Numerical Study on Landau Damping

Tie Zhou1

School of Mathematical Sciences

Peking University, Beijing 100871, China

Yan Guo2

Lefschetz Center for Dynamical Systems

Division of Applied Mathematics

Brown University, Providence, RI 02912, USA

and

Chi-Wang Shu3

Division of Applied Mathematics

Brown University, Providence, RI 02912, USA

Abstract

We present a numerical study of the so-called Landau damping phenomenon in

the Vlasov theory for spatially periodic plasmas in a nonlinear setting. It shows that

the electric �eld does decay exponentially to zero as time goes to in�nity with gen-

eral analytical initial data which are close to a Maxwellian. The time decay depends

on the length of the period as well as the closeness between the initial data and the

Maxwellian. Similar pattern is observed if the Maxwellian is replaced by other alge-

braically decaying homogeneous equilibria with a single maximum, or even by some

homogeneous equilibria with small double-humps. The numerical method used is a

high order accurate hybrid spectral and �nite di�erence scheme which is carefully cali-

brated with the well-known decay theory for the corresponding linear case, to guarantee

a reliable resolution free of numerical artifacts for a long time integration.

1E-mail: zht@sxx0.math.pku.edu.cn. Research supported by NSF Grant INT-9601084 and ARO Grant

DAAG55-97-1-0318 while in residence at the Division of Applied Mathematics, Brown University.2E-mail: guoy@cfm.brown.edu. Research supported in part by NSF Grant DMS-9971306 and an Sloan

Fellowship.3E-mail: shu@cfm.brown.edu. Research supported by ARO Grant DAAG55-97-1-0318 and DAAD19-00-

1-0405, NSF grants DMS-9804985 and ECS-9906606, and AFOSR Grant F49620-99-1-0077.

1

1 Introduction and Notations

The simplest model to describe a hot dilute electrons moving in a �xed ion back-

ground is the following one dimensional Vlasov-Poisson system. Let F (t; x; v) denote

the density of electrons in a collisionless plasma, E denote its electric �eld, then the

Vlasov-Poisson system is

Ft + vFx +E(t; x)Fv = 0; (1.1)

Ex =R1�1 F (t; x; v)dv � 1: (1.2)

Here we have normalized all physical constants to be one. A simple steady-state solu-

tion (equilibrium) is the Maxwellian distribution

F (t; x; v) =1

p2�

exp(�v2

2) � m(v); (1.3)

and E(t; x) � 0. We may reformulate the Vlasov-Poisson system (1.1) and (1.2) as

equations for the perturbations f and e of the equilibrium (1.3) so that

F = m(v) + f; E = 0 + e:

We deduce that they satisfy

ft + vfx + e(t; x)fv = �e(t; x)m0(v); (1.4)

ex =

Z 1

�1f(t; x; v)dv; (1.5)

with its corresponding linearized equation (by dropping the term \e(t; x)fv"):

ft + vfx = �e(t; x)m0(v); (1.6)

ex =

Z 1

�1f(t; x; v)dv: (1.7)

In 1946, L. D. Landau discovered that waves in a plasma should be damped even in

the absence of collisions. More precisely, he has shown that the macroscopic (collective)

electric �eld e(t; x) to the linearized Vlasov-Poisson system (1.6) and (1.7) decays

exponentially to zero as time tends to in�nity. The e�ect of the Landau damping, as

it was subsequently called, plays a fundamental role in the study of the plasma physics

ever since, and it is highly signi�cant from both physical as well as mathematical points

of view: Although the Vlasov-Poisson system is time reversible on the particle level,

their collective e�ect is time irreversible.

Unfortunately, strictly speaking, the Landau damping is still a linear phenomenon

up to now. Despite many signi�cant theoretical, numerical, and experimental work

along this direction [15], no rigorous justi�cation of the Landau-damping has been

given in a nonlinear, dynamical sense. As a matter of fact, recently, there have been

quite some renewed interests [3] as well as controversy about the Landau damping. In

[7] and [8], it was proven that even in the linear case, there is no decay at all if the

physical space is the whole line. Even in a �xed spatial period, it is of fundamental

importance to determine if the nonlinear e�ect could take over eventually and destroy

the decay property of the electric �eld. It is the purpose of this article to use a highly

2

accurate and carefully calibrated numerical scheme to simulate the Vlasov-Poisson

system (1.4) and (1.5) over a long time, and to study the time-decay of its electric �eld

je(t; �)j1 = jE(t; �)j1

in a fully nonlinear and dynamical setting.

In order to extensively study the Landau damping phenomenon, we mainly consider

the following type of initial 2a�-periodic perturbation

f(0; x; v) = � cos(x=a) exp(sin(x=a))1

p2�

exp(�v2

2): (1.8)

We �rst reduce this problem to a 2�-periodic problem. By a direct computation,

we notice that if [F (t; x; v); E(t; x; v)] is one of the solutions of (1.1) and (1.2) with

initial condition (1.8), so is the rescaled pair

F (t; x; v) = aF (t; ax; av); E(t; x) =1

aE(t; ax): (1.9)

Notice that the pair of [F (t; x; v); E(t; x)] satis�es

F (0; x; v) = [1 + � cos x exp(sinx)]

�a

p2�

exp(�(av)2

2)

�; (1.10)

or f(0; x; v) = � cos x exp(sinx)h

ap2�

exp(� (av)2

2)i, and

jE(t; �)j1 =1

ajE(t; �)j1: (1.11)

This implies that we can recover all the decay information of the solution to the 2a�-

periodic problem (1.8) by studying the standard 2� periodic problem (1.10) around a

rescaled Maxwellian (depending on a)

m(av) =a

p2�

exp(�(av)2

2): (1.12)

Throughout this article, we shall only compute 2�-periodic problems with this

scaled Maxwellian. Our numerical evidence shows that Landau damping does exist

for the nonlinear Vlasov-Poisson system (1.1), (1.2) with analytical initial data such as

(1.8) which is close to a Maxwellianm(av). The decay rate depends on the parameter a

(or equivalently, the length of the period): The larger a is, the slower is the decay rate.

For such cases, our numerical simulations indicate very similar results between the

nonlinear problem and the linear problem, until machine zero is reached. On the other

hand, no exponential decay is observed if the initial data is far from m(av). We also

observe that the Landau damping phenomenon is robust: the same conclusions hold

if one replaces the Maxwellian by other algebraically decaying equilibria with a single

maximum in v, or even by some equilibria close to m(av) with small double-humps!

This implies that those double-humped equilibria may be dynamically stable. It is

also well-known that many large double-humped equilibria which satisfy the Penrose

instability criterion are indeed unstable [6], and there are arbitrarily small BGK waves

close to them [5]. This implies no Landau damping is possible in this case.

3

In the literature there have been developments of numerical methods to solve the

Vlasov-Poisson system (1.1), (1.2) and Landau damping has been used as a test case

[2, 4, 10]. These methods split (1.1) into two one dimensional systems, i.e., �rst solve

Ft + vFx = 0 (1.13)

for half a time step, and then solve

Ft +E(t; x)Fv = 0 (1.14)

for the second half time step. Characteristic based method has been used in each of

the split steps. Unfortunately, such splitting can be at most second order accurate.

About Landau damping, exponentially decay in the linear cases is observed in [2] and

some no-decaying phenomenon in the nonlinear cases is observed in [2, 4, 10].

The emphasis of this paper is not to develop a new numerical method, rather it

is to study Landau damping in the full nonlinear, dynamic setting, by using a high

order accurate hybrid spectral and �nite di�erence method, to be described in detail

in section 2, which is carefully calibrated with the well-known decay theory for the

corresponding linear case, to guarantee a reliable resolution free of numerical artifacts

for a long time integration. We carefully apply the principle that any computed feature

which disappears after a grid re�nement is very likely to be a numerical artifact rather

than a phenomenon relevant to the solution of the original PDE.

2 A Description of the Numerical Method

To discretize the Vlasov equation (1.4), (1.5), we use a Fourier collocation spectral

method in the x direction, a ninth order upwind-biased �nite di�erence method in the

v direction to obtain a method-of-lines ODE in t, and then discretize this ODE by the

classical fourth order explicit Runge-Kutta method. Several remarks are in order:

1. This method is based on a successful WENO (weighted essentially non-oscillatory,

[9]) algorithm to solve the kinetic equations in semiconductor device simulations

[1]. As the solutions for the Vlasov equation is quite smooth, the weights in the

WENO schemes can be frozen to be the linear weights, resulting in a upwind-

biased �nite di�erence approximation in the v direction. The ninth order method

we use involves the ten grid points xi�5 to xi+4 to compute the derivative fv at

the grid xi, if the coeÆcient e(t; xi) is positive. Otherwise, the upwind-biasing

would be to the right and the ten points used would be from xi�4 to xi+5. The

linear weights can be found in [14].

2. Since the numerical solution is periodic in x and the solution is quite smooth, a

Fourier spectral method is the most natural choice to discretize the x derivative.

Fast Fourier Transform (FFT) can be handily used to make the computation

eÆcient.

3. The un-split method of lines approach coupled with a fourth order Runge-Kutta

method in time, and a small time step required by the CFL stability condition,

guarantees a global high order accuracy in space and time. We have performed

careful calibrations of the numerical method with the well-known decay theory for

the corresponding linear case, to guarantee a reliable resolution free of numerical

4

artifacts for a long time integration. Grid re�nement study has been performed

to make sure that any observed decay or non-decay of the electric �eld is not a

numerical artifact.

We will use a rectangular mesh to represent the x-v phase plane with the compu-

tational domain f(x; v)j0 � x � 2�; jvj � vmaxg. The cut-o� vmax is carefully chosen

and closely monitored to make sure that the numerical solution is well below round-

o� zero there for all t. In fact, it is found out that an erroneous choice of vmax may

lead to spurious numerical artifacts such as an increase in the magnitude of e, which

seems to converge with a mesh re�nement study but would go away when vmax is

enlarged. We use a uniform mesh in both x and v directions and denote the grid

points as (xi; vj). The numerical solutions are denoted by eni and fnij, for i = 1; 2; :::; N

and j = �M;�M + 1; :::;M . Periodic boundary conditions are enforced in the x di-

rection, namely fni+N;j = fnij and eni+N = eni . Neumann boundary conditions (zero

normal derivatives) are imposed in the v direction, which do not a�ect the accuracy of

the method as we choose vmax to guarantee that the numerical solution is well below

round-o� zero there for all t.

We now outline the details of the computational procedure:

1. Given the data fnij, for each �xed i, compute the concentration cni (the integral

on the right hand side of (1.5)) by the rectangular rule in v, which is in�nitely

high order accurate since f is fast decaying in v.

2. Use FFT in x to �nd the Fourier coeÆcients of the concentration cni .

3. Scale the Fourier coeÆcients of the concentration cni (divided it by i), and then

use inverse FFT to �nd the point values of the electric �eld eni .

4. In order to compute the x-derivative, for �xed n and j, use FFT in x to �nd the

Fourier coeÆcients of fnij.

5. Scale the Fourier coeÆcients of fnij (multiplied it by i), and then use inverse FFT

to get the point values of the x-derivative fx.

6. Compute the v-derivative by the ninth order upwinding-biased �nite di�erence

formula.

7. Analytically di�erentiate m(v) on the right hand side of (1.4).

8. Find the time step �t by the CFL condition and solve the method of lines ODE

to get fn+1ij by the classical fourth order Runge-Kutta method.

All the computations in this paper were carried out on SUN ULTRA-30 worksta-

tions with a \f77 -fast -r8" compile option. We use the FFT subroutine in the IMSL

library for the spectral method in x.

3 Numerical Results

The main numerical results are presented in this section. We have performed many

more numerical tests, including many tests for calibrating purpose to make sure that

what we present are not numerical artifacts. However we will show only a selected

group of representative results.

In all the �gures we plot the maximum amplitude, over x, of the electric �eld jeni jas a function of time tn, in a logarithm scale.

5

T

log(EMAX)

010

2030

4050

10-17

10-15

10-13

10-11

10-9

10-7

10-5

10-3

Fram

e001

29

Jun2000

F

rame

001

29Jun

2000

(a)Linearproblem

.

T

log(EMAX)

010

2030

4050

10-17

10-15

10-13

10-11

10-9

10-7

10-5

10-3

Fram

e001

29

Jun2000

F

rame

001

29Jun

2000

(b)Nonlin

earproblem

.

Figure

3.1:Example3.1,M=1024.

3.1

1-bumpequilib

ria

3.1.1

Maxwellia

nsandscaledMaxwellia

ns

Example

3.1.Wesolvethenonlin

earproblem

(equatio

ns(1.4),(1.5))

with

thefol-

lowinginitia

ldata:

f(0;x;v)=0:01cos(x

)m(v):

(3.1)

Thiscorresp

ondsto

a2�perio

din

xwith

asm

all(0.01)amplitu

depertu

rbatio

n.For

compariso

n,wealso

compute

thelin

earproblem

(equatio

n(1.6),(1.7))with

thesame

initia

ldata

(3.1).

Theresu

ltsare

plotted

inFig.3.1.Wecanobserv

ethattheelectric

�eld

exponentia

llydecay

sinboth

thelin

earandthenonlin

earcases

inasim

ilarfashion,

until

machinezero

isrea

ched.

Example3.2.Wenow

changetheinitia

lconditio

nto

e�ectiv

elyincrea

sethex-perio

d,

bytakinga=2in

(1.12).

Forasm

allamplitu

depertu

rbatio

n

f(0;x;v)=0:0001cos(x

)m(2v);

(3.2)

theelectric

�eld

isstill

observ

edto

exponentia

llydecay

both

forthelin

earandforthe

nonlin

earcases,

Fig.3.2.Thetailin

thenonlin

earcase

after

machinezero

isrea

ched

isanumerica

lartifa

ctwhich

goes

away

with

grid

re�nem

ents.

How

ever,

when

weincrea

sethemagnitu

deofthepertu

rbatio

n

f(0;x;v)=0:5cos(x

)m(2v)

(3.3)

then

theelectric

�eld

does

notseem

todecay

atallforthenonlin

earcase,

seeFig.3.3.

Wehaveperfo

rmed

manymore

numerica

lexperim

entswith

acontin

uum

ofampli-

tudesforExample3.1(a=1)andExample3.2(a=2).Itseem

sthatnumerica

levidence

supports

thefollow

ingplausib

leconclu

sions:

6

T

log(EMAX)

0100

200300

400

10-19

10-17

10-15

10-13

10-11

10-9

10-7

10-5

Fram

e001

29

Jun2000

F

rame

001

29Jun

2000

(a)Linearproblem

.

T

log(E-MAX)

0100

200300

400

10-19

10-17

10-15

10-13

10-11

10-9

10-7

10-5

Fram

e001

3

Jul2000

Fram

e001

3

Jul2000

(b)Nonlin

earproblem

.

Figure

3.2:Example3.2

with

asm

allamplitu

depertu

rbatio

n(3.2),M=1024.

T

log(EMAX)

0100

200

10-15

10-13

10-11

10-9

10-7

10-5

10-3

10-1

Fram

e001

29

Jun2000

F

rame

001

29Jun

2000

(a)Linearproblem

.

T

log(EMAX)

010

2030

40

10-4

10-3

10-2

10-1

Fram

e001

28

Jun2000

F

rame

001

28Jun

2000

(b)Nonlin

earproblem

.

Figure

3.3:Example3.2

with

alarger

amplitu

depertu

rbatio

n(3.3),M=1024.

7

T

log(EMAX)

010

2030

4050

60

10-17

10-15

10-13

10-11

10-9

10-7

10-5

10-3

Fram

e001

28

Jun2000

F

rame

001

28Jun

2000

(a)Linearproblem

.

T

log(EMAX)

010

2030

4050

60

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

Fram

e001

28

Jun2000

F

rame

001

28Jun

2000

(b)Nonlin

earproblem

.

Figure

3.4:Example3.3,M=1024.

�For�xed

x-perio

d2a�,when

theamplitu

deincrea

ses,theelectric

�eld

inthenon-

linearproblem

changes

from

anexponentia

ldecay

similarto

thelin

earproblem

tono-decay.

�For�xed

amplitu

dein

theinitia

lpertu

rbatio

n,when

thex-perio

dincrea

ses,the

electric�eld

inthenonlin

earproblem

decay

sslow

er.Also

itbeco

mes

no-decay

with

much

smaller

amplitu

dein

theinitia

lpertu

rbatio

n.

Thefollow

ingexamples

furth

erverify

these

observ

atio

ns.

Example

3.3.Wenow

changetheform

oftheinitia

lpertu

rbatio

nto

f(0;x;v)=0:01cos(x

)exp(sin

(x))m

(v):

(3.4)

Clea

rlytheelectric

�eld

decay

sexponentia

llyboth

inthelin

earandin

thenonlin

ear

cases,

Fig.3.4.Itseem

sthattheform

oftheinitia

lpertu

rbatio

nhasless

e�ect

onthe

decay

oftheelectric

�eld

thantheamplitu

deorthex-perio

d.

Example3.4.Wenow

changetheinitia

lconditio

nto

e�ectiv

elyincrea

sethex-perio

d,

bytakinga=�in

(1.12).

Forasm

allamplitu

depertu

rbatio

n

f(0;x;v)=0:0001cos(x

)exp(sin

(x))m

(�v);

(3.5)

theelectric

�eld

isstill

observ

edto

exponentia

llydecay

both

forthelin

earandforthe

nonlin

earcases,

Fig.3.5.Thenumerica

lnoises

canbered

uced

byre�

ningthemesh

in

v,Fig.3.6.

How

ever,

when

weincrea

sethemagnitu

deofthepertu

rbatio

n

f(0;x;v)=0:01cos(x

)exp(sin

(x))m

(�v)

(3.6)

then

theelectric

�eld

does

notseem

todecay

forthenonlin

earcase,

Fig.3.7,while

(ofcourse)

itexponentia

llydecay

sin

thelin

earcase.

Wenotice

thatthere

are

some

8

T

log(EMAX)

0200

400600

800

10-13

10-11

10-9

10-7

10-5

Fram

e001

30

Jun2000

F

rame

001

30Jun

2000

(a)Linearproblem

.

T

log(EMAX)

0200

400600

800

10-10

10-9

10-8

10-7

10-6

10-5

10-4

Fram

e001

30

Jun2000

F

rame

001

30Jun

2000

(b)Nonlin

earproblem

.

Figure

3.5:Example3.4

with

asm

allamplitu

depertu

rbatio

n(3.5),M=1024.

T

log(EMAX)

0200

400600

80010

-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

Fram

e001

30

Jun2000

F

rame

001

30Jun

2000

(a)Linearproblem

.

T

log(EMAX)

0200

400600

80010

-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

Fram

e001

30

Jun2000

F

rame

001

30Jun

2000

(b)Nonlin

earproblem

.

Figure

3.6:Example3.4

with

asm

allamplitu

depertu

rbatio

n(3.5),M=2048.

9

T

log(EMAX)

0200

400600

800

10-11

10-9

10-7

10-5

10-3

Fram

e001

30

Jun2000

F

rame

001

30Jun

2000

(a)Linearproblem

.

T

log(EMAX)

0200

400600

800

10-4

10-3

10-2

Fram

e001

30

Jun2000

F

rame

001

30Jun

2000

(b)Nonlin

earproblem

.

Figure

3.7:Example3.4

with

alargeamplitu

depertu

rbatio

n(3.6),M=1024.

T

log(EMAX)

0200

400600

800

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Fram

e001

30

Jun2000

F

rame

001

30Jun

2000

(a)Linearproblem

.

T

log(EMAX)

0200

400600

800

10-4

10-3

10-2

Fram

e001

1

Jul2000

Fram

e001

1

Jul2000

(b)Nonlin

earproblem

.

Figure

3.8:Example3.4

with

alargeamplitu

depertu

rbatio

n(3.6),M=2048.

10

T

log(EMAX)

0200

400600

80010

-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Fram

e001

2

Jul2000

Fram

e001

2

Jul2000

(a)Linearproblem

.

T

log(EMAX)

0200

400600

800

10-4

10-3

10-2

Fram

e001

1

Jul2000

Fram

e001

1

Jul2000

(b)Nonlin

earproblem

.

Figure

3.9:Example3.4

with

alargeamplitu

depertu

rbatio

n(3.6),M=4096.

numerica

lnoises

inthelin

earcase

inFig.3.7.These

noises

are

grea

tlyred

uced

and

eventually

disa

ppearwhen

weperfo

rmgrid

re�nem

entstwice,

seeFig.3.8andFig.3.9.

Example

3.5.In

order

toshow

thelim

itatio

nofournumerica

lapproach,wefurth

er

increa

sethee�ectiv

ex-perio

dbytakinga=2�in

(1.12).

Aninitia

lconditio

n

f(0;x;v)=0:01cos(x

)exp(sin

(x))m

(2�v)

(3.7)

gives

theresu

ltsin

Fig.3.10,where

onecould

notclea

rlyobserv

etheexponentia

l

decay

even

inthelin

earcase,

alth

oughitdoes

seemthatthenonlin

earcase

hasmore

non-m

onotonebehaviorforlarget.

Forthisexample,

even

forthevery

smallpertu

rbatio

n

f(0;x;v)=0:000001cos(x

)exp(sin

(x))m

(2�v)

(3.8)

westill

couldnotclea

rlyobserv

etheexponentia

ldecay

oftheelectric

�eld

,eith

erinthe

linearorinthenonlin

earcase,

seeFig.3.11.In

orderto

verify

thatthisnotnumerica

l,

were�

nethemesh

andget

essentia

llythesamepictu

re,see

Fig.3.12.

3.1.2

Analgebraically

decayingequilib

rium

Inthissectio

n,weshow

anexampleonanalgebraica

llydecay

ingequilib

rium.Itseem

s

thattheresu

ltsare

qualita

tively

similarto

those

obtained

with

theMaxwellia

nsfor

thisexample.

Example

3.6.Weuse

theinitia

ldata

(3.1)with

m(v)rep

laced

byanalgebraica

lly

decay

ingequilib

rium:

m3 (v

)=

8

3�(1+(v)2)3:

(3.9)

InFig.3.13,Fig.3.14andFig.3.15weshow

thetim

ehisto

riesofthemaximum

oftheelectric

�eld

sforthelin

earandnonlin

earproblem

s.It

seemsthatin

both

11

T

EMAX

050

100150

200

0.00

0.00

0.00

0.00

0.00

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

Fram

e001

29

Jun2000

F

rame

001

29Jun

2000

(a)Linearproblem

.

T

E

050

100150

200

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0.011

0.012

0.013

0.014

Fram

e001

28

Jun2000

F

rame

001

28Jun

2000

(b)Nonlin

earproblem

.

Figure

3.10:Example3.5

with

alargeamplitu

depertu

rbatio

n(3.7),M=1024.

T

EMAX

050

100150

200

1E-07

2E-07

3E-07

4E-07

5E-07

6E-07

7E-07

8E-07

9E-07

1E-06

1.1E-06

1.2E-06

1.3E-06

1.4E-06

Fram

e001

29

Jun2000

F

rame

001

29Jun

2000

(a)Linearproblem

.

T

EMAX

050

100150

200

1E-07

2E-07

3E-07

4E-07

5E-07

6E-07

7E-07

8E-07

9E-07

1E-06

1.1E-06

1.2E-06

1.3E-06

1.4E-06

Fram

e001

29

Jun2000

F

rame

001

29Jun

2000

(b)Nonlin

earproblem

.

Figure

3.11:Example3.5

with

avery

smallamplitu

depertu

rbatio

n(3.8),M=1024.

12

T

EMAX

050

100150

200

1E-07

2E-07

3E-07

4E-07

5E-07

6E-07

7E-07

8E-07

9E-07

1E-06

1.1E-06

1.2E-06

1.3E-06

1.4E-06

Fram

e001

3

Jul2000

Fram

e001

3

Jul2000

(a)Linearproblem

.

T

EMAX

050

100150

200

1E-07

2E-07

3E-07

4E-07

5E-07

6E-07

7E-07

8E-07

9E-07

1E-06

1.1E-06

1.2E-06

1.3E-06

1.4E-06

Fram

e001

3

Jul2000

Fram

e001

3

Jul2000

(b)Nonlin

earproblem

.

Figure

3.12:Example3.5

with

avery

smallamplitu

depertu

rbatio

n(3.8),M=2048.

cases

themaximum

oftheelectric

�eld

sdecay

exponentia

lly,sim

ilarto

theresu

ltof

theMaxwellia

ncase

inExample

3.1.Weobserv

esomenumerica

lnoises

butthese

are

\pushed"to

larger

timeformore

re�ned

mesh

es,indica

tingthatthey

are

indeed

numerica

lartifa

cts.

3.2

2-bumpequilib

ria

Inthissectio

n,weshow

someexamples

on2-bumpequilib

ria.Itseem

sthattheresu

lts

are

qualita

tively

similarto

those

obtained

with

the1-bumpequilib

riain

somecases

butdi�eren

tin

someothers.

Example3.7.Weusetheinitia

ldata

(3.1)with

m(v)rep

laced

bya2-bumpequilib

ria

m1 (v

)=

10

11 p

2�(ex

p(�

(v)2

2)+0:1exp(�

((v�5))2

2)):

(3.10)

InFig.3.16,weshow

the2-bumpequilib

ria(3.10)andthetim

ehisto

ryofthemaximum

ofelectric

�eld

forthenonlin

earproblem

.Itseem

sto

decay

exponentia

lly,sim

ilarto

theMaxwellia

ncase

inExample3.1.

Example

3.8.Wenow

changetheinitia

ldata

tobe(3.1)with

m(v)rep

laced

by

another

2-bumpequilib

ria

m2 (v

)=

10a

11 p

2�(ex

p(�

(av)2

2)+0:1exp(�

(a(v�2))2

2));

a=�:

(3.11)

InFig.3.17,weshow

the2-bumpequilib

ria(3.11)andthetim

ehisto

ryofthemax-

imum

ofelectric

�eld

forthenonlin

earproblem

.It

seemsto

decay,

butnotalways

exponentia

lly.Thisisdi�eren

tfro

mtheresu

ltoftheMaxwellia

ncase

inExample3.1.

13

T

log(EMAX)

010

2030

4050

10-7

10-6

10-5

10-4

10-3

10-2

Fram

e001

3

Jul2000

Fram

e001

3

Jul2000

(a)Linearproblem

.

T

log(EMAX)

010

2030

4050

10-7

10-6

10-5

10-4

10-3

10-2

Fram

e001

28

Jun2000

F

rame

001

28Jun

2000

(b)Nonlin

earproblem

.

Figure

3.13:Example3.6,M=1024.

T

log(EMAX)

010

2030

4050

6010

-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Fram

e001

3

Jul2000

Fram

e001

3

Jul2000

(a)Linearproblem

.

T

log(EMAX)

010

2030

4050

60

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Fram

e001

28

Jun2000

F

rame

001

28Jun

2000

(b)Nonlin

earproblem

.

Figure

3.14:Example3.6,M=2048.

14

T

log(EMAX)

020

4060

8010

-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Fram

e001

2

Jul2000

Fram

e001

2

Jul2000

(a)Linearproblem

.

T

log(EMAX)

020

4060

8010

-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Fram

e001

2

Jul2000

Fram

e001

2

Jul2000

(b)Nonlin

earproblem

.

Figure

3.15:Example3.6,M=4096.

v

m(v)

-100

100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Fram

e001

29

Jun2000

F

rame

001

29Jun

2000

(a)2-bumpequilib

ria(3.10).

T

log(EMAX)

010

2030

10-17

10-15

10-13

10-11

10-9

10-7

10-5

10-3

Fram

e001

29

Jun2000

F

rame

001

29Jun

2000

(b)Electric

�eld

,nonlin

earproblem

.

Figure

3.16:Example3.7,M=1024.

15

v

m(v)

-4-2

02

40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1

1.1

Fram

e001

29

Jun2000

F

rame

001

29Jun

2000

(a)2-bumpequilib

ria(3.11).

T

log(EMAX)

0100

200300

400

10-12

10-10

10-8

10-6

10-4

10-2

Fram

e001

30

Jun2000

F

rame

001

30Jun

2000

(b)Electric

�eld

,nonlin

earproblem

.

Figure

3.17:Example3.8,M=1024.

4ConcludingRemarks

Wehaveused

ahighorder

andcarefu

llycalib

rated

numerica

lmeth

odto

solvethe

spatia

llyperio

dicVlasov

-Poisso

nsystem

tostu

dytheso-ca

lledLandaudampingphe-

nomenon,namely

anexponentia

ldecay

ofthemaximum

oftheelectric

�eld

with

time.

Itseem

sthatforthenonlin

earVlasov

-Poisso

nsystem

,Landaudampingex-

istsforanalytica

lpertu

rbatio

nswith

smallamplitu

deto

either

aMaxwellia

n,orto

somepolynomially

decay

ingequilib

ria,even

tosomeequilib

riawith

double

bumps.

Thisdem

onstra

testhatLandaudampingisrobust.

Thelonger

thespatia

lperio

d,the

slower

thedecay

beco

mes.

Forsomelongperio

dcases

ournumerica

lmeth

odisnot

pow

erfulenoughto

detect

wheth

ertheelectric

�eld

decay

sornot.

References

[1]C.Cercig

nani,I.

Gamba,J.Jero

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kinetic,hydrodynamic,andhigh-�eld

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Comput.Meth

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[2]C.Z.ChengandJ.Knorr,

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[3]J.P.Hollow

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Springer, 1998, pp. 325-432.

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17