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COMPUTATIONAL METHODS FOR A ONE-DIMENSIONAL
PLASMA MODEL WITH TRANSPORT FIELD
DUSTIN W. BREWER†
Advisor: Stephen Pankavich†,∗
Abstract. The electromagnetic behavior of a collisionless plasma is described
by a system of partial differential equations known as the Vlasov-Maxwell sys-
tem. From a mathematical standpoint, little is known about this physically
accurate three-dimensional model, but a one-dimensional toy model of the
equations can be studied much more easily. Knowledge of the dynamics of
solutions to this reduced system, which computer simulation can help to deter-
mine, would be useful in predicting the behavior of solutions to the unabridged
Vlasov-Maxwell system. Hence, we design, construct, and implement a novel
algorithm that couples efficient finite-difference methods with a particle-in-cell
code. Finally, we draw conclusions regarding their accuracy and efficiency, as
well as, the behavior of solutions to the one-dimensional plasma model.
1. Introduction
A plasma is a partially or completely ionized gas. Approximately 99.99% of the
visible matter in the universe exists in the state of plasma, as opposed to a solid,
fluid, or a gaseous state. Matter assumes a plasma phase if the average velocity
of particles in a material achieves an enormous magnitude, for example a sizable
fraction of the speed of light. Hence all matter, if heated to a significantly great
temperature, will reside in a plasma state (Figure 1). In terms of practical use
plasmas are of great interest to the energy and aerospace industries among others,
as they are used in the production of electronics, plasma engines, and lasers, as
well as in the operation of fusion reactors [1]. Due to their free-flowing abundance
of ions and electrons, they are great conductors of electricity and widely used in
solid state physics. When a collection of charged particles is of low density or the
characteristic time scales of interest are sufficiently small, the plasma is deemed to
be “collisionless”, as collisions between particles become infrequent. A variety of
collisionless plasmas occur in nature, including the solar wind, Van Allen radiation
belts, and comet tails [9].
2000 Mathematics Subject Classification. Primary: 65M75, 35Q83, 76M28; Secondary: 82D10,
82C22.
Key words and phrases. kinetic theory, Vlasov-Maxwell, particle-in-cell, Lax-Wendroff.
†Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019; email:
dustin.brewer@mavs.uta.edu.
∗Department of Mathematics, United States Naval Academy, Annapolis, MD 21402; email:
pankavic@usna.edu, sdp@uta.edu.
This research was partially supported by a Center for Undergraduate Research in Mathemat-
ics (CURM) mini-grant funded by NSF Grant DMS-063664 and independently by the National
Science Foundation under NSF Grant DMS-0908413.
1
2 D. BREWER
Figure 1. An example of a plasma (right) after heated significantly enough
to become ionized and escape from its gaseous phase (left). After heating,
sufficient energy has been applied to free electrons from their orbitals and form
a plasma of ions and electrons that interact dominantly through self-consistent
electric and magnetic fields.
The basic (non-relativistic) equations of collisionless plasma physics are given
by a coupled system of partial differential equations (PDEs) known as the Vlasov-
Maxwell system:
(1.1)
∂tf+v· ∇xf+E+v
c×B· ∇vf= 0
ρ(t, x) = Zf(t, x, v)dv, j (t, x) = Zvf (t, x, v)dv
∂tE=c∇ × B−j, ∇ · E=ρ
∂tB=−c∇ × E, ∇ · B= 0.
In these equations t≥0, x∈R3, and v∈R3represent time, space, and momentum
respectively, and f=f(t, x, v) is the number density - the distribution of the
number of particles within a specific spatial and velocity element - of a particular
species of ion in the plasma. In addition, ρand jare the charge and current densities
of the plasma, while Eand B, the electric and magnetic fields, are generated by
the individual charges of the ions. The constant cdenotes the speed of light, and
the first equation of (1.1), known as the Vlasov equation, describes the motion
of ions and electrons in the system due to their free-streaming velocities and the
force from the electric and magnetic fields. The distribution of ions in fgives
rise to charge and current densities which, in turn, drive the fluctuations of the
electric and magnetic fields. Hence, the Vlasov equation is nonlinearly coupled to
Maxwell’s equations which model the behavior of the fields. A complete derivation
of the system and further discussion can be found in the standard references [9]
and [17].
A given plasma can be composed of a variety of (say, N) species of differing
charge; for example H+ions, electrons, and O2−ions. In fact, most plasmas contain
at least two species - ions and electrons. Thus, a separate Vlasov equation must be
imposed for each particle density fα(t, x, v) with charge eα, indexed by α= 1, .., N,
and the densities of charge and current are written as sums over α. For simplicity,
we will discuss and investigate monocharged plasmas only; hence eα= 1 and we
omit the superscript αas in (1.1). Only in the verification of the computational
PARTICLE METHOD FOR 1D PLASMA MODEL 3
method and the resulting construction of a steady state will we consider a two-
species plasma. In addition to ions of differing charges, a number of particle masses
can be involved, but throughout we will consider the mass mof each ion in the
plasma to be normalized to one.
To complete the initial-value problem on the whole space, the system (1.1) is
supplemented with given, smooth initial conditions f(0, x, v) = f0(x, v), E(0, x) =
E0(x), ∂tE(0, x) = E1(x), B(0, x) = B0(x), and ∂tB(0, x) = B1(x). Unfortu-
nately, many questions concerning this problem remain open from a mathematical
viewpoint. For example, it is unknown as to whether a smooth (continuously dif-
ferentiable) particle distribution fand vector fields Eand Bresult for all t≥0
from imposing the same smoothness properties on their initial values. While this
problem is currently at the forefront of research within kinetic theory, much more
is known about the system if magnetic effects are removed. That is, if one assumes
that B(t, x)≡0 or if one takes the limit as c→ ∞, then the resulting equations
(cf. [15]) are given by the Vlasov-Poisson system which models only electrostatic
effects of the plasma:
(1.2)
∂tf+v· ∇xf+E· ∇vf= 0
ρ(t, x) = Zf(t, x, v)dv
E=∇U, ∆U=ρ.
Now, it is well known that the field equations in (1.1) can be written as wave
equations, meaning the smooth vector fields Eand Bsatisfy the above formulation
of Maxwell’s equations if and only if
∂ttE−c2∆E=−c2∇ρ−∂tjand ∂tt B−c2∆B=c∇ × j.
Thus, the field equations in the full electromagnetic problem are hyperbolic, or
“wave-like” in nature. Contrastingly, the equation for the electrostatic field in
(1.2) has no time derivatives and no “wave-like” behavior. Instead, its behavior is
determined by a solution to Poisson’s equation (i.e., ∆U=ρ) which is a prototypical
elliptic partial differential equation. Hence, the field equations in (1.2) tend to
yield smoother solutions than those in (1.1), and this has led to the majority of the
progress in the analysis of the system.
Since little is known about the three-dimensional electromagnetic problem, it
makes sense to pose it in one dimension first (i.e., one component each for space
and momentum) and attempt to solve the dimensionally-reduced problem. Unfor-
tunately, this destroys the cross product terms in the Vlasov equation of (1.1) and
again magnetic effects are lost. Instead, the natural one-dimensional reduction of
(1.1) is the 1D Vlasov-Poisson system:
(1.3)
∂tf+v ∂xf+E(t, x)∂vf= 0,
∂xE=Zf(t, x, v)dv.
If one wishes to maintain the full electromagnetic structure, a second velocity vari-
able is required in the formulation, giving rise to the “one-and-one-half dimensional”
4 D. BREWER
Vlasov-Maxwell system [7]:
(1.4)
∂tf+v∂xf+ (E1+v2B)∂v1f+ (E2−v1B)∂v2f= 0
∂xE1=Zfdv, ∂tE1=−Zv1fdv
∂t(E2+B) + ∂x(E2+B) = −Zv2fdv
∂t(E2−B)−∂x(E2−B) = −Zv2fdv
where x∈R,v∈R2, and the speed of light chas been normalized to one. Sur-
prisingly, the question of the C1regularity of f, E , and Bremains open even in
this case. The noticeable difference between (1.4) and (1.2) is the introduction of
electric and magnetic fields E2and Bthat are solutions to transport equations,
which propagate values of the fields with speed c= 1. Mathematically speaking,
the transport equations may create problems with intersecting characteristics due
to the unbounded nature of particle velocities vthat may travel faster than the
propagation of signals from the electric and magnetic fields. Simply put, it is pos-
sible for v=c= 1 and hence the Vlasov and Maxwell characteristics can intersect
in space-time. This phenomena has been referred to in a previous work [6] as “res-
onant transport”, and its effects on the regularity of solutions are still unknown.
Thus, in order to study the behavior of such a system, but keep the problem posed
in a one-dimensional setting, we consider the following nonlinear system of PDEs
which couples the Vlasov equation to a transport field equation:
(1.5)
∂tf+v ∂xf+B(t, x)∂vf= 0,
∂tB+∂xB=Zf(t, x, v)dv
with given initial conditions f(0, x, v) = f0(x, v) and B(0, x) = B0(x).This nonlin-
ear (as Bdepends upon fand vice versa) system of PDEs serves as a toy model
of plasma dynamics in which the effects of the field are transported in space-time.
Though it does not truly represent a magnetic field, we have used Bto denote the
field variable of (1.5) in order to distinguish it from the field of (1.2). Addition-
ally, we note that the model, though not obtained directly from (1.1) and hence not
completely physical, does retain the aforementioned property of resonant transport.
Thus, studying the behavior of (1.5) will allow us to gain intuition about the effects
of resonance within (1.4) and (1.1).
In [13] the initial-value problem corresponding to (1.5) with given f0and B0was
considered, and the local-in-time well-posedness of smooth solutions was shown.
As this aspect of the system has been investigated and no numerical studies of
(1.5) exist in the literature, we turn our attention to the design, construction,
and implementation of computational methods for approximating solutions to the
system. Since analytical solutions are often so difficult to derive, numerical methods
are crucial to understanding the behavior of these quantities. Therefore, the main
focus of our paper is to design and construct accurate and efficient computational
methods to analyze solutions of (1.5). Additionally, we wish to determine (by
computational means) the impact that the hyperbolic structure of the field has on
the properties of solutions, as well as, the possible influence of different boundary
conditions.
PARTICLE METHOD FOR 1D PLASMA MODEL 5
This paper proceeds as follows. In the next section, we will provide a detailed de-
scription of the numerical methods that we utilize in order to compute approximate
solutions to (1.5). Section 3 will contain a discussion of different boundary condi-
tions, as well as, the validation of our numerical methods using representative test
runs. In Section 4, we present numerical results involving different initial particle
distributions and boundary conditions. This will allow us to draw some conclu-
sions about the utility and feasibility of these methods and the general behavior of
solutions.
2. Description of Computational Methods
One of the main computational tools for simulating the behavior of collisionless
plasmas is the particle method (also known as a “particle-in-cell” or “PIC” method),
which combines finite-difference approximations with interpolation and averaging
techniques in order to track representative “superparticles” formed by a conglom-
eration of thousands or millions of ions. For problems that use a kinetic (rather
than a continuum) description of plasma, particle methods are often utilized to
approximate solutions numerically and can be less expensive than other traditional
approximation techniques for partial differential equations, such as finite-difference
or finite-element methods. Other advantages of PIC methods include their relative
ease to construct and their cost-effective nature in simulating higher-dimensional
problems. Standard references on the subject include the books [1] and [10]. Other
works regarding PIC methods include [3], [4], [5], [8], and [14].
2.1. Overview. The general structure of a particle method can be described with-
out great difficulty. As for other numerical methods, the simulation phase space,
which is (x, v) for (1.5), is discretized into grids of finite length. Therefore, we
must divide the spatial domain of total length Lxinto Ncells, each of length ∆x
and indexed by i= 1,2, ..., N so that Lx=N·∆x. Similarly, we divide the ve-
locity domain of total length Lvinto Mcells, each of length ∆vand indexed by
j= 1,2, ..., M so that Lv=M·∆v. Also, the simulation of total time-length Tis
decomposed into Ptimesteps, each of length ∆tand indexed by n= 1,2, ..., P , so
that T=P·∆t. We assume here that Lx,Lv,T, ∆x, ∆v, and ∆tare arbitrary,
positive values known prior to performing the simulation and used to determine
N,M, and Pexplicitly. When the discretized dimension of time is added to the
particle domain, we arrive at what is referred to as the particle grid or particle
mesh. The left and right ends of each cell in the spatial domain at timestep nare
called grid points. Hence, there are N+ 1 total grid points that form the spatial
component of the mesh with the first grid point representing the left boundary and
the last grid point representing the right boundary of the domain.
Upon constructing the particle mesh, we place representative “superparticles”
within the phase space domain. These particles are not individual ions in the
plasma, but rather represent a group of many charged particles and are used to
track particle interactions, positions, and velocities. To begin the simulation, the
particles are initialized with starting positions and velocities at time t= 0, with
representative particles placed within cells, often at grid points or at the center of
each cell. We place one at the center of each cell and thus choose the total number
of representative particles to be K=N·M. Upon computing and applying the
associated fields and forces at each time, the particles will be allowed to move freely
through the spatial and velocity domain. Thus, the position and velocity of the kth
6 D. BREWER
particle at timestep n, denoted xn
kand vn
krespectively, are allowed to move off of
the mesh. In fact, the velocity grid is only used to initialize the particles; after time
zero no velocity grid points are needed to track the particles. Contrastingly, we will
see that spatial grid points will be crucial in the computation of field values and
forces on particles. Once the particle distribution is known, we use it to compute
macroscopic quantities such as the charge density
ρ(t, x) = Zf(t, x, v)dv,
and then the field B(t, x) by approximating the solution of the field equation of
(1.5), namely
∂tB+∂xB=ρ.
Finally, we calculate the force exerted by the field and “move” the particles by
changing their respective positions and velocities accordingly. Since the trajecto-
ries have now been calculated for the next timestep, the process repeats and this can
be continued until the stopping time t=Tis reached. The particle-in-cell method
combines differential approximations with particle-tracking, where the positions
and velocities of the particles are used to calculate the macro-quantities in each
cell; hence the name “particle-in-cell”. As we will soon describe, weighting schemes
play a large role in these calculations, specifically because particle charges, posi-
tions, and velocities must be recorded “at the particle”, whereas densities, fields,
and forces are indexed by static grid points.
2.2. PIC Method. With the preliminaries out of the way, the particle method
can be precisely described. In what follows, we will construct the method assuming
that the spatial and velocity intervals of interest begin at the origin, though any
interval can be discretized in the following fashion using simple translations and
dilations of the grid. For much of the construction we will follow [1], which describes
a similar particle method for an electrostatic plasma governed by (1.2).
Let the initial data f0∈C1([0, Lx]×[0, Lv]) and B0∈C1[0, Lx] be given. We
begin the simulation by choosing ∆x, ∆v > 0 and define for every i= 1, ..., N and
j= 1, ..., M with k=i·j,
x0
k=2i−1
2∆x,
v0
k=2j−1
2∆v,
qk=f0(x0
k, v0
k)∆x∆v.
These quantities represent the initial particle positions, initial particle velocities,
and the total charge of each particle included in the simulation, respectively. Notice
that the charge of a particle is a conserved quantity as it does not vary over time,
and hence qkis not time-dependent. This attribute of the particle-in-cell method
guarantees that the law governing the conservation of total charge is preserved
throughout the simulation. Choose ∆t > 0 and define tn=n·∆tfor n∈N. The
functions that represent particle positions and velocities for other times xn
kand vn
k
will be defined later for n∈N. Once these are known, the approximation of the
PARTICLE METHOD FOR 1D PLASMA MODEL 7
Figure 2. The leap-frog algorithm, in which values of the particle velocities
and positions are advanced at opposing timesteps in order to create a more
stable method.
continuous particle distribution is then given by
(2.1) f(tn, x, v) =
K
X
k=1
qk
∆x∆vˆ
δ(x−xn
k)δ(v−vn
k)
where δis the Dirac delta function and ˆ
δis the first-order weighting function defined
by
(2.2) ˆ
δ(x) =
1−|x|
∆x,if |x|<∆x
0,else.
Next, define the grid points xi=i·∆xfor i= 0, ..., N . We will write
Bn
i=B(tn, xi)
for field values at the nth timestep and ith grid point and define the function Bn(x)
by linear interpolation of the grid point values Bn
i. A leap-frog scheme is utilized for
the particle trajectories and first-order averaging methods are used to interpolate
the field and charge density values, which we shall now discuss in greater detail.
2.3. Newton’s Equations and the Leap-Frog Algorithm. When performing
a simulation, the calculations of the velocity and position of each particle at a
new timestep cannot be simultaneously performed since the two quantities are
not simultaneously known. In order to correct for this, the calculations must be
offset from each other so that the position can be calculated from the velocity at a
previous timestep and vice versa. A well-known finite-difference method of this type
is commonly referred to as the “leap-frog” method. To initialize this process, we
first use the force on the particles calculated from their charge and initial positions
to find the velocity at a half timestep earlier; that is, we calculate v−1/2
k. Beginning
with this value, we may alternatively advance the particle positions and velocities
in future timesteps while maintaining this structure. Hence, the velocity of each
particle will be offset from its position by a half timestep and the algorithm can
calculate each value at the same time. This scheme is said to be time-centered
since the values of each are determined based on the value of their counterpart in
the center of each timestep (see Figure 2). Additionally, leap-frog possess other
8 D. BREWER
Figure 3. The particle weighting which allows the simulation to store the
effects of the ions at grid points so that the field can be computed at these
points. Here, θn
i,k is defined by (2.4).
nice properties such as second-order accuracy and time-reversibility. To initiate the
leap-frog scheme, we first use initial field values and particle positions to define
v−1/2
k=v0
k−B0(x0
k)·∆t
2.
In order to find the velocity and position of each particle at the next timestep, we
simply apply standard, one-sided finite-difference approximations to solve Newton’s
equations of motion. From these equations, we know that F=ma =mdv
dt , where
Frepresents a force, and since we have normalized the mass to one, the equation
is reduced to dv
dt =F. From here, a forward-difference approximation is utilized.
Thus, assume for some n∈Nthat the positions and velocities xn
kand vn−1/2
kare
known for all k= 1, ..., K, and that Bn+1
iis known for all i= 0, ..., N. After a linear
interpolation of the field values at grid points, we obtain the function Bn+1(x) and
employ the approximation
vn+1/2
k−vn−1/2
k
∆t=Bn+1(xn
k).
Here, the force is determined by values of the discretized field, thus in Newton’s
equation F=Bn+1(xn
k). Solving the above equation for vn+1/2
k, the velocity at the
next timestep, we find
vn+1/2
k=vn−1/2
k+ ∆t·Bn+1(xn
k).
Now that we know vn+1/2
k, we can invoke the evolution equation for the position of
particles, namely dx
dt =v. Using the same explicit approximation of the derivative,
we find
xn+1
k−xn
k
∆t=vn+1/2
k
and solving as before for the approximation at the next timestep, this becomes
xn+1
k=xn
k+ ∆t·vn+1/2
k.
At the time of initialization, the field B0
kis known a priori for all x∈Rand hence
leap-frog can be utilized without an initial field computation. This crucial portion
of the process is called the “particle-mover” for obvious reasons.
PARTICLE METHOD FOR 1D PLASMA MODEL 9
Figure 4. A flow chart of the processes involved in one timestep of the
particle-in-cell method. Such an algorithm is also discussed in [1].
2.4. Weighting and Interpolation. Now that we have a value for the position
and velocity of each particle, we need to calculate the field that is created by the
movement of their associated charges. This is where one of the key components of
the particle-in-cell method comes into play; namely weighting or averaging. Rather
than computing the force imposed on each particle from the field of every other
particle in the simulation, the fields will be calculated at grid points on the boundary
of each cell. The first step in this process is to calculate the charge density at each
grid point. If xn
kand vn−1/2
khave been computed by the particle-mover, we can
now compute the density at the nth timestep and hence the field at timestep n+ 1.
We define for n∈Nand all i= 0, ..., N ,
(2.3) ρn
i=Zf(tn, xi, v)dv.
As we cannot compute the integral in (2.3) exactly due to the discretization of
the domain, we use a summation of discrete elements and a first-order weighting
scheme to perform the interpolation of the value of ρat grid points. This integral
can be computed by weighting the effect that the charge of the ions (indexed by
particle) has at each of their closest grid points. Thus, we divide the charge of each
particle into two pieces, with one piece affecting the spatial grid point to the left
of the particle and the other piece associated to the grid point located directly to
the right. As we have chosen to implement a first-order weighting scheme (using
ˆ
δ), the level or degree to which the charge is split then depends linearly upon the
proximity to either grid point. Hence, we compute
ρn
i=
K
X
k=1
qk
∆xθn
i,k + (1 −θn
i−1,k)
10 D. BREWER
where the proportion of charge θn
i,k ∈[0,1] is given by the first-order weighting
function
(2.4) θn
i,k =ˆ
δ(xi−xn
k)
and ˆ
δis defined by (2.2). This allows us to average the charge over all particles
and determine the density at spatial grid points (see Figure 3).
Though it is not immediately clear at present, it will be advantageous for us
later to utilize the function
(2.5) ψ(t, x) = Z(1 + v)f(t, x, v)dv.
Further, we will denote the value of this function at grid points by ψn
i=ψ(tn, xi).
Then, a similar weighting scheme can be utilized to compute these values from qk
and vn
k, namely
ψn
i=
K
X
k=1
qk
∆x(1 + vn
k)θn
i,k + (1 −θn
i−1,k).
Once ρn
iand ψn
ihave been computed, we define Bn+1
iusing a finite-difference
method to solve the field equation of (1.5), which depends upon knowledge of these
quantities. At this stage of the process, many options are available to compute
the field values. As we will discuss in greater detail within the next section, the
method we have chosen utilizes an explicit, higher-order finite-difference scheme to
compute Bn+1
ifor i= 0, ..., N .
Finally, after the values of the field have been determined at grid points by Bn+1
i,
we must use them to determine the force on each particle at timestep (n+ 1) so
that the particles can then be moved. However, particle quantities such as position
and momentum need not reside at gripoints and hence we need a similar weighting
procedure to generate forces “at the particle” from field values at meshpoints. As
previously mentioned, this is done by linear interpolation, or more precisely by
using
Fn+1
k=
N
X
i=1θn
i,k ·Bn+1
i+ (1 −θn
i,k)·Bn+1
i+1
where θn
i,k ∈[0,1] is determined by the proximity of the particle’s position to the
ith spatial grid point using (2.4). The process may then continue by advancing the
particle positions and velocities, xn
kand vn−1/2
k, to the next timestep, using the
particle-mover to compute vn+1/2
kand then xn+1
k. A visualization of the sequence
of calculations for each timestep is shown in Figure 4.
2.5. Finite-Difference Methods and Lax-Wendroff. Though the particle method
is utilized to approximate the number density that satisfies the Vlasov equation of
(1.5), we must still compute the values of Bin the field equation. To do so we
utilize a traditional finite-difference method. An introductory summary of such
methods can be found in [11]. Assuming the values of Bn
iare known for every
i= 0, ..., N and some n∈N, we attempt to find the values of Bat timestep
(n+ 1), namely Bn+1
i. Applying the most elementary of these approximations, the
one-sided forward-difference method, to the space and time derivatives in the field
equation of (1.5):
∂tB+∂xB=ρ,
PARTICLE METHOD FOR 1D PLASMA MODEL 11
we arrive at
Bn+1
i−Bn
i
∆t+Bn
i+1 −Bn
i
∆x=ρn
i.
Therefore, the value of Bn+1
idepends upon values of Bnat exactly two other grid
points, Bn
iand Bn
i+1. Since we are looking to find the value for Bat the next
timestep, we solve for Bn+1
iwhich yields
Bn+1
i=1 + ∆t
∆xBn
i−∆t
∆xBn
i+1 + ∆tρn
i.
The forward-difference method is easy to implement since it is explicit in time;
that is, it calculates values for timestep (n+ 1) from known values via a finite-
difference approximation at the nth timestep. Unfortunately, this method is known
to be numerically unstable. Additionally, it is well known that the error term of this
approximation is only first order. To find a more accurate and stable approximation
scheme, we look to a higher-order method.
What we desire is a method that incorporates the ease and expense of an ex-
plicit forward-difference method, yet possesses better accuracy and stability prop-
erties. Intuitively, since the error terms are calculated from the difference of the
approximations using a Taylor series, it would make sense to derive a higher-order
approximation starting from the first few terms of the Taylor expansion, thereby
reducing the error term. The so-called Lax-Wendroff method does just this. It
begins with a Taylor series expansion over one timestep and converts each of the
terms in the approximation to a different, but equivalent, form that contains no
time derivatives. Using an idea similar to this, we expand the field Bin time to
find
(2.6) B(t+ ∆t, x) = B(t, x)+∆t·∂tB(t, x) + (∆t)2
2∂ttB(t, x) + O(∆t3).
From the field equation, we know
∂tB=ρ−∂xB.
Hence, we can remove time derivatives of B, and therefore the need to know values
of Bat later timesteps, by replacing them with spatial derivatives of Band values
of ρ. For the second-order term in the expansion, we need to find the second time
derivative of B, and again the field equation yields
∂ttB=∂tρ−∂t(∂xB) = ∂tρ−∂x(ρ−∂xB) = ∂tρ−∂xρ+∂xx B.
If we assume the particle distribution is periodic in vor possesses compact v-
support, we can use the Vlasov equation of (1.5) to represent ∂tρ−∂xρwithout
time derivatives as
∂tρ−∂xρ=Z(∂tf−∂xf)dv
=Z(∂tf+v∂xf−(1 + v)∂xf)dv
=−Z∂v(Bf )dv −Z(1 + v)∂xf dv
=−∂xZ(1 + v)f dv
=−∂xψ.
12 D. BREWER
Here R∂vf dv = 0 due to the assumption of compact velocity support or periodicity
in v. Recall ψwas defined earlier in (2.5), and we can now see that this function is
useful in representing ∂ttBwithout time derivatives as
∂ttB=−∂xψ+∂xx B.
Using these terms in the Taylor series expansion (2.6), we find
B(t+ ∆t, x) = B(t, x)+∆t(ρ(t, x)−∂xB(t, x))
+(∆t)2
2(∂xxB(t, x)−∂xψ(t, x)) + O(∆t)3
(2.7)
Notice that the truncation of this approximation is accurate up to order (∆t)3.
From this expression, we can apply central-difference approximations to the spatial
derivatives. In general, such an approximation can be derived by subtracting the
Taylor series expansions for an arbitrary smooth function µ(x) about the points
x+ ∆xand x−∆x, with ∆x > 0. Doing this yields
µ(x+ ∆x)−µ(x−∆x) = 2∆xµ0(x) + 2
3!(∆x)3µ000(x) + · · ·
or µ0(x)≈µ(x+∆x)−µ(x−∆x)
2∆xwith an error term of order (∆x)2, assuming µ000 is
bounded. A second-derivative central-difference approximation can also be derived
from the Taylor series by adding the function values of µat x+ ∆xto those at
x−∆xto find
µ(x+ ∆x) + µ(x−∆x) = 2µ(x) + (∆x)2µ00(x) + 2
4!(∆x)4µ(4)(x) + · · ·
or µ00(x)≈µ(x+∆x)−2µ(x)+µ(x−∆x)
(∆x)2with an error term of order (∆x)2, assuming
µ(4) is bounded. Discretizing these difference formulas and applying them to the
∂xB,∂xψ, and ∂xxBterms, we have
(2.8)
∂xB(tn, xi)≈Bn
i+1 −Bn
i−1
2∆x
∂xψ(tn, xi)≈ψn
i+1 −ψn
i−1
2∆x
∂xxB(tn, xi)≈Bn
i+1 −2Bn
i+Bn
i−1
(∆x)2.
Assuming the bounds on higher derivatives above, the approximations in (2.8) are
accurate up to order (∆x)2.
Finally, we use the approximations (2.8) in (2.7) evaluated at t=tnand x=xi
to find our explicit, second-order finite-difference approximation of the field
Bn+1
i=Bn
i+ ∆tρn
i−Bn
i+1 −Bn
i−1
2∆x
+(∆t)2
2Bn
i+1 −2Bn
i+Bn
i−1
(∆x)2−ψn
i+1 −ψn
i−1
2∆x
or grouping terms,
Bn+1
i= 1−∆t
∆x2!Bn
i+ ∆tρn
i−(∆t)2
4∆x(ψn
i+1 −ψn
i−1)
−∆t
2∆x(1 −∆t
∆x)Bn
i+1 −(1 + ∆t
∆x)Bn
i−1.
(2.9)
PARTICLE METHOD FOR 1D PLASMA MODEL 13
This is the finite-difference equation in its final form that is implemented in the
program to compute the field at the next timestep, for every i= 1, ..., N −1 and
n= 0, ..., P −1. The only stipulation to the ensure the stability of this method
is a well-known statement called the Courant-Friedrichs-Lewy (or CFL, cf. [12])
condition which states that we must choose ∆tand ∆xin order to satisfy ν∆t
∆x≤1,
where νis the characteristic speed of the signals generated by the field B(t, x). In
our case, νis the speed of light, which has been normalized to 1. Since the field
equation tells us that such characteristics move to the right with speed ν= 1, we
need only choose ∆t≤∆xto ensure stability of the method.
Hence, in our particle code, we employ a variant of the Lax-Wendroff scheme:
an explicit one-step method which, when solved for at timestep (n+ 1), is given by
(2.9). As shown above this method increases the order of accuracy (to order (∆x)2)
of the one-sided, explicit method and does not require the extra computations that
are needed to utilize an implicit method. In this vein, we now turn our attention to
gauging the accuracy of the entire particle-in-cell method used to model the system.
2.6. Accuracy and a Conserved Quantity. Since the particle method utilizes
a number of mathematical approximations for the differential equations, it can be
difficult to determine a precise order of accuracy analytically, though some error
analysis of (1.2) has been performed [18], [8], [2]. Often a good heuristic for mea-
suring accuracy is the extent to which the PIC method preserves the value of the
conserved energy. We note, however, that in some cases secular changes in such
conserved quantities can be observed during PIC simulations, for example due to
self-heating [1], [10]. In our case, we can derive a conserved, energy-like quantity for
(1.5). Though this quantity technically does not represent the energy of a physical
system, we will still refer to it as the energy for (1.5). As we will see this quantity
is an invariant of the system and is similar to the conserved energy of (1.1) in that
it contains a velocity moment (though first-order instead of second) in the kinetic
portion and the square of the field in the potential portion.
The specific method we will employ is quite simple: the simulation value of
the energy is computed at each timestep and measured against its initial value
to see how well it is conserved throughout. Specifically, this quantity is found by
computing the Hamiltonian of the system (1.5). To derive a formula for this, we
must start by making a few assumptions. Namely, we assume that the particle
distribution tends to zero at infinity; i.e., f(t, x, v)→0 as |x| → ∞ or |v|→∞for
every t > 0. In addition, we assume that a similar statement holds for the field;
i.e., B(t, x)→0 as |x|→∞for every t > 0. Assumptions of compact support or
periodicity (which we shall implement later) for these functions would be sufficient
as well.
Next, we derive the corresponding conservation law. To this end, we define
E(t) = 1
2ZB(t, x)2dx −ZZ vf(t, x, v)dv dx.
Taking the time derivative, integrating by parts, and using (1.5) with the afore-
mentioned boundary or support assumption, it is a straightforward calculation to
show that E0(t) = 0. Hence, E(t) is constant and therefore we write
E(t) = E:= 1
2ZB0(x)2dx −ZZ vf0(x, v)dv dx
14 D. BREWER
for every t≥0. Here, the first term of Erepresents the potential energy in the
system, while the second represents a corresponding kinetic energy term. Notice
that each possesses an integral that must be approximated numerically. Thus,
instead of tracking Ethroughout the particle simulation, we define a discretized
analogue which we deem the simulation energy, namely
(2.10) ¯
E(t) = 1
2
N
X
i=1
(Bn
i)2·∆x−
K
X
k=1
vn
k·qk
where t=n·∆t. Here, vn
kis computed by stepping forward in time by ∆t/2 from
the value of vn−1/2
kin the same way that we initially push back v0
k. Within the
simulations of the next section, we will use this quantity to provide some measure as
to the effectiveness of our computations. For instance, if we find that ¯
Ehas changed
by 3% at the stopping time, then we can generally assume that our computation
of the solution is within 3% of its true value. In addition to this, we will verify the
spatial accuracy of the simulations in a later section by computing a derived steady
state.
3. Boundary Conditions and Particle Method Testing
In order to construct and implement any numerical method for this problem,
the initial state of the plasma (via its initial particle distribution f0and initial field
B0) must be known in advance. Additionally, the aforementioned finite-difference
methods require boundary conditions in order to solve the corresponding equations
near the boundary of the spatial grid. Thus, we now discuss the variety of options
available for the boundary data.
3.1. Boundary Conditions. Since it is important for the equations to model
a variety of situations, and the simulations require knowledge of the behavior of
particles at the boundaries of the spatial domain (recall that the velocity domain
is unconstrained throughout the simulation), we consider many different boundary
conditions (BCs). The most often used of these in such simulations are:
(1) Periodic boundary conditions
(2) Absorbing boundary conditions
(3) Reflective boundary conditions.
A physical picture of these conditions is displayed in Figure 5. The first variety of
boundary condition models the motion of the plasma when moving along a periodic
structure or within a periodic medium. When a particle in the simulation reaches
the spatial boundary, its position is reinitialized to emanate from the opposing
boundary wall by the exact amount from which it escaped the domain. Similarly,
spatial periodicity is imposed on the field. Often the mathematical assumption of
periodicity is made in order to study this phenomena in different geometries such as
crystals or torii, which possess the interesting property of repeating their inherent
structure.
The second condition, namely absorption of the charge at the boundary, models
the situation in which particles that leave a specific spatial region become separated
from the remaining plasma. When a particle in the simulation reaches the bound-
ary, its charge is identically set to zero, thereby “absorbing” its ionic attributes at
the boundary wall. Once a particle exits through one side of the mesh boundary,
it no longer affects the remaining particles, thereby leading to a dissipation of the
PARTICLE METHOD FOR 1D PLASMA MODEL 15
(a) Periodic BCs
(b) Absorbing BCs
(c) Reflective BCs
Figure 5. Graphical representations of different boundary conditions for the
plasma described by (1.5). Vertical lines represent spatial boundaries, solid
horizontal lines represent particle trajectories within the spatial domain, and
dashed horizontal lines represent these trajectories outside of the domain. The
distance adenotes the effect that the particle paths encounter upon reaching
the boundary.
energy. Thus, we naturally expect ¯
Eto decrease significantly during any simula-
tion that utilizes absorbing boundary conditions. Additionally, Dirichlet boundary
conditions are utilized for the field.
The third condition, in which particles are reflected (and therefore not allowed
to escape the simulation), is perhaps the most complicated boundary condition.
In this scenario, we assume that a barrier exists at the spatial boundary that
reflects particles back into the domain of interest. We are careful to note here
that the type of reflection assumed in the simulation is that of a perfectly elastic
collision. More specifically, it is assumed that upon colliding with the barrier at the
boundary, a particle’s inherent properties remain unchanged other than by altering
its direction; that is, we say that no exchange of momentum takes place when
the particle comes in contact with the boundary. Instead, it is perfectly reflected
back into the domain with a velocity of equal magnitude but opposite direction.
Again, Dirichlet boundary conditions are implemented for the field computations.
Since (1.5) is a hyperbolic system, we must also choose the initial data for B
to be compatible with the imposed boundary conditions. Hence, we shall choose
the initial field B0(x) later to satisfy either periodicity or the Dirichlet boundary
conditions, as well.
3.2. Verification and Testing. In order to test the previously described particle
method using a known steady state solution, we follow the explicit example of [14].
We utilize two species of ions: positive charges with density f+and electrons with
density f−. In this situation the steady charge density is determined using the sum
16 D. BREWER
∆xsupi|Bn
i−B(xi)|supi|Bn
i|
0.04 3.20 ×10−30.858
0.02 7.99 ×10−40.858
0.01 2.00 ×10−40.858
Table 1. Error of the field for steady state solution. Here Bn
iis the computed
field and B(xi) is the known steady state solution evaluated at spatial grid
points. The error values in the table are identical over a minimum of 1000
timesteps for a spatial grid of [−1,1]
of the particle distributions by
ρ(x) = Zf+(x, v)−f−(x, v)dv.
We let the electron distribution be defined as a function of the Hamiltonian
f−(x, v) = F1
2|v|2+U(x),
where
F(e) = −eif e≤0
0 if e > 0
and
U(x) = −1
2(1 −x2)3χ[−1,1](x)
is the potential for the steady field B(x). Here,
χA(x) = (1,if x∈A
0,if x6∈ A
is the indicator function for the set A. From U(x) the time-independent field is
calculated by
B(x) = U0(x)=3x(1 −x2)2χ[−1,1](x).
Additionally, the steady charge density can be found by calculating the derivative
of the field
ρ(x) = B0(x) = 3(1 −x2)(1 −5x2)χ[−1,1](x).
Therefore, the distribution of positive charge is determined by ρand f−as
Zf+(x, v)dv =ρ(x) + Zf−(x, v)dv
whence we may take
f+(x, v) = 3(1 −x2)(1 −5x2) + 2
3(1 −x2)9
2χ[−1,1](x)δ(v)
where δis the Dirac delta function.
Using these functions, the method was implemented for several choices of ∆x, ∆v,
and ∆twith periodic boundary conditions in order to ensure convergence to the
correct steady state solution. Table 1 summarizes the results of these runs, listing
the error found by calculating the difference between the known steady field solution
and the computed field at every timestep. Figure 6 displays a comparison between
PARTICLE METHOD FOR 1D PLASMA MODEL 17
Figure 6. The difference between computed field values and the steady state
field for (left) ∆x= 0.04, (center) ∆x= 0.02, and (right) ∆x= 0.01, respec-
tively. The horizontal axis ranges over x∈[−1,1] and the maximum values of
this difference correspond to the stated error values of Table 1. As in Table
1, the computed fields do not change after a minimum of 1000 timesteps for a
grid of length L= 2.
the computed field values and the known steady-state field. We expect that as the
mesh is refined (and the value of ∆xdecreases), the corresponding error should
also decrease at a suitable rate for each time. Evaluating each of the error values
in Table 1, we see that this is the case. As the values of spacings are halved, the
error in computing the field values effectively decreases by a factor of 4. Thus, the
method converges at a quadratic rate, as demonstrated by the approximation of
derivatives in the finite-difference method. In the next section, we will compare and
contrast results from simulations with differing boundary conditions and discuss the
resulting changes in behavior.
4. Numerical Results
Upon verifying the method, we conducted several preliminary simulations. The
particle code was implemented for different choices of initial data and the boundary
conditions discussed in Section 3.1. In this section, we will present results from these
simulations that display the inherent differences amongst their behavior and depend
strongly on the BCs of choice. Throughout the simulations, one similarity must
endure - since the particles are all of like charge, the generated force is repulsion
and the particle distribution must expand quickly. One specific choice of initial
18 D. BREWER
0
10
20
30
40
50
t
0.110
0.115
0.120
0.125
0.130
0.135
0.140
0.145
0.150
EHtL
(a) A graph of ¯
E(t) for t∈[0,50]. Here,
we have taken ∆ := ∆x= ∆v= ∆t=
0.01 and used the initial data of (IC) with
periodic BCs. Notice that the value of ¯
Eis
increasing very slowly over time, changing
by only 4% over 5000 timesteps.
5
10
15
20
25
30
t
0
2
4
6
8
10
EHtL
(b) A graph of the
K
X
k=1
vn
kqkfor t∈[0,10].
Here, we have taken ∆ = 0.01 and used the
initial data of (IC2) with absorbing BCs.
Notice that the values are decreasing dras-
tically over time, thus demonstrating the
dissipation due to absorption of charge at
the spatial boundary.
Figure 7. The change in the energy over time for periodic and absorbing BCs
data represents a smooth, compactly-supported distribution of ions in the system
that quickly decays to zero. The initial field is taken to be sinusodial in order to
represent the wave-like behavior of such physical phenomena. We utilize the phase
space (x, v)∈[−1,1] ×[−1,1] and define the functions
(IC)
f0(x, v) = (1 −4x2)2(1 −4v2)2χ[−1
2,1
2](x)χ[−1
2,1
2](v)
B0(x) = 1
2sin(2πx)χ[−1,1](x).
With these initial conditions, the energy is easily calculated by subtracting
ZZ vf0(x, v)dv dx = 0
from the square norm of the field
1
2ZB0(x)2dx =1
2·1
4= 0.125.
Here, the kinetic portion is zero due to the even symmetry of the initial charge
distribution, and we find that ¯
E(0) = 0.125.
Beginning with the test runs of the previous section, a simulation of (1.5) as-
suming initial conditions (IC) and periodic boundary conditions was conducted for
∆ = ∆t= ∆x= ∆v= 0.01. Figure 7a shows ¯
Ein the computational domain for
times 0 ≤t≤50. As previously mentioned, the deviation of this quantity from
its initial (and conserved) value is seen as a good measure of the accuracy of the
method over time. In this case, the values range between 0.12500, initially, and
0.13011, at time t= 50. In order to measure this difference, we compute the rela-
tive change in values of ¯
Eover time. Table 2 summarizes the results of these runs,
listing the values of the relative change as the mesh values are decreased. This
PARTICLE METHOD FOR 1D PLASMA MODEL 19
∆t= 2 t= 4 t= 6 t= 8 t= 10
0.10 1.36 ×10−23.15 ×10−26.10 ×10−21.02 ×10−11.58 ×10−1
0.05 5.86 ×10−31.04 ×10−21.81 ×10−22.89 ×10−24.27 ×10−2
0.025 2.62 ×10−33.73 ×10−35.70 ×10−38.46 ×10−31.20 ×10−2
0.0125 1.22 ×10−31.48 ×10−31.98 ×10−32.68 ×10−33.58 ×10−3
Table 2. The relative error of the simulation energy RE(t). Here, we take
∆ = ∆t= ∆x= ∆vand use periodic boundary conditions. The initial value
¯
E(0) = 0.125 is identical to E, and hence RE(0) = 0, for every choice of mesh
or grid size.
quantity is found by calculating the difference between the known conserved value
and the computed simulation value at every timestep and dividing by ¯
E, i.e.
RE(t) = ¯
E(t)−¯
E(0)
¯
E(0) .
For instance, from Figure 7a we can see that the relative change for ∆x= ∆v=
∆t= 0.01 at t= 50 is computed to be
RE(50) = ¯
E(50) −¯
E(0)
¯
E(0) =0.13011 −0.125
0.125 ≈0.0409 = 4.09%.
Thus, we generally expect our computations, and hence our approximate particle
distribution and field, to be within 4.1% of their actual value. Notice that the
error begins to creep into the calculations more drastically at time t= 50, which
requires 5000 timesteps to reach. If we limit the computation to a stopping time
of t= 10 with the same data and mesh values, the approximation is seemingly
closer to the actual value since we find a simulation energy of ¯
E(10) = 0.12531 and
thus RE(10) = 2.48 ×10−3≈0.25%. Hence, the method conserves the simulation
energy quite well (within 99.75%) even after 1000 timesteps with a relatively large
mesh spacing. Graphs of the computed particle distribution f(t, x, v) at times
t= 1,2,4,6 are included in Figure 8. From the figure, we can see that the mass of
charge expands quickly with particles being pushed to the right (i.e. both spatial
and velocity values are increased) by their initial velocities and the force of the field.
Upon reaching the boundary particles begin to wrap around the periodic domain,
resulting in greater interaction amongst the charges and the development of long,
string-like structures.
A similar set of simulations with the identical initial particle distribution and
field, but with reflective boundary conditions, was also conducted. Notice that
¯
Efor this system cannot be conserved as the assumptions of our computation of
the Hamiltonian in the previous section are no longer valid. That is, the change
in velocities of particles once they encounter the boundary of the computational
domain will affect the kinetic energy-like term. Hence, additional time-dependent
boundary terms must appear in this quantity and it will not be completely conserved
over time. Graphs of the computed particle distribution f(t, x, v) at times t=
1,2,4,6 for the reflective case are displayed in Figure 9. Again, particles that
encounter the boundary are propelled back into the computational domain. Notice,
20 D. BREWER
(a) Time t= 0 (b) Time t= 1
(c) Time t= 2 (d) Time t= 4
Figure 8. The particle distribution with periodic BCs using (IC) at times
t= 0,1,2,4. Notice that the values of the velocities of particles (y-axis) may
increase outside of the original domain of [−1,1], but spatial values (x-axis)
do not.
however, at time t= 2 a group of particles that attempts to leave the region is
pushed back into the domain with identical spatial values but negative velocities,
in contrast to those of the periodic case that maintain their velocity values but are
spatially transported to the other end of the xinterval.
Finally, we implemented the method to generate simulations with absorbing
boundary conditions and the following initial data:
(IC2)
f0(x, v) = (1 −4x2)2(1 −4|v−0.3|2)2χ[−1
2,1
2](x)χ[−1
2,1
2](|v−0.3|)
B0(x) = 1
2sin(2πx)χ[−1,1](x).
This set of data was specifically chosen so that the particle distribution would not be
symmetric in v, thereby ensuring that the kinetic portion of Eis not identically zero.
As demonstrated in Figure 10, the particles that begin inside the domain disperse
after a very short time period. Those that hit the boundary are absorbed and the
PARTICLE METHOD FOR 1D PLASMA MODEL 21
(a) Time t= 0 (b) Time t= 1
(c) Time t= 2 (d) Time t= 4
Figure 9. The particle distribution with reflective BCs using (IC2) at times
t= 0,1,2,4
charge in the system quickly dissipates. Particles that begin with small velocities
and are initially positioned far from the boundary remain in the simulation but
carry very little charge. Eventually, however, the force of the imposed field must
push them far enough to the right that they interact with the boundary and are
absorbed. After time t= 5, no charged particles remain, and the system ceases to
change. Similar to Figure 7a, a graph of ¯
E(t) is displayed in Figure 7b. Notice that
this quantity quickly decays as the particles are absorbed at the boundaries, and
hence kinetic energy is lost in the system. As in the case of reflective boundary
conditions, Ecannot be conserved as the absorbing boundary alters the computation
of the Hamiltonian in Section 3.
Regardless of the boundary conditions utilized, the particle distribution seems
to develop filaments, or localized “string-like” pieces rather quickly (see Fig 11).
These structures occur along curves in phase space at which the derivatives of
f(t, x, v) grow large. In each case this seems to occur even before additional particle
interaction arises from the imposed boundary conditions and leads one to believe
that it is possible for discontinuities in phase space to occur even from smooth
22 D. BREWER
(a) Time t= 0 (b) Time t= 2
(c) Time t= 4 (d) Time t= 5
Figure 10. The particle distribution with absorbing BCs using (IC2) at times
t= 0,2,4,5
initial data such as (IC) and (IC2). As one can see from Fig 11, the two-dimensional
support of the distribution function fappears to focus into a one-dimensional set.
Such an occurrence of discontinuities for (1.5) would exist in sharp contrast to
simulations of (1.2) in which solutions remain smooth if they begin with a smooth
initial distribution (cf. [14]). A proof of this regularity result has even been obtained
for the much more difficult three-dimensional analogue of (1.2) in [16]. Such a
question regarding the loss of regularity of solutions in the case of a transport field,
i.e. for (1.5), or for the full Vlasov-Maxwell system (1.1), remains unanswered.
However, some limited progress has been made in [13] for (1.5). It is our hope that
our method and results will both lead to the development of new intuition and serve
as the impetus for further study of the properties of solutions to kinetic equations
when coupled to a transported or, more generally an electromagnetic, field.
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PARTICLE METHOD FOR 1D PLASMA MODEL 23
Figure 11. Another view of filaments forming from the initially smooth par-
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Department of Mathematics, University of Texas at Arlington, 411 S Nedderman
Drive, Arlington, Texas 76019
E-mail address:dustin.brewer@mavs.uta.edu
