CC BY
2UDC 517.95:519.6
HIGH-ACCURACY DIFFERENCE SCHEMES FOR THE EQUATION OF ELECTRON WAVES IN COLD PLASMA IN AN EXTERNAL MAGNETIC FIELD
M.M. KAZIMBETOVA
Karakalpak State University named after Berdakh, Nukus, Uzbekistan
Abstract. Difference schemes of the finite difference method of the fourth-order approximation are constructed and studied for all variables for a high-order Sobolev-type equation. The first boundary value problem for the equation of electron waves in cold plasma in an external magnetic field is considered. The approximation is conducted in two stages; at the first stage, the method of straight lines is used, i.e., only spatial variables are approximated. The resulting system of high-dimensional ordinary differential equations is approximated by the finite difference method. To study stability, the method of energy inequalities is used. Theorems for the convergence and accuracy of the constructed difference schemes are proven for the sufficiently smooth solution to the original initial boundary value problem.
Keywords: electron wave equation, Sobolev type equation, difference schemes, finite difference method, stability, convergence, accuracy.
1. Introduction. As is known, high-order Sobolev-type equations are used for mathematical modeling of applied problems in geophysics, oceanology, semiconductor physics, atmospheric physics, and many other fields [1]-[3]. These equations are non-classical and are mainly solved by numerical methods. To numerically solve this type of problem, difference methods or finite element methods based on cubic B- splines were used [4], [5]. At present, the greatest interest is in the construction and study of difference schemes of high accuracy. In particular, in [2], [3] such equations are solved by the finite difference method on quasi-uniform grids. Similar results were obtained in [6], [7] using the finite difference method of the first and second orders of accuracy. In [8]—[10], initial boundary value problems for fourth-order Sobolev-type equations in classes of non-smooth solutions were considered; based on the integro-interpolation method, multiparameter difference schemes of high accuracy were constructed and studied.
This study is devoted to the construction of high-accuracy difference schemes for the first initial boundary value problem for the equation of electron waves in cold plasma in an external magnetic field. Difference schemes of high accuracy are constructed based on the finite difference method for both variables. A priori estimates for solving difference schemes were obtained by the A.A. Samarskii method of energy inequalities. The convergence and accuracy of difference schemes are proven for sufficient smoothness of the solution to the original initial boundary value problem. Algorithms for the numerical implementation of the constructed difference schemes are proposed.
2. Statement of the problem. In domain
QT = x,t):x = (x1,x2,x3) eQ = [0 < xa < la, a = 1,2,3], t e [0,T]}
we consider the initial boundary value problem for the equation of electron waves in cold plasma in an external magnetic field [3]:
a
dt2
a 2 2 dt2 P Be
Л2 u +
f Л2
a 2
at2 Pe
a22
—T + ®B
at2 Be
a2u
ax 3
= f (x, t), (1)
J JVy
k _
= u0,k, k = 0,3, x gQ , (2)
t=0
a , ч
-rU( x, t)
atk
u( X, t )| r = j(t ), t e[0,T], (3)
where A2 =d / dx^ + a2/ ax 2, o is the Langmuir frequency for electrons, (( is the
Pe e
Larmor frequency of electrons, u = ( x, t ) is the generalized potential of electric field.
3. Discretization in space. For further research, we rewrite equation (1) in the following form
a4u a2 u ak —
Lo —u + L — + L2M = f(x, t), —ru(x,0) = uok, k = 0,3, u ( x, t)| r =6(t), (4)
at4 at2 atk r
22 2 222222
where Lo = A2 +8 / 8x3, L^ = (q Lo, L2 = ( a / ax3 , (q = o +( ,
^e e
2 2 2 ( = Co co^, .
Pe e
Let us introduce subspace H h c H that approximates Hilbert space H with the corresponding scalar product and norm. We introduce in Q a grid uniform in each direction (h = C X C X C , where(h = k = 'ahaa = ^ha = lJNa \ «= I2,3.
O
Here (h = (h + Y h ■ We define space H h = W^(Ch ) with norm
il||2 _
N1 N 2 N 3 r
1 222h1h2h3 [(vx )2 + (vx )2 + (vx )2 < M, where constant M does not depend
V i1 i2 i3
on h1, h2, h3. Here v = v(i1h1, i2h2, i3h3 ),
vx = [v(i1h1, i2 h2, i3h3) - v((i1 - 1)h1, i2 h2, i3h3 )]/ h1, vx = [v(i1h1, i2h2, i3h3 ) - v(i1h1, (i2 - 1)h2, i3h3 )]/ h2 ,
v
, = [v(i1h1, i2h2 , i3h3 ) - v(i1h1, i2h2, (i3 - 1)h3 )]/ h3 ,
W2 Ph ) is the space of grid functions vanishing at the boundaries.
Approximating operators L , L and L by corresponding difference relations on the indicated grids, we obtain the Cauchy problem for a system of fourth-order ordinary differential equations:
I. DIS- + B f + Auh(t) = fh, ^(°)= ^Ah, k = 03, dt dt dtk
(5)
where D, B and A are linear constant operators from Hh ^ Hh, D = D > °,
B* = B > °, A* = A > 0, V t > °, uh = uh (t) e Hh, fh = fh (t) e Hh . Here,
D = A, B = PqA , 3
2 _ ^
A = P A3, A=XAa , Amuh =-uh,XX , m =1,2,3, (6)
'mm
a=1
Uh is the value of the function at fixed node X = (Z^, i2h2, i3h3 ),
0
2
uh,xx = (uh ((/1 + 1)h1, hh2, hh3 ) - 2uh (iA, i2h2, hh3 ) + uh ((/1 - 1)hl, i2h2, hh3 Ж h1 ,
2
uh,xx = (uh (/1h1, (/2 + 1)h2, /3h3 ) - 2uh (/1h1, /2h2, /3h3 ) + uh (/1h1, (/2 - 1)h2, /3h3 )) /h2 , uh,xx = (uh (/1A' Z2h2 , (/3 + 1)h3 ) - 2uh ((/1h1 , z2h2 , hh3 )) + uh (/1h1 , z2h2 , (/3 - 1)h3 )) / h3
II. Operators D, B and A approximate operators L0, L and L2 with
the second order, respectively, i.e., O(|h|2), |h| = ^JA + h2 + A .
III. 4. Discretization in time. We introduce uniform grid (r = {tn = nr, n = 0,1,...; r > 0} on segment [0,^] and replace the differential problem
with a difference scheme. Let y approximate ua . In [11], a difference scheme of the fourth-order of approximation was constructed for problem (5) with operators (6) on grid
(r
Dy-t-t + By7t + Ay = Ф, tn e (т, (7)
0 1 __2 __3 _
y = u0^ y = u0,n y = u0,2, y = u0,3 , (8)
where
D = D + (т2 /12)B, B = B + (r2 / 6)A, (9)
yrtrt = (yn+2 - 4yn+1 + 6yn - 4yn-1 + yn-2) / r4, y-t = (yn+1 - 2yn + yn-1) / r2,
yn = y(tn), yn±1 = y(tn ±r), yn±2 = y (tn ± 2r), (r2/6)a2/ / at2,
% = % + 0.5r[E - (r2 /12)D_1B]u0 2 + (r2 / 6)u0>3 + (r3 / 24)D-[/(0) - A^0], «0,2 = u0,2 + ru0,3 + (r2 / 2)D_1[/(0) - Bu0,2 - Au0,0] + +{тъ I A)D-\m-Bu,_2-Au^ = u03 + (3Г / 2)^-1[/(0) - Bw02 - Au0_0] + (5т2 / 4)/Г1[/(0) - Bu0_2 - Au0 J + +(3r2/4)D-1[/(0)-5«0i2-^i0].
IV. 5. Convergence of the scheme. The following theorem was proven in [11].
V. Theorem 1. Let D* = D > 0, B* = B > 0, A* = A > 0 and the stability condition be satisfied
D > (r4 / 4)A . (10)
VI. Then, the solution to the difference scheme (7), (8) with operators (9) converges to a smooth solution to the original problem (5) and the following accuracy estimate holds:
||y(tn) - u(tn)|| < O(r4), tn e(.
VII. Therefore, based on this theorem, we obtain the following result.
VIII. Theorem 2. Let D* = D > 0, B* = B > 0, A* = A > 0 and the stability condition (10) be satisfied. Then the solution to the difference scheme (7), (8) with operators (9) converges to a smooth solution to the original problem (1) - (3) or (4), (2), (3) and for its solution, the following accuracy estimate holds:
||y(x/, tn ) - u(x/, tn < O(|h|2 +A x e(h, tn e(r .
IX. 6. Schemes with weights. Consider a family of difference schemes with weights based on the difference scheme (7), (8) with operators (9):
DyTtTt + ByTt + Ay(CT1,CT2) = q>, tn .
(11)
Here y(G1'G2) = g1 y + (1 — G — G2 )y + G y, where g1, g2 are some constants of schemes with weights, the presence of which allows us to select various explicit and implicit schemes and adjust their accuracy in space.
Similar to [12], we study the stability and convergence of scheme (11) with initial conditions (8). To do this, we reduce (11) to canonical form. We perform the following transformation on scheme (11):
X. Dyn+2 - (4D - T B - tG,A)yn+1 + [6D - 2t2B + (1 - g - g2 )t4A]y
n+1
—(4D — t B — g2tA)yn—1 + Dyn—2 = . Let y = yn+2 in (12). Then (12) has the following form: XI.
where
.2D _4
(12)
B4 yn+4 + B3 yn+3 + B2 yn+2 + B^1 + B0 yn = T4pn,
B4 = D, B3 = -(4D - t2B - TG2 A), B2 = 6D - 2t2B + (1 - g - g2)T4A, B = -(4D - t2B - T4GA), B0 = D. (13)
Next, scheme (12) is written in the following canonical form:
XII.
My- + t2Ry. + T5Py „ + T*Qym + Ay = t4ç ,
(14)
where
M = t(2B4 + B3 — Bl — 2B0 ), R = 2B0 + °.5(^ + B3) + 2B4,
P = °.5(B—B ), Q = — (1/8)(B+B ), a = B4+B3+B2 + B1+B0 .
Hence, considering (13), we obtain
M = t5(G — G) A, R = T B + °.5T4(G + G) A, P = °.5t4(g — g)A, Q = D — (t2 / 4)B — (t4 / 8)(g + g)A, a = r4a.
Let G = G = G, then M = P = 0. Consequently, after elementary calculations, from (14) we obtain
Qym + Ry + Ay = (p, (15)
tt
where
0 = ,D-(r2/4)5-(r4/4)o-^, R = B + t2gA. According to Theorem 2 from [12, p. 276], there is an a priori estimate based on the initial data (^ = 0)
Y
n+1
<
Yr
if the following conditions are met:
ReM > 0, A> 0, R - 4Q-A> 0, A +16Q > 0.
Here
(16) (17)
Yn
A = (1/16)
yn + yn+1 + yn+2 + yn+3
A
+(1/16)
yn+3 + yn+2 - yn+1 - yn
R-4Q-A
+
+
.n+3 „ .n+2
y
y
yn+1 + yn
R-4Q-A
+ (1/16)
.n+3
y
yn+2 + yn+1
y
A+16Q
(18)
n
2
2
Let us check the fulfillment of conditions (17). The first condition Re M > 0 is satisfied since M = 0. The second condition A> 0 is satisfied since A > 0. The following condition R - 4Q - A > 0 will be satisfied if
4D + (1 - 2a)z4A < 2t2B, (19)
and the last condition A + 16Q > 0 will be satisfied if
16D + (1 - 4a)r4A > 4r2B. (20)
Conditions (19) and (20) will be satisfied if a < 1/4, D > (r4 / 8) A or
a< 1/4, D > (r4 / 8) A, (21)
which are conditions for the stability of scheme (11), (8). Thus, the following theorem is proven.
Theorem 3. Let D* = D > 0, B* = B > 0, A* = A > 0 and conditions (21) be satisfied. Then, to solve the difference scheme (11), (8), a priori estimate (16) is valid.
To prove the stability on the right-hand side of scheme (11), (8), we present it in the form of
an equivalent two-layer scheme in space H4 [12]:
Uyt+Uy = t,
where yt = L Ty , (t2 /2)y , (r/ 2)v + (r3 /8)yTt- \ (¡> = {<?, 0,0,0},
^ tt t tt tt j
□
и 0 0 0
0 0 0
R-20-A 0 0
0 R-20-A 0
0
0 A +16Q
□
r M + 0.5rA т( R - 4Q -A) 0 0.5(A + 16Q)^
-t(R-40-A) 0.5T(R- 40- A) 0.5T(R- 40- A) 0 0
-0.5r(R- 4Q - A) 0.5t(R-40- A) 0 ■0.5(A +160) 0 0 0.5(A +160)
Based on Theorem 4 from [12, p. 284], the following statement holds.
Theorem 4. Let D* = D > 0, B* = B > 0, A* = A > 0 and the following operator inequalities be satisfied:
ReM > 0, A> 0, R - 40-A> 0, A +160 > 0. (22)
Then, to solve the difference scheme (15), (8), the following a priori estimate is valid
Y
n+1
<
A
Yn
A
+
P
+
P
A
i +Z тЫ
k=1
where
Yn
A
is calculated according to (18).
Let us check the fulfillment of conditions (22). The first two conditions are satisfied since
M = 0 and A = A* > 0, and the others will be satisfied if (21) holds. Based on Theorems 3 and 4, the following result holds.
A-1
A-1
2
Theorem 5. Let D* = D > 0, B* = B > 0, A* = A > 0 and condition (21) be satisfied. Then, the solution to the difference scheme (15), (8) converges to a smooth solution to the original problem (1)-(3) and the following accuracy estimate holds:
||y(X, t„) - u(x,tn ^ < 0(h2 + r4), x e ah, tn . 7. Higher accuracy on spatial variables. If the solution to the original differential problem has the necessary smoothness in spatial variables, then difference operators of high order of approximation can be built. Difference operators with higher order can be obtained in various ways.
For example, operators of the difference scheme (15), (8) O, R and A are chosen in the following form ( p = 0 ):
2 i , i ? A
T
0 = D~ — ~ 4
- 3 h2
в -vY—A
m=1 12k
, R = B+T2ctX
h2
112k
-A
h2
A = Y A - Y ——A A
^^ m ^^ 1^7 m n
m,n= m^n
il2k
(23)
where Amy = —Amy. Consequently, difference operators O , R and A in (7) approximate differential operators of the fourth-order of approximation error, i.e. 0(|h| ), |h| = h2 + h22 + h2
8. Algorithms for implementing the scheme. Scheme (11) for (=(=( has the following form:
tyjut + Byjt + Ay() = p, tn e ®T,
where y() = ( y + (1 - 2()y + (y. For ( < 1/4 we obtain an explicit scheme (7), (8), which is directly implemented. Numerical calculations can be performed using high-accuracy scheme (15), (8) with operators (23).
9. Conclusions. A boundary value problem for the equation of electron waves in a cold plasma in an external magnetic field was considered in the article. Based on the finite difference method, parametric difference schemes of high-order accuracy in time were built and studied. The presence of parameters in the scheme allows for regularization of schemes to optimize the implementation algorithm and the accuracy of the scheme. The corresponding a priori estimates were obtained and, on their basis, theorems on the degree of convergence and the accuracy of the constructed algorithms were proved with sufficient smoothness of the solutions to the original differential problem.
m=1
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