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functional analytical tools. First, we will discuss the description of a positivesemigroup, which helps to carry out the numerical estimates in the splitting

schemes. Second, a numerical method is discussed with respect to separate dif-ferential and integral parts of the equations. The numerical approximation ofthe abstract splitting scheme is made by applying an iterative splitting methodof the second order.The paper is outlined as follows. In Section 2, we present our mathematical modeland a possible reduced model for further approximations. The functional analyt-ical setting with semigroups is discussed in Section 3. The splitting schemes arepresented in Section 4. The results of some numerical experiments are exhibitedin Section 6. In the conclusions that are given in Section 7, we summarize ourresults.

2 Mathematical Model

In the following a model is presented whose physical motivation is explained in[1], [15] and [16].

The kinetic model considers a fluid dynamical approach to treat the naturalability of plasmas to resonate near the electron plasma frequency pe.

Here we specialise to an abstract kinetic model to describe the dynamics ofthe electrons in the plasma which allows this resonance.

The Boltzmann equation for the electrons is:

f(x,v,t)

t = A[f(x,v,t)] + B[f(x,v,t)], (x, v) x v, t [0, T], (1)

A[f] = v xf(x,v,t)

e

mex vf(x,v,t), (2)

B[f] = (x,v,t)f(x,v,t) +

V

(x,v,v)f(x, v, t)dv , (3)

f(x,v, 0) =f0(x, v), (x, v) P,x P,v, (4)

f(x,v,t) =f1(x, v), (x, v) P,x P,v, (5)

where P,x P,v IR3 IR3 is the six-dimensional phase space. f : P,x P,v [0, T] Xc is the density function and Xc is an appropriate smoothspace, e.g.C2. f0 is the initial function and the f1 is the boundary condition inthe plasma phase space P,x P,v.

We assume, based on the different materials (a plasma and a dielectricum) acomplete reflection of the electrons due to the sheathf(v|| + v) withv||, whichis the parallel and v perpendicular to the surface normal vector. Further theelectric force-field is given as F =ex, where : P,x Xc is the potentialand Xc is an appropriate smooth space, e.g. C2.

Boltzmanns equation is coupled with the electric field , while the potentialis approximated via Poissons equation. The electrostatic approximation of thefield is represented by a potential that is valid on the complete velocity volume

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S P,v and apply Poissons equation:

x () =e(ni(x) f dS) inx P,x

0 inx D,x , (6)

where the full space domain is x = P,x D,x IR3 and the full velocitydomain is v =P,v D,v. The permittivity is equal to0 in the plasmaP,xand 0D in the dielectric D,x. fulfills the boundary conditions Un at anyelectrode En and n = 0 at isolator I, whereas n is the normal vector ofthe isolator surface.

On the surface of the dielectric, a surface charge may accumulate, whichleads to a transition condition:

() = . (7)

We will make the following assumption to modify the model into a simpler andmore tractable system of equations, which allows linearizing and simplifying the

Boltzmann equations, see [20].First we assume we have an analytical solution of the potential, such that

we could reset to a diffusive term as:

F

me v xDx, (8)

whereD is the diffusion tensor andF =eis the external force. This modifiedmodel allows employing the following semi-group theory and deriving splittingschemes for the model.

The modified and splittable model equation is

f(x,v,t)

t = A[f(x,v,t)] + B[f(x,v,t)], (x, v) x v, t [0, T], (9)

A[f] = v xf(x,v,t)xDxf(x,v,t), (10)B[f] = (x,v,t)f(x,v,t) +

V

(x,v,v)f(x, v, t)dv , (11)

f(x,v, 0) =f0(x, v), (x, v) P,x P,v, (12)

f(x,v,t) =f1(x, v), (x, v) P,x P,v. (13)

Next, we will discuss the relevant semigroup theory.

3 Semigroups for Transport Equations

In the following, we derive the exponential growth of the transport semigroupsthat are used in the section on the numerical methods.

We discuss two aspects:

Neutron transport, Electron transport.

The difference of the two transport schemes are:

Neutron transport is not influenced by an electric field, Electron transport is influenced by an electric field.

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3.1 Transport model for the neutrons

For this model we can assume that f(x,v,t) describes the density distributionof the particles at position x Swith speed v V at time t [0, T], see also[3] and [17].

The space S is assumed to be a compact and convex subset of IR3 withnonempty interior, and the velocity space V is

V := {v IR3 :vmin ||||2 vmax}forvmin> 0 and vmax< .

Assumption 31 We make the following assumptions:

Particles move according to their speedv. Particles are absorbed according to the probability given by the function

which depends onx andv.

Particles are scattered according to a scattering kernel depending onx, theincoming speedv, and the outgoing speedv.

Then the neutron transport is

f(x,v,t)

t = v f(x,v,t) (x,v,t)f(x,v,t) (14)

+

V

(x,v,v)f(x, v, t)dv ,

f(x,v, 0) =f0(x, v), (15)

and boundary conditions are included in the transport operator A0 see, in thefollowing, the abstract Cauchy problem.

We next treat the abstract Cauchy problem for this simplified model.

Abstract Cauchy problem: Transport model for the neutrons We havea Banach space X := L1(S V) with Lebesgue measure on S V IR6 anddefine the abstract Cauchy problem as:

du(t)

dt =Bu(t), (16)

du(t)

dt = (A0 M+ K)u(t), (17)

u(0) =u0, (18)

whereu X.We have the following operators:1.) Collisionless transport operator2.) Absorption operator3.) Scattering OperatorImportant results for the further numerical analysis is the fact that the trans-

port semigroup can be estimated by an exponential growth, see [3]:

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Corollary 1. We assume thats(B)> is a dominant eigenvalue and(S(t))t0is irreducible. Then the transport semigroup (S(t))t0 has balanced exponential

growth. There exists a one-dimensional projectionP satisfying0 < P fwhenever0< fsuch that:

|| exp(s(B)t)S(t) P|| Mexp(t), (19)

for allt 0 and appropriateM 1 and > 0.

3.2 Transport model for the electrons or ions

For this model we can assume that f(x,v,t) describes the density distributionof particles at position x S with speed v V at time t [0, T], see also [3]and [17].

The space S is assumed to be a compact and convex subset of IR3

withnonempty interior, and the velocity space V isV := {v IR3 :vmin ||||2 vmax}

forvmin> 0 and vmax< .

Assumption 32 We make the following assumptions:

Particles move according to their speedv. Particles are absorbed according to the probability given by the function

which depends onx andv. Particles are scattered according to a scattering kernel depending onx, the

incoming speedv, and the outgoing speedv. Particles are influenced by the static electric field, which can be derived by

the kinetic theory.

The electron transport is

f(x,v,t)

t = v xf(x,v,t)

e

mex vf(x,v,t)

(x,v,t)f(x,v,t) +

V

(x,v,v)f(x, v, t)dv , (20)

f(x,v, 0) =f0(x, v), (21)

and boundary conditions are included in the transport operators. is the electricfield.

Furthermore, we have Poissons equation,

x () =

e(ni

f dS) inP

0 inD, (22)

the permittivity is equal to 0 in the plasma P and 0D in the dielectric D .

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For the simplification, we assume to solve the Poissons equation analyticallywith:

=

D(f)dx=

P 1P e(ni

f dS)dx dxin P

0 inD, (23)

whereD is the analytical function of the electric field x.We embed the electric field analytically into the transport equation.

f(x,v,t)

t = v xf(x,v,t) xD(f) xf(x,v,t)

(x,v,t)f(x,v,t) +

V

(x,v,v)f(x, v, t)dv , (24)

f(x,v, 0) =f0(x, v), (25)

and boundary conditions are included in the transport operators A0 and A1,see, in the following, the treatment of the abstract Cauchy problem. D(f) is thediffusion parameter that includes the electric field and we assume to approximatevia a constant operator D D(f).

Next we treat the abstract Cauchy problem for a transport model for theelectrons or ions.

Abstract Cauchy problem: Transport model for the electrons or ions.

We have a Banach space X := L1(SV) with Lebesgue measure onSV IR6

and define the abstract Cauchy problem as

du(t)

dt =Bu(t), (26)

du(t)

dt = (A0+ A1 M+ K)u(t), (27)

u(0) =u0, (28)

whereu X.We have the operators1.) Collisionless transport operator2.) Diffusion operator3.) Absorption operator4.) Scattering OperatorAn important result for the further numerical analysis is the fact that the

transport semigroup can be estimated by an exponential growth. Due to theanalytical embedding of the electric field, we could also estimate such operator.

Corollary 2. We assume thats(B)> is a dominant eigenvalue and(S(t))t0is irreducible Then the transport semigroup (S(t))t0 has balanced exponential

growth. There exists a one-dimensional projectionP satisfying0 < P fwhenever0< fsuch that:

|| exp(s(B)t)S(t) P|| Mexp(t), (29)

for allt 0 and appropriate M 1 and > 0.

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In the next section we discuss the splitting schemes.

4 Splitting Schemes

In general, operator splitting methods are used to solve complex models in geo-physical and environmental physics. They have been developed and applied in[19], [21] and [22].

4.1 Sequential splitting method for nonlinear problems

For our problems, nonlinear splitting schemes are necessary, see [13]. We coulduse the result for the general formulation of nonlinear ordinary differential equa-tions, which are given by

c(t) =F1(t, c(t)) + F2(t, c(t)), (30)

where the initial conditions are cn =c(tn).As before, we can decouple the above problem into two (usually simpler)

sub-problems, namely,

c(t)

t =F1(t, c

(t)) withtn t tn+1 andc(tn) =cn , (31)

c(t)

t =F2(t, c

(t)) withtn t tn+1 andc(tn) =c(tn+1), (32)

where the initial values are given by cn =c(tn) and the split approximation onthe next time level is defined as cn+1 =c(tn+1).

For this case, the splitting error can be determined by using the Jacobiansof the non-linear mappings F1 and F2:

n= 1

2[

F1c

F2, F2

c F1](t

n, c(tn)) + O(2n). (33)

Hence, for the general case, the splitting error is of first order, i.e., O(n).

Remark 1. Higher order splitting methods are given in [10]. Based on the Strangsplitting, higher order nonlinear splitting methods are also possible, see [2].

In the next subsection we present the iterative splitting method.

4.2 Iterative splitting method

Alternatives are iterative approaches to nonlinear splitting schemes.We concentrate again on nonlinear differential equations of the form

du

dt =A(u(t))u(t) + B(u(t))u(t), withu(tn) =un, (34)

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whereA(u), B(u) are matrices with nonlinear entries and densely defined, wherewe assume that the entries involve the spatial derivatives ofc, see [23]. In the

following we discuss the standard iterative operator splitting method as a fixed-point iteration method to linearize the operators.

We split our nonlinear differential equation (34) by applying

dui(t)dt

=A(ui1(t))ui(t) + B(ui1(t))ui1(t), with ui(tn) =cn, (35)

dui+1(t)dt

=A(ui1(t))ui(t) + B(ui1(t))ui+1(t), with ui+1(tn) =cn, (36)

where the time step is = tn+1 tn. The iterations are i = 1, 3, . . . , 2m+ 1.u0(t) = cn is the starting solution, where we assume that the solution c

n+1 isnearcn, oru0(t) = 0. Thus we have to solve the local fixed-point problem. c

n isthe known split approximation at time level t = tn.

The split approximation at time level t= tn+1 is defined as cn+1 =u2m+2(tn+1).

We assume that the operators A(ui1

(tn+1)), B(ui1

(tn+1)) are constant fori= 1, 3, . . . , 2m + 1. Here the linearization is done with respect to the iterations,such thatA(ui1), B(ui1) are at least non-dependent operators in the iterativeequations, and we can apply the linear theory. For the linearization we assumeat least in the first equation A(ui1(t)) A(ui(t)), and in the second equationB(ui1(t)) B(ui+1(t)), for small t.We have

||A(ui1(tn+1))ui(tn+1) A(un+1)u(tn+1)|| ,for sufficient iterations i {1, 3, . . . , 2m + 1}.

Remark 2. The linearization with the fixed-point scheme can be used for smoothor weakly nonlinear operators, but otherwise we lose the convergence behavior,not converging to a local fixed point, see [14].

5 Numerical Integration of the Integro-Part

We treat the following integro-differential equation:

u

t =

t0

u(s)ds, (37)

u(0) =u0, (38)

The integration part is done numerically withTrapezoidal rule:

ba

f(x) dx b a

n

f(a) + f(b)

2 +

n1k=1

f

a + k

b a

n

(39)

where the subintervals have the form [kh, (k+ 1)h], with h = (ba)/n and k =0, 1, 2,...,n1.

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Degree Common name Formula Error term

1 Trapezoid rule ba2 (f0+f1) (ba)3

12 f(2)()

2 Simpsons rule ba

6 (f0+ 4f1+f2)

(ba)5

2880 f(4)

()3 Simpsons 3/8 rule ba8 (f0+ 3f1+ 3f2+f3)

(ba)5

6480 f(4)()

4 Booles rule ba90 (7f0+ 32f1+ 12f2+ 32f3+ 7f4) (ba)7

1935360 f(6)()

Table 1. Numerical integration formula (closed NewtonCotes formula)

The higher order formulas are given as closed NewtonCotes formulas:

wherefiis a shorthand forf(xi), withxi= a + i(ba)/n, andn the degree.

We obtain the following formula for the Trapezoid rule,

u

t

=t/2(u(0) + u(t))ds, (40)

u(0) =u0, (41)

and obtain the analytical result

u(t) = 2

2exp(

t2

4)u(0)

1

2u(0), (42)

For a higher order formula like Simpsons rule, we have

u

t =t/6(u(0) + 4u(t/2) + u(t))ds, (43)

u(0) =u0. (44)

We apply the idea of a polynomial solution:

u(t) =a0+ a1t + a2t2 + a3t3 + . . .

and we obtain, after differentiating the coefficients,

a1+ 2a2t + 3a3t2 + . . . (45)

=t/6

a0+ 4(a0+ a1t/2 + a2t2/4 + a3t

3/8 + . . .)

+a0+ a1t + a2t2 + a3t

3 + . . .

,

a0= u0. (46)

Now, comparing coefficients yields

a0= u0 (47)

a1= a3= a5= . . .= 0 (48)

a2= 3a0 (49)

a4= 1

12a2, . . . . (50)

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6 Numerical Experiments

We present the results of our numerical experiments based on the neutron trans-port. A simplified one-dimensional model is given by

t c + vxc Dxxc + c=

(x,v,v)c(x, v, t)dv .

The velocity v and the diffusion D are given by the plasma model. The initialconditions are given by c(x, 0) =c0(x) and the boundary conditions are trivial,nc(x, t) = 0.

A first integral operator is

(x,v,v)c(x, v, t)dv =

T

0

c(x, t)dt.

A second integral operator isWe assume a simple collision operator: (x,v,v) =q(v)(1 + v2),

whereq(v) is the potential, e.g.,v 2.

We deal with the first integral operator and define the following operators:A= v 1

x[1 1 0]ID 1

x2[1 2 1]I

B= (+ t)I,while

exp(Bt) = exp((t + t2/2)I),whereIis the identity matrix.

In the following, the simplified real-life problem for a neutron transport equa-

tion, which includes the gain and loss of a neutron, will be presented.We concentrate on the computational benefits of a fast computation of the

iterative scheme, given with matrix exponentials.

The equation is

tc + Fc= 1c+

t0

2c(x, t)dt, in [0, t], (51)

F= v D, (52)

c(x, t) =c0(x), on, (53)

c(x, t) =c1(x, t), on [0, t]. (54)

In the following, we deal with the semidiscretized equation given by thematrices

tC= (A 1+ 2) C, (55)

whereC = (c1, . . . , cI)T is the solution of the species in the mobile phase in each

spatial discretization point (i = 1, . . . , I).

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We have the following two operators for the splitting method:

A= D

x2

2 11 2 1

. . . . . .

. . .

1 2 11 2

(56)

+ v

x

11 1

. . . . . .

1 11 1

IR

II (57)

whereIis the number of spatial points.

1 =

1 00 1 0

. . . . . .

. . .

0 1 001

IR

II (58)

For the integral term we have the following ideas:Case 1:t0

2c(x, t)dt 2tc(x, t),and we obtain the matrix

2=

2t2/2 0

02t2/2 0. . .

. . . . . .

02t2/2 002t2/2

IRII. (59)

For the operator splitting scheme, we apply A and B = 1+ 2 and weapply the iterative splitting method, given in Equations (35)(36).

Case 2:

We integrate the operator B with respect to the previous solutions Ci12(Ci1) and we obtain the matrix

2(Ci1)

=

t02c1,i1(x, s)ds 0 0

0t0

2c2,i1(x, s)ds 0...

. . . . . .

...

0 0t0

2cI,i1(x, s)ds

IRII. (60)

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We obtain B(C) =2(Ci1) + 1C.The iterative scheme is given by

For i = 1, 2, . . .

Ci(t) = exp(A(t tn))C(tn)

+ttn

exp((t s)A)B(Ci1(s))ds, t (tn, tn+1].

(61)

For the reference solution, we apply a fine time- and spatial scale withoutdecoupling the equations. Figure 1 presents the numerical errors between theexact and the numerical solution. Here we obtain the optimal results for one-sided iterative schemes on the operator B , meaning that we iterate with respectto B and use A as the right hand side.

Remark 3. For all iterative schemes, we can reach faster results than with thestandard schemes, due to the fact that the iterative schemes benefit from their

fast computations of the exponential matrices. With from four to five iterativesteps, we obtain more accurate results than we did with the expensive stan-dard schemes. With one-sided iterative schemes, we obtain the best convergenceresults.

7 Conclusions and Discussion

We presented the coupled model for the transport of deposition species in aplasma environment. We assumed the flow field could be computed by the plasmamodel and the transport of the deposition species by a transportreaction model.

Such a first model can help understand the important modeling of the plasmaenvironment in a CVD reactor.

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102

101

100

107

106

105

104

103

102

101

100

101

AB, Strang, oneside with A

t

errL1

AB

Strang

c1

c2

c3

c4

c5

c6

102

101

100

1010

108

106

104

102

100

102

AB, Strang, oneside with B

t

errL1

AB

Strang

c1

c2

c3

c4

c5

c6

102

101

100

108

106

104

102

100

102

AB, Strang, twoside

t

errL1

AB

Strang

c1

c2

c3

c4

c5

c6

Fig. 1.Numerical errors of the one-sided splitting scheme withA (upper figure), theone-sided splitting scheme with B (middle figure), and the iterative schemes with

1, . . . , 6 iterative steps (lower figure).