{(0, 2π)}

, there exists

θ ∈ (0, 2π)

su h that

Z

2π

0 g(θ) dθ −

N

X

θ

−

1 k=0

h

θ

g(θ

k

)

= h

2 θ

π

6 ∂

2 _g(θ)

∂θ

2 .

(3.23)

The two previous Lemmata allow us to estimate the terms involving

I

1

and

I

3

in (3.22).

For

I

2

, we need rst toestimate the error

T − T

h

with respe t to

f − f

h

.

Lemma 3.7 Let assume thatAssumption 3.2 holds. Let

T

be denedin (1.3)(1.4) and

T

h

be

its approximation omputed with (3.14). Then, there exists

C > 0

su h that for

n = 1, . . . , N

t

,

we have

k sup

α∈[0,2π]

|T (t

n

_{, ·, α) − ΠT}

h

(t

n

, ·, α)|k

L

2 _(ω)

≤ C(∆t + max{h

_x

, h

_y

} +

max{h

x

, h

y

}

∆t

+ h

2 θ

+

+kf(t

n

, ·, ·) − f

h

(t

n

, ·, ·)k

L

2 _{(ω×[0,2π])}

).

Proof. Let

n ∈ {1, . . . , N

t

}

,

x

∈ ω

and

α ∈ [0, 2π]

. We have from(3.14) that

|T (t

n

, x, α) − ΠT

h

(t

n

, x, α)| = |φ(∂

t

S(t

n

, x) + v

α

· ∇

x

S(t

n

, x)) − ΠT

h

(t

n

, x, α)|

≤ kφ

′

k

L

∞

(|∂

_t

S(t

n

, x) − S

_th

(t

n

, x)| + |v

_α

· ∇

_x

S(t

n

, x) −

1 ∆t

(S

h

(t

n

, x) − ΠS

h

(t

n

, x − v

α

∆t))|),

(3.24)

where

S

th

is dened in (3.13). We will estimateseparately ea h term of the sum of the right

hand side. Letus introdu e

S

e

t

aweaksolution of

(17)

ompleted with Neumann boundary onditions, where

J

h

is dened in (3.12) at the nodes of

the mesh and extended on

ω

thanks tothe linear interpolationoperator

Π

. From Proposition

3.1, we dedu e that

k e

S

t

− S

th

k

L

2 _(ω)

≤ C max{h

_x

, h

_y

}k e

S

_t

k

_W

2,2

_(ω)

≤ C max{h

_x

, h

_y

}k∇ΠJ

_h

k

_L

2 _(ω)

,

where the ellipti regularity on equation (3.25) is used. And Lemma 3.3 allows to bound the

term

k∇ΠJ

h

k

L

2 (ω)

. Moreover, from (3.11)and (3.25), we dedu e that

k∂

t

S − e

S

t

k

L

2 _(ω)

≤ CkJ − ΠJ

_h

k

_L

2 _(ω)

≤ C(h

2 θ

+ kf − Πf

h

k

L

2 _{(ω×[0,2π])}

),

where wehave used the errorestimate given by the trapezoidalrule (3.23)to estimate

J − J

h

for

f ∈ W

2,2

_{(ω × [0, 2π])}

. We on lude then that

k∂

t

S−S

th

k

L

2 _(ω)

≤ k∂

_t

S− e

S

_t

k

_L

2 _(ω)

+k e

S

_t

−S

_th

k

_L

2 _(ω)

≤ C(h

2 θ

+kf −Πf

h

k

L

2 +max{h

_x

, h

_y

}).

(3.26) Weintrodu e

S

e

weak solutionof the ellipti problem

−∆ e

S + e

S = Πρ

h

,

(3.27)

ompleted with Neumannboundary onditions. Therefore, wehave that

k sup

α∈[0,2π]

|v

α

· ∇

x

S − v

α

· ∇

x

e

S|k

L

2 _(ω)

≤ V kS − e

Sk

_W

1,2

_(ω)

≤ Ckρ − Πρ

_h

k

_L

2 _(ω)

≤ C(h

2 θ

+ kf − Πf

h

k

L

2 _{(ω×[0,2π])}

),

(3.28)

whereweusetheellipti regularityforequation(3.27)andtheerrorestimateforthetrapezoidal