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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. Havl´ıˇckov´a Charles University, Faculty of Mathematics and , V Holeˇsoviˇck´ach 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. model describes macroscopic plasma phenomena and reveals how the statistical 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 macro- scopic quantities such as density, flux, average velocity, , or flux. Governing PDEs can be derived from the Boltzmann equation (BE) by taking velocity moments. Zero moment of ∞ + 3 the Boltzmann equation ( −∞ (BE) d v) yields for the particle density R ∂n Qρ + ∇ · (n~u) = (1) ∂t m ∞ ∞ + 3 1 + 3 Density and average velocity are defined as n = −∞ f d v and ~u = n −∞ ~vf d v. The source term on the right side of the equation (1) correspondsR to the term ofR the Boltzmann equation and ∞ ρ + 3 describes mass production and annihilation. It is defined as Q = m −∞ (∂f/∂t)c d v. Similarly, the transport equation (equation of motion) (2, 3) and heat transporR t equation (energy equation) ∞ ∞ + 3 1 + 2 3 (4, 5) can be found as first moment (m −∞ ~v (BE) d v) and second moment ( 2 m −∞ v (BE) d v) of the Boltzmann equation. R R ∂(n~u) ↔ m + ∇ P + m∇(n~u~u) − nq(E~ + ~u × B~ ) = ~uQρ + Q~p (2) ∂t ∂~u ↔ mn + mn(~u · ∇)~u + ∇ P − nq(E~ + ~u × B~ ) = Q~p (3) ∂t

∂ 1 1 1 ↔ 1 1 p + mnu2 + ∇ · p~u + ~u P + mnu2~u + L~ − nqE~ · ~u = u2Qρ + ~u · Q~p + QE (4) ∂t µγ − 1 2 ¶ µγ − 1 2 ¶ 2

1 ∂p ↔ + ∇ · (p~u) + (P ·∇)~u + ∇ · L~ = QE (5) γ − 1 µ ∂t ¶ ∞ ~ 1 + − − 2 3 Mean quantities are defined by the following equations. L = 2 m −∞ (~v ~u)(~v ~u) f d v defines heat ↔ ∞ ↔ ~ + 3 R flux L, P = m −∞ (~v − ~u)(~v − ~u)f d v defines pressure tensor P and Pij = p δij defines scalar pressure p. γ in (4) andR (5) is the ratio of specific . Source terms in (2) and (4) describe transport of

180 HAVL´ICKOVˇ A:´ FLUID MODELLING IN PLASMA PHYSICS

∞ ~p + − 3 momentum and due to and they are defined as Q = m −∞ (~v ~u)(∂f/∂t)c d v, ∞ E 1 + − 2 3 R Q = 2 m −∞ (~v ~u) (∂f/∂t)c d v. We canR 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 equa- tions 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 ( and ) plasma one obtains the so-called two-fluid model. , 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 collision with neutral, the source term for the formation of ρ species k can be evaluated as Qk = mk r lrkkr(Te)nenn, where lrk is the number of particles of species k created or lost per collision type r [BukowskiP 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 mkml E 3 2mkml ~ − − − 2 − Qk = l mk+ml nk ( ~uk ~ul) νkl and Qk = l 2 k nk (mk+ml) (Tk Tn) νkl, where νkl is the mean collisionP frequency [Golant, 1980; Chen et al., 2004].P Fluid equations as used in simulations are often simplified by neglecting some terms: - neglecting 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 contribution compared to the thermal one, which leads to the significant simplification of the energy equation [Bukowski et al., 1996; Chen et al., 2004]. ~ 5 ~ − 5 ∇ - neglecting the first term in the thermal flux closure for Le = 2 ΓekTe 2 kDene 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 drift- diffusion approximation [Herrebout et al., 2001; Herrebout et al., 2002; Chen et al., 2004; Boeuf, 1987; Donk´o, 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. ∂n Qρ k + ∇ · Γ~ = k (6) ∂t k m

Γ~k = ± µknkE~ − ∇(Dknk) (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

181 HAVL´ICKOVˇ A:´ 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 con- servation, 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 SYSTEMOF EQUATION APPROXIMATE PARTIAL DISCRETIZATION ALGEBRAIC SOLVER SOLUTION DIFF.EQS. EQUATIONS

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 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 Crank- Nicholson 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 -diffusion type ∂n ∂Γ + = 0 (8) ∂t ∂x ∂n Γ = v n − D (9) D ∂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; Mareˇset al., 2002], provides an optimum way to discretize the drift-diffusion equation for particle transport. It has been developed in the frame of device simulations [Scharfetter and Gummel, 1969]. k+1 k+1 k+1 k 1 − 1 n − n Γi+ Γi− i i + 2 2 = 0 (10) ∆t ∆x

182 HAVL´ICKOVˇ A:´ FLUID MODELLING IN PLASMA PHYSICS

− vD∆x vD ni+1 ni exp ( D ) Γ 1 = (11) i+ 2 £ − vD∆x ¤ 1 exp ( D ) − vD∆x vD ni ni−1 exp ( D ) Γ − 1 = (12) i 2 £ − vD∆x ¤ 1 exp ( D ) 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 (do- main) 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

N

u = ujφj(x, y, z) (13) Xj=1

In equation (13) u is approximative solution, uj are unknown coefficients and φj are basis (shape) func- tions. 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

Wm(x, y, z) R dx dy dz = 0 m = 1, 2, ..., N (14) Z 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

183 HAVL´ICKOVˇ A:´ FLUID MODELLING IN PLASMA PHYSICS method [Fletcher, 1991]. The computational domain is subdivided into a set of cells that cover the whole domain. The 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 b c cell-centeredvolume cell-vertexvolume cell-vertexvolume cell-vertexvolume (non-overlapping) (overlapping) (ontriangularcells)

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

∂u ∂f ∂g + + = 0 (15) ∂t ∂x ∂y

Integrating the equation (15) over the control volume ABCD (Fig. 4) and applying Green’s theorem yields the following equation d u dV + (f dy − g dx) = 0 (16) dt Z Z 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 du S j,k + (f ∆y − g ∆x) = 0 (17) ABCD dt XAB 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].

184 HAVL´ICKOVˇ A:´ 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 [Bartoˇset 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 elemen- tary 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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