numerical study on landau damping - brown university...landau-damping has b een giv en in a...
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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: [email protected]. 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: [email protected]. Research supported in part by NSF Grant DMS-9971306 and an Sloan
Fellowship.3E-mail: [email protected]. 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
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(a)Linearproblem
.
T
log(EMAX)
010
2030
4050
10-17
10-15
10-13
10-11
10-9
10-7
10-5
10-3
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rame
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(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
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400600
800
10-13
10-11
10-9
10-7
10-5
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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.
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17