- Email: [email protected]
The conditions for convergence of the particles-in-a-cell and stream tube methods
U.S.S.R. Comput. Maths. Math. Phys. Vol. 19, pp. 178- 192 0 Pergamon Press Ltd. 1980. Printed in Great Britain.
THE CONDITIONS FOR CONVERGENCE OF THE PARTICLES-IN-A-CELL AND STREAM TUBE METHODS* Yu. I. MOKIN Moscow (Received
18 Janti~
1978)
THE CONSTRUCTION of discrete models for the problem of the motion of a beam of charged particles in a cylindrical vacuum diode is explained. The functional characteristics of discrete models are investigated. In the solution of a number of physical and technical problems we encounter the problem of investigating the motion of intense beams of charged particles in external and self electromagnetic fields. The complexity of the experimental investigation of these problems and the strong non-linearity of the processes taking place lead to the necessity for numerical experiments using modem computers. The use of computers for the formulation and conduct of numerical experiments in turn gives rise to the need to develop various forms of adequate mathematical description of the processes studied, and also the problem of constructing difference algorithms of the specifred quality [l] , permitting the numerical experiments to be performed at a high level of economy and accuracy. For the development of numerical methods of solving problems of the motion of beams of charged particles, and also for the investigation of the quality of the numerical algorithms corresponding to them, their theoretical and numerical validation becomes of central value. In this paper we study discrete algorithms for the solution of a characteristic problem of electron optics, which includes all the fundamental features of the class of problems considered: on the motion of a beam of relativistic electrons in a cylindrical vacuum diode [2]. For this problem various discrete models are constructed, generalizing the method of stream tubes and the particles-in-a-cell method, which are widely used in computational practice. The models constructed are conservative and differ from the method of stream tubes and the particles-m-a-cell method by the order of approximation of the system of integro-differential equations in [2]. The fundamental result of this paper is the determination of sufficient conditions for the convergence of the stream tube and particles-in-cell methods. The proof of convergence depends considerably on the results obtained in [3], where it is shown that the stream-tube method is one of the difference schemes for the system of integro-differential equations of [2 ] . We note that although the discussion in this paper deals with the example of a specific problem, all the results obtained and the method of proving them are of a general nature.
*Zh. vjkhisl. Mat. mat. Fiz.. 19,2,444-457,
1979.
178
Particles-in-a-celland stream tube methods
1, Co~t~ction
179
of the initial family of discrete models
In this section we construct a family of discrete models describing the motion of relativisitic electrons in a cylindrical vacuum diode. This family is described by the choice of the functions fix. u) and e(x), and also by the type of quadrature formula used to appro~mate the integrals in the system of integrodifferential equations in [2]. The function e(x) defines the shape and size of the unit volume associated with the point x at which the charge density and current density are calculated. The function fix, y) determines the method of calculating the charge and current at the point x. The choice of the quad~ture formula fmes the dist~bution of the macrop~icles in the discrete model. i-i
c
rz
1 D
1. We introduce the fundamental notation and formulate some facts used in the paper. The motion of a beam of relativisitic electrons will be studied in the domain D shown in the figure. The motion of the electrons begins at points s of the emission zone I’, 0 < s < u, and ends on part of the boundary C of the domain D. From each point of the emission zone there emerges a particle path whose coordinates and velocity-vector components in a cylindrical coordinate system are represented, as functions of time t and the parameter s, by x(s, t) = (r(s, t), z(s, t)) and v(s, t) = (Yr (s, t), v, (s, t)), respectively. The vector p=mu ( l-v2 / c*)-“b is the impulse of the electron. We wiIl describe the value of the emission of electrons from I’ by the density 1(s) of the emission stream. For the transit time of the electrons in the domain D we will use the notation r(s), s E r. Therefore, in accordance with the defmition of the function t(s) there exists the inclusion 5(s, t) ED
if @=I-,
Oa
x(s, t(s) )=Z.
Let 51 be some set of the Euclidean space R,. We will denote by P (a), 0 < Q < 1, the aggregate of functions Ax), x f SZ,such that
2. We will explain the defuiiion of the class of approximate methods of solving the problem of the motion of a beam of relativistic electrons in a cylindrical vacuum diode. We will describe and fix, y). We suppose that the function this class by the functions E (z) - (e, (51, e, (5) > e(x) satisfies the following conditions:
180
Yu. I. Mokin
2) a constant eg > 0 exists such that 8, (2) 3%
e. (5) ho.
We put (r, 2) , r>eo, z>eo}.
Q- {ZER, 1z-
We will assume that the function Ax, u) satisfies the conditions: 3) f(s, y)=C”
(&XQ) ,
O
f(T v) >o;
4) there exists a neighbourhood B of zero in the space R2 such that f(s, y) >O, *B, e) -1. (t) , e (5) ==e=const,
Some examples of the functions f(s, properties enumerated are given in [3 ] .
~4.
satisfying all the
Using the functions e(x) and fix, u) we form the function g(x, A) by the formula
g(z,:,)-_(s-L,e(t))
[
Jf(z-~,e(t))~]-‘, z,a~~. D
LetC(x, y),
CO (x, u) and Cl&, y), x, y e D, be singular kernels of the integral expressions in [2]. We consider the functions
G(s,Y;
+jwwh,y)dh,
(1)
D
(2) We will investigate the properties of the functions g(x, u), G(x, y; g) and Gk (x, y; g). Lemma 1 Let conditions 1)-4) be satisfied. Then we have
2) Gk y; g), G,(s, Y; gP=C=(~X~L
k-0,
1.
Moreover, for each futed value of the variable y E D the following inclusion holds: G(z, Y; g), ‘Sk
Y; g>=C-‘(~),
k-0,
1.
Proof: Under the conditions of the lemma 5 D
f(s-h;
e(z))dzay>O
for all h45. Consequently, assertion 1) follows from the corresponding smoothness of the functions E(.x,JJ). Assertion 2) is proved in [3]. The lemma is proved.
Particles-in-a-cell and stream tube methods
181
To abbreviate the formulas we agree to omit the letter g in formulas (1) and (2) and in those cases when this does not cause confusion, to write G(x, y), Gk(x, v) instead of G (3, y; g) , G,(s, Y; g), k--O, 1.
=D,
3. We define the scalar potential cp(x) and the vector potential by the formulas cp(z)=cp&)+
jr(s)
[ ‘j”G(M,t))dt]
0
0,A,(5)) ,
(3)
ds,
0
a.(x)=&s)[&(~,t)G~(x,x(s,t))dt 0
A (2) = (A,(z),
cl
ds, 1
(4)
(5)
in which cpg(x) is the solution of the equation Am = 0, assuming the specified values of V(u) on the boundary of the domain D.We put
E(x) --grad v (4,
(6)
H(x)=rotA(x).
We will determine the function x(0, t), (I E I’ from the system of integro-differential equations of the motion of particles in the self-consistent electromagnetic fields E(x), H(x):
--e
$
[ E(x(a,t))+
C
x(o,O)=a~r;
(6 t)= v(cJ,9,
(7)
v(u,0)=u0(u).
The system of integro-differential equations (7) together with the equation (see [2],)
~(u)+jl(s) [~)~(o;x(s,t))dt]dS=O, 0
u=r,
0
is closed. It permits the functions x(u, t) and Z(u), u E F to be determined. The electromagnetic fields are determined by these functions in accordance with (3)-(6).
(8)
182
Yu I. hfokin
4.We now consider the co~t~ction of the class of ~onse~ative discrete methods of the stream-tube type for the system of equations (7), (8). To clarify and simplify the technical aspect of the discussion we suppose that FZ(5) ~0, z=fs. With this assumption the system of equations (7), (8) is equivalent to the system of equations of electrostatics
AT==-4np= (9) E (2) =-grad
rp(z) ,
z-
(r, 2) ED,
and the system of ordinary differential equations for determining the function x(u, b)
f 5
(a,t)=-eE(x(o,t)), (c&O)=m=r,
g- (~,~)==~(~,O,
(10)
u (a, 0) =uo (a).
We will describe the class of discrete models by the type of quadrature formula used to approximate the integral with respect to the variable S in formulas (8) and (9). Let ol, ..., wt. **t CYN;ak > 0 be the collection of coefficients of the quadrature formula, and - Sk. bI (8), (9) We replace the mtegI?IL Sl,..., Sk,Sk+l,..., SNitSnOdeS.WepUthk=Sk+l with respect to the variable s by the ~o~espon~g quadrature formula. As a result we obtain an integrodifferentiai equation for determining the scalar potential
and the system of algebraic equations for dete~~g
the numbers 11, , , IN
(12)
The functions xk(t) =x(Sk, r) are determined from the system of equations (IO), considered at the nodes Sk. k = 1,2, . . . , N of the quadrature formula. Therefore, the initial family of discrete models is described by the system of equations (X0)-( 12). It is completely defined by the class of functions fix, JJ), by the class of functions e(x) and by the cIass of quadrature formulas, whose coefficients are positive. We formulate one of the fundamental algorithmic characteristics of the family of discrete methods constructed.
Particles-in-a-celland stream tube methods
183
Lemma 2 Any discrete model of the initial family is conservative. In other words, the functions x(u, r), cp(x) and p(x) satisfy the equations
(
Uk2 0)
I--
C2
-‘I*
=-.Lrp(s(t))+(
1
c P (2) ax= c.
J D
Y h-i
adkTkhl,
1 -yq
-% ,
k=l, 2,. . . , N.
(13)
(14)
Proof: Equation (13) is a consequence of the system of equations (10). Equation (14) follows from the definition of the function p(x) and from the formula
D
The lemma is proved. We will give some examples of the stream tube method, based on the use of various quadrature formulas. Example 1. For the trapeziodal formula the numbers ok, k = 1,2, . . . , N, are determined
from the equations a2=.
. . =aN+
-1,
h,=a I N.
ai=aN=‘/2,
This quadrature formula is of second-order approximation in the quantity 1/N. The stream tube method corresponding to the trapezoidal formula was used for the numerical determination of the function p(x) in [4]. Example 2. We suppose that the number N is odd. Then we can calculate p(x) by Simpson’s formula. The numbers ok, k = 1,2, . . . , N, are determined from the equations
al='aN='/5, a5-5=.
&“a&‘.
. .
=aN-_l=‘i5,
. . =%-2=2/S.
This quadrature formula has fourth-order approximation in the quantity 1/N Example 3 (the Gauss formula). We can calculate the function p(x) and determine the numbers 1k by using a quadrature formula of a higher degree of accuracy. In this case the nodes of the quadrature formula must be taken at the roots of a Legendre polynomial of degree N. The coefficients ark of .the Gauss formula are tabulated in [S] .
184
Yu. I. Mokin
5. Discrete models based on the particle-in-cell method are obtained from Eqs. (lo)-( 12) by replacing the integrals with respect to the variable t by a suitably chosen quadrature formula. The details of this substitution and the general form of the resulting equations are explained in [3].
2. The sufficient conditions for the particle-in-cell method to converge For a whole series of modern physical and technical problems an urgent problem is the calculation of the self electromagnetic fields of charged particles moving in some domain D of space. The fields are calculated by Maxwell’s equations in which the functions p(x) and f(x) represent respectively the charge and current per unit volume. Therefore, the self electromagnetic fields of the particles will be uniquely determined from Maxwell’s system of equations if with each point of the domain D there is associated some unit of volume and if the method of calculating the charge and current in this volume along the trajectories of the charged particles is indicated. As shown in [3] , the question of the choice of the function g(x, u) in Eqs. (9), (10) is also closely connected with the choice of the closeness of the approximate methods of describing the motion of the charged particles to the method based on the kinetic equations of Vlasov. Fixing the function g(x, JJ), and consequently the size and shape of the unit volume, we obtain some degree of closeness of these methods to each other. To calculate the functions p(x) and f(x) we use a discrete method consisting of the subdivision of the whole flow of charged particles into a small number of suitably chosen parts. The number of parts in this method, called the particles-in-cell method, is at the present time chosen from considerations of the technical possibilities of modern computen. The particle-in-cell method is used to calculate the charge density and current density as a component element of the algorithm for the solution of specific problems, in which their solvability is known in advance or is assumed. Also, since the particle-in-cell method is not intended for the isolation and subsequent calculation of one or all of the possible solutions of the problem considered, therefore when studying the qualitative features of this method it is necessary to assume not only the solvability but also the unique solvability of the problem. This means that the convergence of algorithms based on the particle-in-ceil method must be studied on the assumption of the unique solvability of the corresponding problems, and also on the assumption that the function f(x, v) is fmed. 1. We will briefly explain the main features of the particle-in-cell method. By the proof of some number of prior estimates it will be shown that the sequence of solutions of any model of the initial family is compact in the corresponding functional space. From this face and from ArzelB’s theorem the corollary that some subsequence of solutions converges to the solution of problem (7), (8) is obtained. From the unique solvability of this problem the conclusion will be drawn that the whole sequence of solutions of the discrete models converges to the solution of problem (7), (8). 2. The convergence of the solutions of the discrete methods, based on the particle-m-cell method and the stream tube method, is proved very similarly. Therefore, it is natural for us to confine ourselves to the proof of convergence of only the latter of these methods. Moreover, in order to simplify the technical aspect of the discussion, we will prove the convergence of the solution of problem (1 O)-( 12) to the solution of problem (8~( 10).
185
Particles-in-a-ceiland stream tube methods
We formulate the concept of the solution of the system of equations (1 O)-( 12). For this we fm the number iV and consider the vector x(t) = (xl (I), . . . , x&)) and the vector I= (Zl, . , . , IN). D$%niti~~~.The two vectors x(t) and I will be said to be the solution of system (lo)-{ 12) if the following conditions are satisfied: 1) the components xk(t), k = 1,2, . . . , IV, of the vector x(t) are twice continuously differentiable with respect to the variable t, 0 G t < Tk; moreover, xk (t) ED for 0 < t < Tk, and zA(TJ&, k-&Z,. . . , N; 2) Constants Zo and To exist such that
3) the vectors x(t) and Z satisfy the system of equations (IO)-(12). 3. To prove the compactness of the family of solutions of problem (lo)-( 12) we present some additional reasoning. Let the vectors x(t) and f be a solution of the system of equations (10)-(12). We construct from these vectors the function E (5) =-grad cp(5) , ZED, and note that the smoothness of ;Cand Lemma 1 imply the inclusion J!?(X)E O(D). We determine the function x(N)@, t) as the solution of the system of ordinary differential equations
+4(x),
f-z?,
(l-p
pm?2
.-‘A
)
with initial conditions z (s, 0) --s,(s) =I’, u (s, 0) =u,, (s) . It is obvious that the function x(~)(s, r) is uniquely determined and that at the points s = Sk the equations zor) (s, t) =x&i : k-l, 2,..., N are satisfied. We denote by n=n (a), UE& the outward normal to the boundary of the domain D. Lemma 3 We suppose that the inequalities
(h CT,) n) --0=-o, 1
Ial,paz,
k--l, 2,*.., N.
are satisfied uniformly with respect to N Then: 1) uniformly in N the function E(x) has a bounded gradient at points of the set 0; 2) a function
g(W)(s, t(N)(s) ) EC;
P’ (s) , OeGz,
exists such that
a9
(s, t) ED, Ott dN’
(S),
Yu. I. Mokin
186
3) the function tCN)(s) ELip [0, al ; can be chosen independently of N
the Lipschitz constant of the function P)(s)
Roof: Statement 1) of the lemma follows from the inclusion fix) E P(R2)
and the estimate
We prove statement 2). The uniform boundedness of the gradient of the function E(x), the boundedness of the numbers Tk. k = 1,2, . . . , N, and known results of the theory of ordinary differential equations imply the existence of a constant A4such that
1dN) (s, t) -dN) (so, t) I+ 1dN) (s, t) -dN) (so, t) 1,cMI s-so I,
(1%
s, SoE[O, al. We consider the points s = Sk, s = Sk+1 Sk < S < Sk+ 1. we put Zh ( Tk) -q@.
and prove statement 2) of the lemma for the case where
In our case the boundary of the domain D in some &neighbourhood of each point or r==@(s), (rg, zo) of it can be represented in one of two ways: z=P(r) , jr-r0 I (8 the function P being twice continuously differentiable. For definiteness we will Ia-GI+ z-P(r), Ir-rh(Tr) 1~6 holdsinaS-neighbourhood consider that the representation of the point ok. In this neighbourhood we pass to the new coordinate g=r, q-z-P(t) , Ir--r& (T,,) I (6 and note that for points (r, z) situated within Z the inequality 9 < 0 holds. It is easy to see that in this neighbourhood of the point Uk the normal n is a function of the variable r and that along the particle path Q(S,t), &j, t) the following differential equation is satisfied: -“dI,- Vt.
$-(i+p”)‘b(o’“‘(i,t),n),
We suppose that n(s, t) < 0 for T,G t< T,+min ( v. / 8M, 6 I c) . In other words we assume that in this interval of variation of the variable c the point x(N)(s, t) will be situated inside the domain D.Then In(rtN’(s,
Tk))-n(r(N)(~,
t)) IGf(t-T,).
Consequently, from inequality (15) considered for I = Tk, we have I (dN)(s,
-(u(“)(s, GW(
Tk), n(rcN’(s, T,)))-(dN’(s, t)), n(rtN)(s, T,)))
t), n) I<1 (dN)(s, Tn)
I+1 (dN)(s, t),
n--n(r(“)(S,
t-Tk) eve / 4.
This and (15) imply the estimate
1(dN)(sh, T,), ~(r@‘A))-WN%
t), n) i
TJ))
I
Partkle&-a-cell and stream tube methods
2M 1s~+~-s~]
Therefore, choosing N such that
Tj’(s, t)=(P(s,
/ 4,
187
we finally obtain
t), n) (l+P)?QV,/2.
Therefore, the function ~(s, t) satisfies the relations
tl’avo 12,
tl
(s, Th)--Eo--QM1S-Sk I,
OG--T,Gmin(v,/8M,
&o>O,
(16)
S/c).
Finally, this implies the estimate
(s,+)
q(S,t)2q(s,TJ+$min >+min(s,$) if
2Jqsn+i-s,I~(V,/2)min(u,/8M,
-2Mls-s,lM, Thi s contradiction proves the existence of
6/c).
the function P)(s). We turn to the proof of statement 3) of the lemma. From (16) we have the estimate
T
It(N) (S)-ttN) (Sk) laws-&I.
Consequently,
It’N’(S)-t(N)(Slr) I
GE IS-srl. UO
The lemma is proved. onto the set of points x E R2 with cp(z) We continue the function E (5) =-grad preservation of the norm O(R2). For the newly obtained function we retain the previous notation E(x). We solve the Cauchy problem with the function E(x). Thereby we continue the function x(~)(s, r) to all values of the parameter C,0 < t < To : dP -=--E(x), a.!
dX -=u, at
5
b, 0) 2x0 b) ,
u (s, 0) =uc
(s).
For this function we retain the notation _@)(s, f). Corollary 1. Let the assumptions of Lemma 3 be satisfied. Then the family of functions x(~)(s, t) is compact in the metric Cp ( [ 0, a] X [ 0, To] ) , and the family of functions t(N)(~) is compact in the metric CIO, a].
Proof: The assertions 1) and 3) of Lemma 3 imply the uniform boundedness and ArzelP’s theorem implies the equicontinuity of the family of functions x (N)(s, r) and @)(s). compactness of these sequences.
188
metric
Yu. I. Mokin
We extract from the sequence of functions .#)(s, C”J( [O, a]X[O, To]) and put
t) a subsequence convergent in the
Iim P”) (8, t) =5(s,
t).
Nk -COD
We wiIl consider that the subsequence t @‘k)(s) also converges to some function t(s). In order not to introduce new notation due to the passage to subsequences, we wilI assume that the sequences x@)(s, t) and t(m(s) themselves already converge. Lem??U 4 Let the assumptions of Lemma 3 be satisfied. Then x(s, t) ED for 0 < t < t(s). Rooof: We assume that the statement of the lemma is not true. Then a T, 0 < r < r(s) such that x(s, r) E Z. In view of the uniform convergence of the sequence S)(s, t) to the function x(s, t) for any e > 0 we can find an No(e) such that ]z(“)(s,t)--~(s,t) O
I+IrAN1(s, t)-u(s,t)
NS%(e),
]
OGtGTo.
This implies that the point x(~)(s, t) lies in an e-neighbourhood of the point x(s, r) of the boundary C. As in the proof of statement 2) of Lemma 3, in a S-neighbourhood of the point x(s, 7) we consider the change of variables E = r, 7)=z - fir). We put rr(N)(s, t) = n(t(N)(s, t)). The estimate
1(U(N)(Y, t) , rQN)(s, t) i G I W”)(s,
a
e,
4,
4 >I
n(N)(&N-~b(~, 9, 4% t) >I
+I (u(s, t), n(s; t)-(u(s, implies that for O
w,
(uo/8M, 6/2)
T), n(s, IT)) ~<.2Me+2M~t-zl
and
I t--z1 +nin
(v&M,
612~)
the
(u(N) (s, t) , ?zCN) (s, t) ) >u,/4. Moreover, for all values of the variable t of the interval indicated the point x(~)(s, t) does not go outside the G-neighbourhood of the point x(s, 7). We now show that I ttN) (s) --z 1Gmin For this we consider the two cases: r > fiNI and T Q dN)(s). Both cases (ud4M, 6/2c). are treated identically, therefore we will only discuss the first one. For t--z-min (z%/&f, 6/k) we have rl(N’(t)C~(N)(r)-%min
(2,$)<2Me-+min
($$,a,
if eQ ( uo/8M) min (u&M, 6/2c). We have therefore proved that I ttN).( s) -T I Gmin (v,/4M, 6/2c) . This in turn implies that
3
It(“)(s)-~lGIq(~)(s,
T) 162Me.
Particles-in-o-celiand stream tube methods
189
Therefore, we finally obtain
This estimate implies that r = r(s). This contradiction proves the lemma. Let us consider the function
Q (5) = ( QI (z) , . . . , Qr (5) , . , . , QN (s) ) ,
QiJz)=j~~Vk(f))d~~
where
x=Rz.
0
From the method of co~t~ct~g
Q(N)(s, x) =
x(~‘(s, t) it follows that the function
s g(x,
xcN)(s, t) ) dt,
O~sez,
x=R,,
0
assumes the values Q&x) at the nodes of the quadrature formula s = Sk, k = 1,2, . . . , A? CoroZZav 2. Let the assumptions of Lemma 4 be satisfied. Then QtN)(s, 5) OcaGf, and
EC@([0,a]XR,)
The convergence is uniform for s E [O, a] for any fared x E R2. Z%oo$ The inclusion zfN) (8, t) EC'(IO, a]X [ 0,To]) follows from the inchtsion E(x) E Cl(D). The smoothness of the function M)(s) follows from the smoothness of the boundary of the domain D and the smoothness of x(N)@, t). The limiting values of these sequences x(s, t) and f(s) are also smooth functions. This and Lemma 1 imply the smoothness of Q(s, x). The uniform convergence of QtN)(s, x) to Qfs, x) is obtained from the uniform convergence of #)(s, t) to x(s, f) and r(~){s) to f(s). The statement is proved. We denote by fiNI the piecewise-constant complement of the mesh function CulZll..., oh+. The boundedness of the numbers (~1, . . . , aN implies that the family of functions fiNI is uniformly bounded in the metric L(0, a) and for any measurable set E 5 [0, a] the knit
is uniform with respect to the index N. In this formula &5’) denotes the Lebesgue measure of the set E. Froin the above we obtain that the sequence of functions I(N)(s) is weakly compact in the space L(0, a) (see [6] ). From this sequence of func~ons we extract a weakly convergent subsequence. In order not to introduce new notation due to passing to a subsequence, we will consider that the sequence fiNI itself already converges to some function Z(s) E L(0, a).
,
190
Yu. I. Mohn
Lemma 5
Let the assumptions of Lemma 4 be satisfied. Then the sequence of functions
converges to the function
at each point x ED. Proof: We denote by o( 1) terms tending to zero-as N + ~0.Lemma 4 implies the equation t(9)
N
p’N’(z)-~I(N)(s,,)
s
g(z,s(s,t))dth,+o(1).
0
h-1
Let t(N)(s) be the piecewiseconstant completion of the mesh function r(sl), . . . , f(sN) on the set of points s E [0, a], and let FtN)(s, t) be for every fured t f [0, TO] the piecewise-constant completion of the function x(sl , t), . . . , X(SN, r) from the set of nodes of the quadrature formula on the segment [0, a]. We have the formula W)(s)
PcN)
(z) =
5I’N’(s)[ 0
5 g(z, ZfN)(s, t)) dt]
The uniform boundedness of the sequence fiNI with respect to the variable s imply the equation
p”‘(+~l(ll(a)
ds + o (1).
0
and the smoothness of the function Q(s, x)
[ Ij’lg(s,s(s,f))dt]da+o(l).
0
0
Taking into account the weak convergence of the sequence I(N)(s) to the function I(s), we pass in this equation to the limit as N + 00. As a result we obtain the formula
p(z-)=j
I(s) [ ~‘g(l,s(a,t))dt]ds. 0
0
The lemma is proved. 4. We will now formulate and prove the fundamental result of the section.
Let the vectors x(r), I be the solution of the discrete model of the initial family, based either on the particle-in-cell method, or on the stream tube method. Then:
Particles-in-a-celland stream tube methods
191
1) the sequences of functions x cN)(s, t) and t(N)(s) converge as N + 00,uniformly on the set [O,x] X [0, To], to the functions x(s, t) and t(s) respectively; moreover, x(s, t) ED for 0 < t < t(s), and x(s, t(s)) f C; 2) the sequence of functions P(~)(X) and In@) converge uniformly on D to some functions p(x) and g(x), the function cp(x) satisfying the differential equation
AT=-4np
jl(s)[ tj’ig(x,x(s, t))dt]ds
(xl =-4n
0
0
and the boundary conditions
3) let Ljjv)==rot Pk the sequence of functions
and the vectors Pk (2, y, t) - (v, 164 (3,
gi), 0, vr, A& (2,
pi) 1;
HcN) (x) bi
0
converges as N-t 00,uniformly in 5, to the function
where
L(x, x(s, t~)=rot(v~(s,t)G~(x, ~(8, t)), 0, v~(s,t)Go(x,ds, 0)); 4) the following equations are satisfied: dpldt=-
e grad cp(CC)- +
&,‘dt=u(s,
t) t
uXH,
OGsGa, Oast(s).
Roofi We will prove the theorem for discrete models based on the stream tube method. The proof of the theorem is similar in the general case, Moreover, the assumed unique solvability of problem (7) implies that it is sufficient to establish ah the statements of the theorem for some subsequence of indexes Ni, i = 1,2, . . . . To prove statement 1) of the theorem we choose a sequence of indexes Ni such that the corresponding sequences of functions xtN)(s, r) and @)(s) converge utiormly on the set [0, a] X [0,7’o]. In the proof of the theorem we will consider that this sequence of indexes is fured. Then Lemmas 2-4 imply statement 1) of the theorem. From Lemma 5 there follows the convergence, uniform on 5, of the sequence of functions pN(x) to the function p(x). From this and from well-known results of the theory of
192
I. F. Potapenkoand K A. Chuyanov
elliptic equations, we conclude that rp(X’*,Q, (x) in 6. Also, since &) E @(is> and &p(x) /an-o for x E I’. This reasoning established part 2) of the theorem. Part 3) is proved in the same way as Lemma 5. The validity of statement 4) of the theorem is obvious. The author considers it his duty to thank A. A. Samarskii for suggesting the problem, for his interest and for discussing the results. Translated by J. Berry.
REFERENCES 1.
SAMARSKII, A. A. Theory of dtffctence schemes CTeoriyaraznostnykh&hem), “Nat&a”, Moscow, 1977.
2.
MOKIN, Yu. I. On two models of the steady motion of charged particles in a vacuum diode. Nutem. sb.,
106 (148),2,234-264,1978. 3.
MOKIN, Yu, I. Some methods for the approbate description of the steady Sow of electrons in a vacuum diode. Zh. vj%isf Ma. mat. Fit., l&S, 1186-1195,1978.
4.
GOLOVIN, G. T. et ui. Formulation of problems of the steady motion of heavy-current beams of reIatIvistIc electrons in a cylindrical vacuum diode. In: Numcrlcol methods bf rmrrhmwticul phydcr (Chid. metody matem. fia.), 4-11, NI VTs MGU, Moscow, 1978.
5.
KRYLOV, V. 1.27re approximateewluationof integr& (PrIblIzhennoe vychislenie integralov), “Nauka”, Moscow, 1967.
6.
DUNFORD, N. and SCHWARTZ, J. T. Lineuroperators (LIneinye operatory), Part 1, “Mir”, Moscow, 1964.
U.S.S.R. Comput. Maths. Muth. Phys. Vol. 19, pp. 192- 198 6 Pergamon Press Ltd. 1980. PrInted in Great Britain.
COMPLETELY CONSERVATIVE DIFFERENCE SCHEMES FOR A SYSTEM OF LANDAU EQUATIONS* 1. F. ~TAP~~O
and V. A. C~~OV Moscow
(Received 12 January 1978 ; revised 24 April 1978)
DIFFERENCE schemes are constructed for which the number of particles and the energy in the system of Landau kinetic equations is conserved. Examples are given of the calculation of relaxation in an infinite medium and of the outflow of plasma from a magnetic mirror. In this paper we consider the probiem of ratify-homogeneous relaxation in a system of charged particles isotropic in respect of velocities. In view of the great difference between the masses of ions and electrons the later stages of the relaxation of the energies of these particles proceeds very slowly. In order to avoid an accumulation of errors in the numerical solution it is desirable to use completely conservative difference schemes. The concept of a completely conservative difference scheme for problems of gasdynamics was introduced in [ 1,2] . The property of complete conservativeness for a system of Landau equations, consists of the fact that the difference equation, written in the form of the law of conservation of the number of particles, *Zh. vjkhisl. Mat. mat. Fiz., 19,2,458-463,
1979.
THE CONDITIONS FOR CONVERGENCE OF THE PARTICLES-IN-A-CELL AND STREAM TUBE METHODS* Yu. I. MOKIN Moscow (Received
18 Janti~
1978)
THE CONSTRUCTION of discrete models for the problem of the motion of a beam of charged particles in a cylindrical vacuum diode is explained. The functional characteristics of discrete models are investigated. In the solution of a number of physical and technical problems we encounter the problem of investigating the motion of intense beams of charged particles in external and self electromagnetic fields. The complexity of the experimental investigation of these problems and the strong non-linearity of the processes taking place lead to the necessity for numerical experiments using modem computers. The use of computers for the formulation and conduct of numerical experiments in turn gives rise to the need to develop various forms of adequate mathematical description of the processes studied, and also the problem of constructing difference algorithms of the specifred quality [l] , permitting the numerical experiments to be performed at a high level of economy and accuracy. For the development of numerical methods of solving problems of the motion of beams of charged particles, and also for the investigation of the quality of the numerical algorithms corresponding to them, their theoretical and numerical validation becomes of central value. In this paper we study discrete algorithms for the solution of a characteristic problem of electron optics, which includes all the fundamental features of the class of problems considered: on the motion of a beam of relativistic electrons in a cylindrical vacuum diode [2]. For this problem various discrete models are constructed, generalizing the method of stream tubes and the particles-in-a-cell method, which are widely used in computational practice. The models constructed are conservative and differ from the method of stream tubes and the particles-m-a-cell method by the order of approximation of the system of integro-differential equations in [2]. The fundamental result of this paper is the determination of sufficient conditions for the convergence of the stream tube and particles-in-cell methods. The proof of convergence depends considerably on the results obtained in [3], where it is shown that the stream-tube method is one of the difference schemes for the system of integro-differential equations of [2 ] . We note that although the discussion in this paper deals with the example of a specific problem, all the results obtained and the method of proving them are of a general nature.
*Zh. vjkhisl. Mat. mat. Fiz.. 19,2,444-457,
1979.
178
Particles-in-a-celland stream tube methods
1, Co~t~ction
179
of the initial family of discrete models
In this section we construct a family of discrete models describing the motion of relativisitic electrons in a cylindrical vacuum diode. This family is described by the choice of the functions fix. u) and e(x), and also by the type of quadrature formula used to appro~mate the integrals in the system of integrodifferential equations in [2]. The function e(x) defines the shape and size of the unit volume associated with the point x at which the charge density and current density are calculated. The function fix, y) determines the method of calculating the charge and current at the point x. The choice of the quad~ture formula fmes the dist~bution of the macrop~icles in the discrete model. i-i
c
rz
1 D
1. We introduce the fundamental notation and formulate some facts used in the paper. The motion of a beam of relativisitic electrons will be studied in the domain D shown in the figure. The motion of the electrons begins at points s of the emission zone I’, 0 < s < u, and ends on part of the boundary C of the domain D. From each point of the emission zone there emerges a particle path whose coordinates and velocity-vector components in a cylindrical coordinate system are represented, as functions of time t and the parameter s, by x(s, t) = (r(s, t), z(s, t)) and v(s, t) = (Yr (s, t), v, (s, t)), respectively. The vector p=mu ( l-v2 / c*)-“b is the impulse of the electron. We wiIl describe the value of the emission of electrons from I’ by the density 1(s) of the emission stream. For the transit time of the electrons in the domain D we will use the notation r(s), s E r. Therefore, in accordance with the defmition of the function t(s) there exists the inclusion 5(s, t) ED
if @=I-,
Oa
x(s, t(s) )=Z.
Let 51 be some set of the Euclidean space R,. We will denote by P (a), 0 < Q < 1, the aggregate of functions Ax), x f SZ,such that
2. We will explain the defuiiion of the class of approximate methods of solving the problem of the motion of a beam of relativistic electrons in a cylindrical vacuum diode. We will describe and fix, y). We suppose that the function this class by the functions E (z) - (e, (51, e, (5) > e(x) satisfies the following conditions:
180
Yu. I. Mokin
2) a constant eg > 0 exists such that 8, (2) 3%
e. (5) ho.
We put (r, 2) , r>eo, z>eo}.
Q- {ZER, 1z-
We will assume that the function Ax, u) satisfies the conditions: 3) f(s, y)=C”
(&XQ) ,
O
f(T v) >o;
4) there exists a neighbourhood B of zero in the space R2 such that f(s, y) >O, *B, e) -1. (t) , e (5) ==e=const,
Some examples of the functions f(s, properties enumerated are given in [3 ] .
~4.
satisfying all the
Using the functions e(x) and fix, u) we form the function g(x, A) by the formula
g(z,:,)-_(s-L,e(t))
[
Jf(z-~,e(t))~]-‘, z,a~~. D
LetC(x, y),
CO (x, u) and Cl&, y), x, y e D, be singular kernels of the integral expressions in [2]. We consider the functions
G(s,Y;
+jwwh,y)dh,
(1)
D
(2) We will investigate the properties of the functions g(x, u), G(x, y; g) and Gk (x, y; g). Lemma 1 Let conditions 1)-4) be satisfied. Then we have
2) Gk y; g), G,(s, Y; gP=C=(~X~L
k-0,
1.
Moreover, for each futed value of the variable y E D the following inclusion holds: G(z, Y; g), ‘Sk
Y; g>=C-‘(~),
k-0,
1.
Proof: Under the conditions of the lemma 5 D
f(s-h;
e(z))dzay>O
for all h45. Consequently, assertion 1) follows from the corresponding smoothness of the functions E(.x,JJ). Assertion 2) is proved in [3]. The lemma is proved.
Particles-in-a-cell and stream tube methods
181
To abbreviate the formulas we agree to omit the letter g in formulas (1) and (2) and in those cases when this does not cause confusion, to write G(x, y), Gk(x, v) instead of G (3, y; g) , G,(s, Y; g), k--O, 1.
=D,
3. We define the scalar potential cp(x) and the vector potential by the formulas cp(z)=cp&)+
jr(s)
[ ‘j”G(M,t))dt]
0
0,A,(5)) ,
(3)
ds,
0
a.(x)=&s)[&(~,t)G~(x,x(s,t))dt 0
A (2) = (A,(z),
cl
ds, 1
(4)
(5)
in which cpg(x) is the solution of the equation Am = 0, assuming the specified values of V(u) on the boundary of the domain D.We put
E(x) --grad v (4,
(6)
H(x)=rotA(x).
We will determine the function x(0, t), (I E I’ from the system of integro-differential equations of the motion of particles in the self-consistent electromagnetic fields E(x), H(x):
--e
$
[ E(x(a,t))+
C
x(o,O)=a~r;
(6 t)= v(cJ,9,
(7)
v(u,0)=u0(u).
The system of integro-differential equations (7) together with the equation (see [2],)
~(u)+jl(s) [~)~(o;x(s,t))dt]dS=O, 0
u=r,
0
is closed. It permits the functions x(u, t) and Z(u), u E F to be determined. The electromagnetic fields are determined by these functions in accordance with (3)-(6).
(8)
182
Yu I. hfokin
4.We now consider the co~t~ction of the class of ~onse~ative discrete methods of the stream-tube type for the system of equations (7), (8). To clarify and simplify the technical aspect of the discussion we suppose that FZ(5) ~0, z=fs. With this assumption the system of equations (7), (8) is equivalent to the system of equations of electrostatics
AT==-4np= (9) E (2) =-grad
rp(z) ,
z-
(r, 2) ED,
and the system of ordinary differential equations for determining the function x(u, b)
f 5
(a,t)=-eE(x(o,t)), (c&O)=m=r,
g- (~,~)==~(~,O,
(10)
u (a, 0) =uo (a).
We will describe the class of discrete models by the type of quadrature formula used to approximate the integral with respect to the variable S in formulas (8) and (9). Let ol, ..., wt. **t CYN;ak > 0 be the collection of coefficients of the quadrature formula, and - Sk. bI (8), (9) We replace the mtegI?IL Sl,..., Sk,Sk+l,..., SNitSnOdeS.WepUthk=Sk+l with respect to the variable s by the ~o~espon~g quadrature formula. As a result we obtain an integrodifferentiai equation for determining the scalar potential
and the system of algebraic equations for dete~~g
the numbers 11, , , IN
(12)
The functions xk(t) =x(Sk, r) are determined from the system of equations (IO), considered at the nodes Sk. k = 1,2, . . . , N of the quadrature formula. Therefore, the initial family of discrete models is described by the system of equations (X0)-( 12). It is completely defined by the class of functions fix, JJ), by the class of functions e(x) and by the cIass of quadrature formulas, whose coefficients are positive. We formulate one of the fundamental algorithmic characteristics of the family of discrete methods constructed.
Particles-in-a-celland stream tube methods
183
Lemma 2 Any discrete model of the initial family is conservative. In other words, the functions x(u, r), cp(x) and p(x) satisfy the equations
(
Uk2 0)
I--
C2
-‘I*
=-.Lrp(s(t))+(
1
c P (2) ax= c.
J D
Y h-i
adkTkhl,
1 -yq
-% ,
k=l, 2,. . . , N.
(13)
(14)
Proof: Equation (13) is a consequence of the system of equations (10). Equation (14) follows from the definition of the function p(x) and from the formula
D
The lemma is proved. We will give some examples of the stream tube method, based on the use of various quadrature formulas. Example 1. For the trapeziodal formula the numbers ok, k = 1,2, . . . , N, are determined
from the equations a2=.
. . =aN+
-1,
h,=a I N.
ai=aN=‘/2,
This quadrature formula is of second-order approximation in the quantity 1/N. The stream tube method corresponding to the trapezoidal formula was used for the numerical determination of the function p(x) in [4]. Example 2. We suppose that the number N is odd. Then we can calculate p(x) by Simpson’s formula. The numbers ok, k = 1,2, . . . , N, are determined from the equations
al='aN='/5, a5-5=.
&“a&‘.
. .
=aN-_l=‘i5,
. . =%-2=2/S.
This quadrature formula has fourth-order approximation in the quantity 1/N Example 3 (the Gauss formula). We can calculate the function p(x) and determine the numbers 1k by using a quadrature formula of a higher degree of accuracy. In this case the nodes of the quadrature formula must be taken at the roots of a Legendre polynomial of degree N. The coefficients ark of .the Gauss formula are tabulated in [S] .
184
Yu. I. Mokin
5. Discrete models based on the particle-in-cell method are obtained from Eqs. (lo)-( 12) by replacing the integrals with respect to the variable t by a suitably chosen quadrature formula. The details of this substitution and the general form of the resulting equations are explained in [3].
2. The sufficient conditions for the particle-in-cell method to converge For a whole series of modern physical and technical problems an urgent problem is the calculation of the self electromagnetic fields of charged particles moving in some domain D of space. The fields are calculated by Maxwell’s equations in which the functions p(x) and f(x) represent respectively the charge and current per unit volume. Therefore, the self electromagnetic fields of the particles will be uniquely determined from Maxwell’s system of equations if with each point of the domain D there is associated some unit of volume and if the method of calculating the charge and current in this volume along the trajectories of the charged particles is indicated. As shown in [3] , the question of the choice of the function g(x, u) in Eqs. (9), (10) is also closely connected with the choice of the closeness of the approximate methods of describing the motion of the charged particles to the method based on the kinetic equations of Vlasov. Fixing the function g(x, JJ), and consequently the size and shape of the unit volume, we obtain some degree of closeness of these methods to each other. To calculate the functions p(x) and f(x) we use a discrete method consisting of the subdivision of the whole flow of charged particles into a small number of suitably chosen parts. The number of parts in this method, called the particles-in-cell method, is at the present time chosen from considerations of the technical possibilities of modern computen. The particle-in-cell method is used to calculate the charge density and current density as a component element of the algorithm for the solution of specific problems, in which their solvability is known in advance or is assumed. Also, since the particle-in-cell method is not intended for the isolation and subsequent calculation of one or all of the possible solutions of the problem considered, therefore when studying the qualitative features of this method it is necessary to assume not only the solvability but also the unique solvability of the problem. This means that the convergence of algorithms based on the particle-in-ceil method must be studied on the assumption of the unique solvability of the corresponding problems, and also on the assumption that the function f(x, v) is fmed. 1. We will briefly explain the main features of the particle-in-cell method. By the proof of some number of prior estimates it will be shown that the sequence of solutions of any model of the initial family is compact in the corresponding functional space. From this face and from ArzelB’s theorem the corollary that some subsequence of solutions converges to the solution of problem (7), (8) is obtained. From the unique solvability of this problem the conclusion will be drawn that the whole sequence of solutions of the discrete models converges to the solution of problem (7), (8). 2. The convergence of the solutions of the discrete methods, based on the particle-m-cell method and the stream tube method, is proved very similarly. Therefore, it is natural for us to confine ourselves to the proof of convergence of only the latter of these methods. Moreover, in order to simplify the technical aspect of the discussion, we will prove the convergence of the solution of problem (1 O)-( 12) to the solution of problem (8~( 10).
185
Particles-in-a-ceiland stream tube methods
We formulate the concept of the solution of the system of equations (1 O)-( 12). For this we fm the number iV and consider the vector x(t) = (xl (I), . . . , x&)) and the vector I= (Zl, . , . , IN). D$%niti~~~.The two vectors x(t) and I will be said to be the solution of system (lo)-{ 12) if the following conditions are satisfied: 1) the components xk(t), k = 1,2, . . . , IV, of the vector x(t) are twice continuously differentiable with respect to the variable t, 0 G t < Tk; moreover, xk (t) ED for 0 < t < Tk, and zA(TJ&, k-&Z,. . . , N; 2) Constants Zo and To exist such that
3) the vectors x(t) and Z satisfy the system of equations (IO)-(12). 3. To prove the compactness of the family of solutions of problem (lo)-( 12) we present some additional reasoning. Let the vectors x(t) and f be a solution of the system of equations (10)-(12). We construct from these vectors the function E (5) =-grad cp(5) , ZED, and note that the smoothness of ;Cand Lemma 1 imply the inclusion J!?(X)E O(D). We determine the function x(N)@, t) as the solution of the system of ordinary differential equations
+4(x),
f-z?,
(l-p
pm?2
.-‘A
)
with initial conditions z (s, 0) --s,(s) =I’, u (s, 0) =u,, (s) . It is obvious that the function x(~)(s, r) is uniquely determined and that at the points s = Sk the equations zor) (s, t) =x&i : k-l, 2,..., N are satisfied. We denote by n=n (a), UE& the outward normal to the boundary of the domain D. Lemma 3 We suppose that the inequalities
(h CT,) n) --0=-o, 1
Ial,paz,
k--l, 2,*.., N.
are satisfied uniformly with respect to N Then: 1) uniformly in N the function E(x) has a bounded gradient at points of the set 0; 2) a function
g(W)(s, t(N)(s) ) EC;
P’ (s) , OeGz,
exists such that
a9
(s, t) ED, Ott dN’
(S),
Yu. I. Mokin
186
3) the function tCN)(s) ELip [0, al ; can be chosen independently of N
the Lipschitz constant of the function P)(s)
Roof: Statement 1) of the lemma follows from the inclusion fix) E P(R2)
and the estimate
We prove statement 2). The uniform boundedness of the gradient of the function E(x), the boundedness of the numbers Tk. k = 1,2, . . . , N, and known results of the theory of ordinary differential equations imply the existence of a constant A4such that
1dN) (s, t) -dN) (so, t) I+ 1dN) (s, t) -dN) (so, t) 1,cMI s-so I,
(1%
s, SoE[O, al. We consider the points s = Sk, s = Sk+1 Sk < S < Sk+ 1. we put Zh ( Tk) -q@.
and prove statement 2) of the lemma for the case where
In our case the boundary of the domain D in some &neighbourhood of each point or r==@(s), (rg, zo) of it can be represented in one of two ways: z=P(r) , jr-r0 I (8 the function P being twice continuously differentiable. For definiteness we will Ia-GI+ z-P(r), Ir-rh(Tr) 1~6 holdsinaS-neighbourhood consider that the representation of the point ok. In this neighbourhood we pass to the new coordinate g=r, q-z-P(t) , Ir--r& (T,,) I (6 and note that for points (r, z) situated within Z the inequality 9 < 0 holds. It is easy to see that in this neighbourhood of the point Uk the normal n is a function of the variable r and that along the particle path Q(S,t), &j, t) the following differential equation is satisfied: -“dI,- Vt.
$-(i+p”)‘b(o’“‘(i,t),n),
We suppose that n(s, t) < 0 for T,G t< T,+min ( v. / 8M, 6 I c) . In other words we assume that in this interval of variation of the variable c the point x(N)(s, t) will be situated inside the domain D.Then In(rtN’(s,
Tk))-n(r(N)(~,
t)) IGf(t-T,).
Consequently, from inequality (15) considered for I = Tk, we have I (dN)(s,
-(u(“)(s, GW(
Tk), n(rcN’(s, T,)))-(dN’(s, t)), n(rtN)(s, T,)))
t), n) I<1 (dN)(s, Tn)
I+1 (dN)(s, t),
n--n(r(“)(S,
t-Tk) eve / 4.
This and (15) imply the estimate
1(dN)(sh, T,), ~(r@‘A))-WN%
t), n) i
TJ))
I
Partkle&-a-cell and stream tube methods
2M 1s~+~-s~]
Therefore, choosing N such that
Tj’(s, t)=(P(s,
/ 4,
187
we finally obtain
t), n) (l+P)?QV,/2.
Therefore, the function ~(s, t) satisfies the relations
tl’avo 12,
tl
(s, Th)--Eo--QM1S-Sk I,
OG--T,Gmin(v,/8M,
&o>O,
(16)
S/c).
Finally, this implies the estimate
(s,+)
q(S,t)2q(s,TJ+$min >+min(s,$) if
2Jqsn+i-s,I~(V,/2)min(u,/8M,
-2Mls-s,lM, Thi s contradiction proves the existence of
6/c).
the function P)(s). We turn to the proof of statement 3) of the lemma. From (16) we have the estimate
T
It(N) (S)-ttN) (Sk) laws-&I.
Consequently,
It’N’(S)-t(N)(Slr) I
GE IS-srl. UO
The lemma is proved. onto the set of points x E R2 with cp(z) We continue the function E (5) =-grad preservation of the norm O(R2). For the newly obtained function we retain the previous notation E(x). We solve the Cauchy problem with the function E(x). Thereby we continue the function x(~)(s, r) to all values of the parameter C,0 < t < To : dP -=--E(x), a.!
dX -=u, at
5
b, 0) 2x0 b) ,
u (s, 0) =uc
(s).
For this function we retain the notation _@)(s, f). Corollary 1. Let the assumptions of Lemma 3 be satisfied. Then the family of functions x(~)(s, t) is compact in the metric Cp ( [ 0, a] X [ 0, To] ) , and the family of functions t(N)(~) is compact in the metric CIO, a].
Proof: The assertions 1) and 3) of Lemma 3 imply the uniform boundedness and ArzelP’s theorem implies the equicontinuity of the family of functions x (N)(s, r) and @)(s). compactness of these sequences.
188
metric
Yu. I. Mokin
We extract from the sequence of functions .#)(s, C”J( [O, a]X[O, To]) and put
t) a subsequence convergent in the
Iim P”) (8, t) =5(s,
t).
Nk -COD
We wiIl consider that the subsequence t @‘k)(s) also converges to some function t(s). In order not to introduce new notation due to the passage to subsequences, we wilI assume that the sequences x@)(s, t) and t(m(s) themselves already converge. Lem??U 4 Let the assumptions of Lemma 3 be satisfied. Then x(s, t) ED for 0 < t < t(s). Rooof: We assume that the statement of the lemma is not true. Then a T, 0 < r < r(s) such that x(s, r) E Z. In view of the uniform convergence of the sequence S)(s, t) to the function x(s, t) for any e > 0 we can find an No(e) such that ]z(“)(s,t)--~(s,t) O
I+IrAN1(s, t)-u(s,t)
NS%(e),
]
OGtGTo.
This implies that the point x(~)(s, t) lies in an e-neighbourhood of the point x(s, r) of the boundary C. As in the proof of statement 2) of Lemma 3, in a S-neighbourhood of the point x(s, 7) we consider the change of variables E = r, 7)=z - fir). We put rr(N)(s, t) = n(t(N)(s, t)). The estimate
1(U(N)(Y, t) , rQN)(s, t) i G I W”)(s,
a
e,
4,
4 >I
n(N)(&N-~b(~, 9, 4% t) >I
+I (u(s, t), n(s; t)-(u(s, implies that for O
w,
(uo/8M, 6/2)
T), n(s, IT)) ~<.2Me+2M~t-zl
and
I t--z1 +nin
(v&M,
612~)
the
(u(N) (s, t) , ?zCN) (s, t) ) >u,/4. Moreover, for all values of the variable t of the interval indicated the point x(~)(s, t) does not go outside the G-neighbourhood of the point x(s, 7). We now show that I ttN) (s) --z 1Gmin For this we consider the two cases: r > fiNI and T Q dN)(s). Both cases (ud4M, 6/2c). are treated identically, therefore we will only discuss the first one. For t--z-min (z%/&f, 6/k) we have rl(N’(t)C~(N)(r)-%min
(2,$)<2Me-+min
($$,a,
if eQ ( uo/8M) min (u&M, 6/2c). We have therefore proved that I ttN).( s) -T I Gmin (v,/4M, 6/2c) . This in turn implies that
3
It(“)(s)-~lGIq(~)(s,
T) 162Me.
Particles-in-o-celiand stream tube methods
189
Therefore, we finally obtain
This estimate implies that r = r(s). This contradiction proves the lemma. Let us consider the function
Q (5) = ( QI (z) , . . . , Qr (5) , . , . , QN (s) ) ,
QiJz)=j~~Vk(f))d~~
where
x=Rz.
0
From the method of co~t~ct~g
Q(N)(s, x) =
x(~‘(s, t) it follows that the function
s g(x,
xcN)(s, t) ) dt,
O~sez,
x=R,,
0
assumes the values Q&x) at the nodes of the quadrature formula s = Sk, k = 1,2, . . . , A? CoroZZav 2. Let the assumptions of Lemma 4 be satisfied. Then QtN)(s, 5) OcaGf, and
EC@([0,a]XR,)
The convergence is uniform for s E [O, a] for any fared x E R2. Z%oo$ The inclusion zfN) (8, t) EC'(IO, a]X [ 0,To]) follows from the inchtsion E(x) E Cl(D). The smoothness of the function M)(s) follows from the smoothness of the boundary of the domain D and the smoothness of x(N)@, t). The limiting values of these sequences x(s, t) and f(s) are also smooth functions. This and Lemma 1 imply the smoothness of Q(s, x). The uniform convergence of QtN)(s, x) to Qfs, x) is obtained from the uniform convergence of #)(s, t) to x(s, f) and r(~){s) to f(s). The statement is proved. We denote by fiNI the piecewise-constant complement of the mesh function CulZll..., oh+. The boundedness of the numbers (~1, . . . , aN implies that the family of functions fiNI is uniformly bounded in the metric L(0, a) and for any measurable set E 5 [0, a] the knit
is uniform with respect to the index N. In this formula &5’) denotes the Lebesgue measure of the set E. Froin the above we obtain that the sequence of functions I(N)(s) is weakly compact in the space L(0, a) (see [6] ). From this sequence of func~ons we extract a weakly convergent subsequence. In order not to introduce new notation due to passing to a subsequence, we will consider that the sequence fiNI itself already converges to some function Z(s) E L(0, a).
,
190
Yu. I. Mohn
Lemma 5
Let the assumptions of Lemma 4 be satisfied. Then the sequence of functions
converges to the function
at each point x ED. Proof: We denote by o( 1) terms tending to zero-as N + ~0.Lemma 4 implies the equation t(9)
N
p’N’(z)-~I(N)(s,,)
s
g(z,s(s,t))dth,+o(1).
0
h-1
Let t(N)(s) be the piecewiseconstant completion of the mesh function r(sl), . . . , f(sN) on the set of points s E [0, a], and let FtN)(s, t) be for every fured t f [0, TO] the piecewise-constant completion of the function x(sl , t), . . . , X(SN, r) from the set of nodes of the quadrature formula on the segment [0, a]. We have the formula W)(s)
PcN)
(z) =
5I’N’(s)[ 0
5 g(z, ZfN)(s, t)) dt]
The uniform boundedness of the sequence fiNI with respect to the variable s imply the equation
p”‘(+~l(ll(a)
ds + o (1).
0
and the smoothness of the function Q(s, x)
[ Ij’lg(s,s(s,f))dt]da+o(l).
0
0
Taking into account the weak convergence of the sequence I(N)(s) to the function I(s), we pass in this equation to the limit as N + 00. As a result we obtain the formula
p(z-)=j
I(s) [ ~‘g(l,s(a,t))dt]ds. 0
0
The lemma is proved. 4. We will now formulate and prove the fundamental result of the section.
Let the vectors x(r), I be the solution of the discrete model of the initial family, based either on the particle-in-cell method, or on the stream tube method. Then:
Particles-in-a-celland stream tube methods
191
1) the sequences of functions x cN)(s, t) and t(N)(s) converge as N + 00,uniformly on the set [O,x] X [0, To], to the functions x(s, t) and t(s) respectively; moreover, x(s, t) ED for 0 < t < t(s), and x(s, t(s)) f C; 2) the sequence of functions P(~)(X) and In@) converge uniformly on D to some functions p(x) and g(x), the function cp(x) satisfying the differential equation
AT=-4np
jl(s)[ tj’ig(x,x(s, t))dt]ds
(xl =-4n
0
0
and the boundary conditions
3) let Ljjv)==rot Pk the sequence of functions
and the vectors Pk (2, y, t) - (v, 164 (3,
gi), 0, vr, A& (2,
pi) 1;
HcN) (x) bi
0
converges as N-t 00,uniformly in 5, to the function
where
L(x, x(s, t~)=rot(v~(s,t)G~(x, ~(8, t)), 0, v~(s,t)Go(x,ds, 0)); 4) the following equations are satisfied: dpldt=-
e grad cp(CC)- +
&,‘dt=u(s,
t) t
uXH,
OGsGa, Oast(s).
Roofi We will prove the theorem for discrete models based on the stream tube method. The proof of the theorem is similar in the general case, Moreover, the assumed unique solvability of problem (7) implies that it is sufficient to establish ah the statements of the theorem for some subsequence of indexes Ni, i = 1,2, . . . . To prove statement 1) of the theorem we choose a sequence of indexes Ni such that the corresponding sequences of functions xtN)(s, r) and @)(s) converge utiormly on the set [0, a] X [0,7’o]. In the proof of the theorem we will consider that this sequence of indexes is fured. Then Lemmas 2-4 imply statement 1) of the theorem. From Lemma 5 there follows the convergence, uniform on 5, of the sequence of functions pN(x) to the function p(x). From this and from well-known results of the theory of
192
I. F. Potapenkoand K A. Chuyanov
elliptic equations, we conclude that rp(X’*,Q, (x) in 6. Also, since &) E @(is> and &p(x) /an-o for x E I’. This reasoning established part 2) of the theorem. Part 3) is proved in the same way as Lemma 5. The validity of statement 4) of the theorem is obvious. The author considers it his duty to thank A. A. Samarskii for suggesting the problem, for his interest and for discussing the results. Translated by J. Berry.
REFERENCES 1.
SAMARSKII, A. A. Theory of dtffctence schemes CTeoriyaraznostnykh&hem), “Nat&a”, Moscow, 1977.
2.
MOKIN, Yu. I. On two models of the steady motion of charged particles in a vacuum diode. Nutem. sb.,
106 (148),2,234-264,1978. 3.
MOKIN, Yu, I. Some methods for the approbate description of the steady Sow of electrons in a vacuum diode. Zh. vj%isf Ma. mat. Fit., l&S, 1186-1195,1978.
4.
GOLOVIN, G. T. et ui. Formulation of problems of the steady motion of heavy-current beams of reIatIvistIc electrons in a cylindrical vacuum diode. In: Numcrlcol methods bf rmrrhmwticul phydcr (Chid. metody matem. fia.), 4-11, NI VTs MGU, Moscow, 1978.
5.
KRYLOV, V. 1.27re approximateewluationof integr& (PrIblIzhennoe vychislenie integralov), “Nauka”, Moscow, 1967.
6.
DUNFORD, N. and SCHWARTZ, J. T. Lineuroperators (LIneinye operatory), Part 1, “Mir”, Moscow, 1964.
U.S.S.R. Comput. Maths. Muth. Phys. Vol. 19, pp. 192- 198 6 Pergamon Press Ltd. 1980. PrInted in Great Britain.
COMPLETELY CONSERVATIVE DIFFERENCE SCHEMES FOR A SYSTEM OF LANDAU EQUATIONS* 1. F. ~TAP~~O
and V. A. C~~OV Moscow
(Received 12 January 1978 ; revised 24 April 1978)
DIFFERENCE schemes are constructed for which the number of particles and the energy in the system of Landau kinetic equations is conserved. Examples are given of the calculation of relaxation in an infinite medium and of the outflow of plasma from a magnetic mirror. In this paper we consider the probiem of ratify-homogeneous relaxation in a system of charged particles isotropic in respect of velocities. In view of the great difference between the masses of ions and electrons the later stages of the relaxation of the energies of these particles proceeds very slowly. In order to avoid an accumulation of errors in the numerical solution it is desirable to use completely conservative difference schemes. The concept of a completely conservative difference scheme for problems of gasdynamics was introduced in [ 1,2] . The property of complete conservativeness for a system of Landau equations, consists of the fact that the difference equation, written in the form of the law of conservation of the number of particles, *Zh. vjkhisl. Mat. mat. Fiz., 19,2,458-463,
1979.