WDS'06 Proceedings of Contributed Papers, Part III, 180–186, 2006.
ISBN 80-86732-86-X © MATFYZPRESS
Fluid Model of Plasma and Computational Methods for
Solution
E. Havlı́čková
Charles University, Faculty of Mathematics and Physics, V Holešovičkách 2, 180 00 Prague, Czech
Republic.
Abstract. This contribution gives basic insight into the fluid modelling in plasma
physics. The main aim of the paper is to offer brief conspectus of equations
that form the fluid model of plasma and to point out the differences between the
theoretical model in general form as derived from the Boltzmann equation and the
fluid model which is commonly used in practical simulations. Second part of the
contribution summarizes computational methods which are often used for solution
of the described model, as they were the most widely mentioned in literature.
Introduction
The simulation of plasma processes can be based on various approaches. The microscopic description
of plasma is for many practical purposes too detailed or computationally too complicated and in some
applications we need not to know the information about the behaviour of individual particles. Fluid model
describes macroscopic plasma phenomena and reveals how the statistical plasma parameters evolve in
time and space.
Fluid model of plasma
Theoretical description of plasma
Fluid model of plasma is based on partial differential equations (PDEs) which describe the macroscopic quantities such as density, flux, average velocity, pressure, temperature or heat flux. Governing
PDEs can be derived from the Boltzmann equation (BE) by taking velocity moments. Zero moment of
R +∞
the Boltzmann equation ( −∞ (BE) d3 v) yields continuity equation for the particle density
Qρ
∂n
(1)
+ ∇ · (n~u) =
m
∂t
R +∞
R +∞
Density and average velocity are defined as n = −∞ f d3 v and ~u = n1 −∞ ~v f d3 v. The source term
on the right side of the equation (1) corresponds to the collision term of the Boltzmann equation and
R +∞
describes mass production and annihilation. It is defined as Qρ = m −∞ (∂f /∂t)c d3 v. Similarly, the
momentum transport equation (equation of motion) (2, 3) and heat transport equation (energy equation)
R +∞
R +∞
(4, 5) can be found as first moment (m −∞ ~v (BE) d3 v) and second moment ( 12 m −∞ v 2 (BE) d3 v) of
the Boltzmann equation.
m
↔
∂(n~u)
~ + ~u × B)
~ = ~uQρ + Q~p
+ ∇ P + m∇(n~u~u) − nq(E
∂t
mn
↔
∂~u
~ + ~u × B)
~ = Q~p
+ mn(~u · ∇)~u + ∇ P − nq(E
∂t
(2)
(3)
¶
↔
1
1
2
~
~ · ~u = 1 u2 Qρ + ~u · Q~p + QE (4)
p~u + ~u P + mnu ~u + L − nq E
+∇·
2
2
γ−1
¶
µ
↔
∂p
1
~ = QE
+ ∇ · (p~u) + (P ·∇)~u + ∇ · L
(5)
γ − 1 ∂t
R
~ = 1 m +∞ (~v − ~u)(~v − ~u)2 f d3 v defines heat
Mean quantities are defined by the following equations. L
2
−∞
R
↔
↔
~ P = m +∞ (~v − ~u)(~v − ~u)f d3 v defines pressure tensor P and Pij = p δij defines scalar pressure
flux L,
−∞
p. γ in (4) and (5) is the ratio of specific heats. Source terms in (2) and (4) describe transport of
∂
∂t
µ
1
1
p + mnu2
2
γ−1
¶
µ
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HAVLÍČKOVÁ: FLUID MODELLING IN PLASMA PHYSICS
R +∞
momentum and heat transfer due to collisions and they are defined as Q~p = m −∞ (~v −~u) (∂f /∂t)c d3 v,
R +∞
QE = 12 m −∞ (~v − ~u)2 (∂f /∂t)c d3 v.
We can continue to derive the moments for high order terms, however the equation chain must be
truncated somewhere. In many practical problems this is made in the first order by substituting the
energy equation by an equation of the state, or in the second order by using algebraic expression for the
heat flux as closure approximation. Therefore the equations (1), (2) and (4) establish the typical set
of fluid equations which are used in many simulations. Moreover, to complete the system of governing
equations and to obtain the self-consistent description of plasma, we must consider the Maxwell’s equations describing the electromagnetic behaviour of plasma. The meaning of individual terms in the fluid
equations and their derivation is explained in Golant et al. [1980], Chen [1984] or Braginskii [1965] in
more detail.
Specific application of the fluid theory leads to various forms of the model. Generally we can use
one set of fluid equations for each plasma species and considering a simple two component (ion and
electron) plasma one obtains the so-called two-fluid model. Magnetohydrodynamics, the most widely
known plasma theory, is another modification - plasma is considered as a single fluid in the center of mass
frame [Chen, 1984; Vold et al., 1991]. A choice of the model depends on the character of the simulated
plasma.
Classical fluid formulation
Practical use of the fluid model of plasma is not based on the general form of the equations (1),
(2) and (4) as they were derived from the Boltzmann equation, but it is always connected with various
approximations. Therefore let’s introduce the classical fluid modelling approach used most commonly in
simulations [Chen et al., 2004; Bukowski et al., 1996].
Approximative form of the equations is connected especially with a simplification of the source
terms which are very complicated functions of velocity. The mass balance collision term gives the
rates of creation and loss of species. The productivity of the reactions is defined by the reaction rate
coefficients kr corresponding to the collision of type r. The coefficients kr are the input parameters
of the simulation. Assuming an electron
P collision with neutral, the source term for the formation of
species k can be evaluated as Qρk = mk r lrk kr (Te )ne nn , where lrk is the number of particles of species
k created or lost per collision type r [Bukowski et al., 1996]. The useful approximation of the collision
term in the momentum transport equation (2) and the energy equation (4) is the Krook’s approximation
P 3
P
2mk ml
ml
nk (u~k − u~l ) ν kl and QE
Q~pk = − l mmkk+m
k = −
l 2 k nk (mk +ml )2 (Tk − Tn ) ν kl , where ν kl is the mean
l
collision frequency [Golant, 1980; Chen et al., 2004].
Fluid equations as used in simulations are often simplified by neglecting some terms:
- neglecting viscosity effect in the transfer equation characterizing the anisotropic part of the pressure,
which yields the reduction of pressure tensor to the scalar pressure [Golant et al., 1980].
- neglecting kinetic energy contribution compared to the thermal one, which leads to the significant
simplification of the energy equation [Bukowski et al., 1996; Chen et al., 2004].
- neglecting the first term in the thermal flux closure for electrons L~e = 52 Γ~e kTe − 52 kDe ne ∇Te , which
is known as Fourier’s approximation [Chen et al., 2004]. The expressions for thermal flux of various
species can be found in Golant et al. [1980].
The most widely mentioned approximation in the classical fluid formulation is the so-called driftdiffusion approximation [Herrebout et al., 2001; Herrebout et al., 2002; Chen et al., 2004; Boeuf, 1987;
Donkó, 2001; Passchier et al., 1993]. This approximation reduces the number of partial differential
equations included in model by the use of the algebraic expression for particle flux Γ~k (7) instead of full
equation of motion (2). The equation (7) describes the transport of species due to density gradient and
the transport of charged species under the influence of the electric field.
Qρ
∂nk
+ ∇ · Γ~k = k
m
∂t
(6)
~ − ∇(Dk nk )
Γ~k = ± µk nk E
(7)
Coefficients of mobility µ and diffusion D are input parameters of the simulation. The expression (7)
is equivalent to the equation of motion (3) after neglecting unsteady (first term) and inertial (second
term) contributions. Conditions under which the drift-diffusion approximation and the approximations
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HAVLÍČKOVÁ: FLUID MODELLING IN PLASMA PHYSICS
mentioned above are fulfilled can be found in Golant et al. [1980] and Chen et al. [2004]. These
publications also include more detailed characteristics of the individual terms in the equations and the
definition of all quantities.
Computational methods of solution
The physical aspects of any fluid flow are governed by three fundamental principles - mass conservation, Newton’s second law and energy conservation. These principles can be expressed in terms of
mathematical equations, which are usually partial differential equations. This section briefly resumes the
methods of computational fluid dynamics (CFD) that give the solution of PDEs describing the plasma
as a fluid. The basic computational techniques are based on the construction of a discrete grid and the
replacement of individual differentiated terms in PDEs by algebraic expressions connecting nodal values
on a grid (Fig. 1).
GOVERNING
PARTIAL
DIFF. EQS.
DISCRETIZATION
SYSTEM OF
ALGEBRAIC
EQUATIONS
EQUATION
SOLVER
APPROXIMATE
SOLUTION
Figure 1. Overview of the computational solution procedure.
The most common choices for converting the PDE to the algebraic one are finite difference, finite
element and finite volume methods. In practice, time derivations in the time-dependent equation are
discretized almost exclusively using the finite difference method and spatial derivatives are discretized
by either the finite difference, finite element or finite volume method, typically.
Finite difference method
The method of finite differences (FDM) is widely used in CFD [Meeks et al., 1993; Montierth et
al., 1992; Baboolal, 2002]. Finite difference representation of derivatives is based on Taylor’s series
expansions. The discretization process introduces an error dependent on the order of terms in the
Taylor’s series which are truncated. Derivation of elementary finite differences of various order accuracy
is described in Wendt [1992].
To represent the derivatives by the differences, a number of choices is available, especially when the
dependent variable appearing in the governing equation is a function of both coordinates and time. In
this case, the finite difference approaches can be divided into implicit and explicit ones [Wendt, 1992].
The explicit approaches are relatively simple to set up and program but there are stability constraints,
given by Von Neumann method, which can result in long computer running times. On the other hand
the implicit approaches are stable even for larger values of the time step, however a system of algebraic
equations must be solved at each time step and implicit techniques are more complicated to implement.
Publications devoted to fluid modelling in plasma mention a variety of methods based on the finite
difference scheme, namely the explicit Lax-Wendroff and MacCormack’s methods or implicit CrankNicholson scheme, all described in Wendt [1992] in more detail. The summary of some techniques is
given in Vold et al. [1991].
As mentioned in the first section, the drift-diffusion approximation is widely used to simplify the
fluid model of plasma. Then the continuity equation is of convection-diffusion type
∂n ∂Γ
=0
+
∂x
∂t
(8)
∂n
(9)
∂x
In equation (9) vD = ±µE and µ and D are transport coefficients. Standard discretization of the
convection-diffusion equation requires very small grid spacing ∆x to be stable, therefore several special
discretization schemes have been developed. Scharfetter-Gummel implicit scheme given by equations
(10), (11) and (12), which is the most popular one [Boeuf, 1987; Chen et al.; 2004, Passchier et al.,
1993; Mareš et al., 2002], provides an optimum way to discretize the drift-diffusion equation for particle
transport. It has been developed in the frame of semiconductor device simulations [Scharfetter and
Gummel, 1969].
− Γk+1
Γk+1
nk+1
− nki
i− 12
i+ 12
i
=0
(10)
+
∆x
∆t
Γ = vD n − D
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HAVLÍČKOVÁ: FLUID MODELLING IN PLASMA PHYSICS
Γi+ 12 =
¤
£
vD ni+1 − ni exp ( vDD∆x )
(11)
Γi− 12 =
¤
£
vD ni − ni−1 exp ( vDD∆x )
(12)
1 − exp ( vDD∆x )
1 − exp ( vDD∆x )
Examing the limiting cases of the scheme, for diffusion-dominated problems the limit gives classical
central difference second-order scheme and for convection-dominated problems the limit produces the
upwind difference form of the first-order accuracy [Passchier et al., 1993].
Finite element method
First essential characteristic of finite element method (FEM) is to divide the continuum field (domain) into non-overlapping elements. The elements have either a triangular or a quadrilateral form, they
can be curved and cover the whole domain. The grid formed by elements need not be structured, in
opposite to FDM. The basic philosophy of FEM is that an approximate solution of the discrete problem
is assumed a priori to have prescribed form, the solution has to belong to a function space
u=
N
X
uj φj (x, y, z)
(13)
j=1
In equation (13) u is approximative solution, uj are unknown coefficients and φj are basis (shape) functions. The basis functions are chosen almost exclusively from low-order piecewise polynomials and are
assumed to be non-zero in the smallest possible number of elements associated to the index of the basis
function (Fig. 2), which is computationally advantageous.
Finite element method does not look for a solution of the PDE itself, but looks for a solution of
integral form of the PDE obtained from a weighted residual formulation [Wendt, 1992; Fletcher, 1991].
The unknown coefficients uj are determined by requiring that the integral of the weighted residual of the
partial differential equation R over the computational domain Ω is zero
Z
Wm (x, y, z) R dx dy dz = 0
m = 1, 2, ..., N
(14)
Ω
Different choices for the weight function Wm in (14) give rise to different methods in the class of methods
of weighted residuals. In FEM the weight functions are chosen from the same family as the basis functions
Wm = φm , which is the most popular choice. Equation (14) results in a system of algebraic equations
for coefficients uj , or a system of ordinary differential equations for time-dependent problems. The finite
element approach is applied in Charrada et al. [1996] for example.
Figure 2. One-dimensional linear approximating functions [Fletcher, 1991].
Finite volume method
The basic idea of the finite volume method (FVM) is to discretize the integral form of the equations
instead of the differential form and it can be thought as a special case of the so-called subdomain
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HAVLÍČKOVÁ: FLUID MODELLING IN PLASMA PHYSICS
method [Fletcher, 1991]. The computational domain is subdivided into a set of cells that cover the whole
domain. The volumes on which the integral forms of conservation laws (such as continuity equation or
momentum transport equation) are applied need not coincide with the cells of the grid and they can
even be overlapping. Different choices of volumes (Fig. 3) determine different formulations of the finite
volume technique.
nodes
a
c
b
cell-centered volume
cell-vertex volume cell-vertex volume
(non-overlapping)
(overlapping)
cell-vertex volume
(on triangular cells)
Figure 3. Typical choice of volumes in FVM [Wendt, 1992].
Let’s consider the cell-vertex formulation (Fig. 4) and the PDE of the form of equation
∂g
∂u ∂f
=0
+
+
∂x ∂y
∂t
(15)
Integrating the equation (15) over the control volume ABCD (Fig. 4) and applying Green’s theorem
yields the following equation
Z
Z
d
u dV +
(f dy − g dx) = 0
(16)
dt
ABCD
Discretizing the equation (16) to obtain an approximate evaluation of the integral form of the governing
PDE, one ends with equation (17) for each nodal point (j,k). The equation (17) then leads to the
discretization scheme of the cell-vertex formulation of FVM.
DA
SABCD
duj,k X
+
(f ∆y − g ∆x) = 0
dt
(17)
AB
Detailed analysis of steps that result in the discretization scheme is described in Fletcher [1991] and
Wendt [1992]. Application of the finite volume approach can be found in Maruzewski et al. [2002] for
instance.
Figure 4. Two-dimensional finite volume [Fletcher, 1991].
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HAVLÍČKOVÁ: FLUID MODELLING IN PLASMA PHYSICS
Discussion
Both finite difference techniques and the methods based on the weighted residual formulation are
widely used in simulations. The advantage of FDM is its relatively simple implementation, especially
in the case of problems which don’t require to transform the coordinates. By contrast, practical use of
FEM and FVM in simulations is usually connected with commercial solvers such as Fluent [Freton et
al., 2000; Gonzalez et al., 2002; Blais et al., 2003] or Comsol Multiphysics [Bartoš et al., 2005].
Comparison of the individual methods can be deduced from Wendt [1992] and Fletcher [1991], who
demonstrate various applications of the methods by particular examples. The most important advantage
of FEM and FVM is given by the possibility to use unstructured grid. Due to the unstructured form,
very complex geometries can be handled with ease. This geometric flexibility is not shared by FDM
[Wendt, 1992]. In addition, FVM provides a simple way of discretization without the need to introduce
generalized coordinates even when the global grid is irregular [Fletcher, 1991]. The use of FVM is also
supported by situations, where the conservation laws can not be represented by PDEs but only the
integral forms are guaranteed (discontinuities, etc.). However, the main problem of FVM are difficulties
in the definition of derivatives which can not be based on a Taylor-expansion. FVM becomes more
complicated when applied to the PDE containing the second derivatives, therefore the FVM is best
suitable for flow problems in primitive variables where the viscous terms are absent or are not very
important [Wendt, 1992].
Accuracy of the particular methods is analyzed by Fletcher [1991] and Wendt [1992]. As a rough
guide, the use of linear approximating functions in FEM generates solutions of about the same accuracy
as second-order FDM and the accuracy of FEM with quadratic approximating functions is comparable
with third-order FDM [Fletcher, 1991]. The accuracy of the finite volume techniques is determined by
the formulation type and can depend on irregularity of the grid. Generally, finite volume approaches are
first-order or second-order accurate in space [Wendt, 1992].
Conclusion
Presented contribution provides a brief description of the fluid model of plasma and reviews elementary computational approaches.
Only the basic aspects of the methods were described with emphasis on the comparison of the
individual techniques and with a view to accentuate their essential advantages. Extended analysis of
both the fluid model of plasma and the computational methods for solution is included in the referenced
publications.
Acknowledgments. The work is a part of the research plan MSM0021620834 that is financed by the
Ministry of Education of the Czech Republic and was partly supported by the Grant Agency of Charles University
Prague, Grants No. 296/2004 and 220/2006.
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