Вычислительная математика и математическое...
TRANSCRIPT
-
I
2000
-
, 75- . . 20-
..
I
, 19{22 2000 .
-
519.6 00{01{1023
: - . I/ . ... | .: , 2000. | 256 .
, " ", - 75- .. 20- - .
, - .. . { , , , , - , . -, , .
- , .
This collection of works contains the reports presented to the Jubilee Conference "Nu-merical Mathematics and Mathematical Modeling" dedicated to the seventy fth birthdayof Academician G.I.Marchuk and to the twentieth anniversary of the Institute of NumericalMathematics.
Themes of the reports presented are associated with the scientic directions, in whichG.I.Marchuk achieved the most outstanding results: the methods for calculating nuclearreactors, numerical mathematics, models and methods in the physics of the atmosphereand ocean, mathematical models of immune processes, problems of environment protection,parallel computations and the architecture of computers. Most of reports presented in themain, the rst volume of the conference proceedings, are of a review character.
The conference proceedings are certainly interesting to the specialists in numerical math-ematics and mathematical modeling and also to the students and postgraduate students ofrelated specialization.
The publication is supported by the Russian Foundation for Basic Research, project00{01{10023
ISBN 5{201{08806{6c , 2000.
-
75- ..
8 2000 75 . 20 , .
.. . .
{ . 1961 .
, , , , - . 1979 . .
, .. , . , . 1975 . . ..
-
4
...
{ . 1981 . .. .
.. , . 1988 . .. .
, { , , , .. .
. { .. , { , , , . , , .
, " ", 75- .. 20- .
-
1
. . , . .
. .. " ",
, .. . - : , - , , - , , - . - .
1. -
. - , -, ., , . - , , , , -. - .
1 ( 00{15{96073, 96{01{00141).
-
6 . . , . .
, .
, , Meredes. : , - , , , "" .
, - , . - ("" -, , , , ..), . , .. { .. { - .
{ { , . , , - 2, , , - , , , , { .
.. - 1955 . [1]. - . 1958 " " [2], { [3]. , , . - .. .. " ", [4] (1971 .) - [4] (1981 .)
., 27 1954 ., .. . .. (1961 .).
1964 . -
2 .. , "... ".
-
7
"" . 1973 . . - - , . (.. ) , .
- . , .. . , .
2.
- ( ) . N(x;; v; t) G S2 R+ R+ (x;; v; t) [2{4, 37]
dN
dt+ v(x; v)N =
Z 10
ZS2W (x; 0; v; v0)v0N(x;0; v0; t)d0dv0 + q (2:1)
S G
N jS = 0; (nx;) < 0; x 2 S; (2:2)
S, -
N jt=0 = N0(x;; v): (2:3) dNdt { :
dN
dt=@N
@t+ v(;r)N; (2:4)
S2 { R3, nx { x S,0(;0), q { .
(2.1) -, , , .. , , . , - q.
-
8 . . , . .
(2.1) (2.4) , - x0 2 G, v
x = x0 +vt: (2:5)
, , (2.1) - , 7- (x;; v; t). - - . , , , (2.1), . : .
, ( ) . . ' =vN { , (x;; v) (2.1)
(;r)' + (x; v)' =Z 10
ZS2W (x; 0; v; v0)'(x;0; v0)d0dv0 + q (2:6)
(2.2) S G
'jS = 0; (nx;) < 0; x 2 S: (2:7)
C (2.6){(2.7)
(;r) + (x; v) =Z 10
ZS2W (x; 0; v0; v) (x;0; v0)dv0d0 + p; (2:6)
jS = 0; (nx;) > 0; x 2 S: (2:7) .. -
[2, 3, 5]. .
{ . . - ( { (3.12)) , . .
-, . - .. , ( 1960 .).
-
9
, : - . (2.1){(2.2), c , - :
@Nk@t
+ vk((;r)Nn + kNk) =Xi
vi
ZS2
W k;i(x; 0)viNi(x;0) d0 + q; (2:8)
1vk
@'k@t
+ (;r)'k + k'k =Xi
vi
ZS2
W k;i(x; 0)'i(x;0) d0 + q; (2:9)
k = 1; 2; : : : ;m, (2.3) (2.2), (2.7). -. (2.9) .
: k W
k;il .
(W k;il ; l = 0; 1; : : : { viWk;i(x; 0)
.)
- - - . - { "", "" "", - , .
. - , [15, 16].
L' = (;r)'+ (x)'; (2:10)
'0 = S' =14
ZS2'(x;0)d0: (2:11)
(2.6)
L' = (x)S'+ q: (2:12)
x = x0 +; x0 2 G -,
d'
d+ (x)' = (x)S' + q: (2:13)
-
10 . . , . .
u(x;) =12['(x;) + '(x;)]; (2:14)
(x) 6= 0 q(x;) = q(x;) [16]
[l(x)(;r)]2u+ u = c(x)Su+ q
(2:15)
S:u l(;r)ujS = 0; (2:16)
l(x) = 1(x), c(x) = (x)=(x) 1. (2.15){(2.16) x0 +
ld
d
2u+ u = c(x)Su+
q
; (2:17)
u l dudj=1 = 0; x1 = x0 +1 2 S; (nx1 ;) < 0; (2:18)
u+ ldu
dj=2 = 0; x2 = x0 +2 2 S; (nx2 ;) > 0: (2:19)
< u; v >=ZG
ZS2uvdxd;
[u; v] =ZS
ZS2j(; n)juvdds+ < l(;r)u; (;r)v > + < u; v > (2:20)
[u] = [u; u]1=2;
u,
G(v) = [v]2 < cSv; v > 2 < v; q= >; (2:21) (2.15)-(2.16) [16].
, , Q
q = fS': (2:22)
(2.7) 1 -
L'+ (x)S' + fS' = ': (2:23) 1 = 0. - kef ,
L'+ (x)S' + 1kef
' = 0: (2:24)
-
11
-
1v
@'
@t+ L' = (x)S' + C + (1 )fS';
(2.25)
dC
dt= C + fS';
C = C(x; t), > 0. -
L'+ (x)S' + C + (1 )fS' = 1v
';
(2.26)
C + fS' = C:
C, . (2.23), (2.24), (2.26) (2.7). (2.26), (2.7) - : -, , = , Re C0 C0 > 0 - [40].
"-" . , - , - '0 = S', , , (2.12), (2.7),
divDr'0 + (x)'0 = q0; (2:27)
'0 + 2D@'0@n
jS = 0; (2:28)
D = 1=3(x); (x) = (x) (x); q0 = Sq: . -
, . - -, ,.. -.
Ip('k) =< 'k; p > (2:29)
-
12 . . , . .
'k (2.6)-(2.7) q = qk, - p 'k . C p (2:6) (2:7)3 - [5]
Ip('k) =< qk; p >; k = 1; 2; : : : ;m; (2:30)
qk p -
Ip('k) = < (L)'k; p >; k = 1; 2; : : : ;m (2:31)( { (3.7) (3.10)). - p, . , p, , Ip('k) - (2.30), (2.6)-(2.7).
- . :
@'
@t= a2'+ V (x)' + f(x; t) (2:32)
{ ( = 1), ( =i)( ),
@2'
@t2= a2'+ b(x)' + f(x; t) (2:33)
{ ( ),
'+ (x)' = f(x) (2:34)
{ ( ) { {
1v
@'
@t+ (;r)'+ (x)' =
4(x)
ZS2'(x;0)d0 + f(x; t;) (2:35)
G (0; T ) S2 R4 S2 (x; t;). (2.35) G - S (.(2.2))
'jS = 0; (; nx) < 0; x 2 S: (2:36)
3 .
-
13
(2.35) (2.32), (2.33),
(;r)'+ (x)' = 4(x)
ZS2'(x;0)d0 + f(x;) (2:37)
{ (2.34). , - (2.37) (2.36) [4, 37]
'0(x) = ZG(y) exp
Z xy(s)ds
'0(y)jx yj2 dy (2:38)
'0 = S'. (2.38)Z xy(s)ds =
Z 10[ty + (1 t)x]dt:
(2.38), jx yj2 x = y, jx yj1. - (2.38).
. (2.37)
@'
@r+1 2r
@'
@+ (r)' = Q(r) (2:39)
(2.36)
'(;R) = 0; 1 0 (2:40)
r = r0 , r20 r2(1 2) = 0, r = r0 .
. r < r0, jj 1:@'
@r= rM(r0)p
r20 r2(1 2)+O(j ln jjj); (2:41)
@'
@=
r2M(r0)pr20 r2(1 2)
+O(1); (2:42)
r > r0; r0; >p1 (r0=r)2:
@'
@r= 2rM(r0)p
r20 r2(1 2)+O(j ln jjj); (2:45)
@'
@=
2r2M(r0)pr20 r2(1 2)
+O(1); (2:46)
M(r0) = '(r0; 0)[](r0) [Q](r0); [f ](r0) = f(r0 + 0) f(r0 0). (2.41){(2.46) '0(r) -
:'0(r) 12M(r0) ln jr r0j+O(1); r ! r0: (2:47)
1952 [17] (1968). - . .. [39]. (2.47) .. 1949 .
- .. [16], { [15].
3. -
, , .. - [2-5].
H { e - < '; > k'k =< ';' >1=2. L 4 D(L), H.
L' = q; ' 2 D(L) (3:1) q 2 H.
L L
< L'; >=< ';L >; ' 2 D(L); 2 D(L): (3:2)
(3.2) . 2 H, ' 2< L'; > H. ( ) g 2 H ,
< L'; >=< '; g >; ' 2 H: (3:3)
4 , -, , - -, G X = (x1; x2; : : : ; xn) , D(L). H L2(G).
-
15
g L , L =g, D(L) L, (3.3) , (3.2).
L , f(';L'); ' 2 D(L)g HH, .. 'k ! ' L'k ! ; k !1 H , ' 2 D(L) L' = .
L L L.
L . L , D(L) H. D(L) H, -
L = L; L = L: (3:4)
' (3.1)
Ip(') =< '; p >; p 2 H: (3:5) , (3.1)
L p = p; 2 D(L): (3:6)
, p D(L); . (3.6), (3.5) (3.2) - Ip(')
Ip(') =< q; p > : (3:7)
, L () ' H
L' = (X X0); (3:8) (XX0) { - , X0. X0. (3.7)
Ip(') = p(X0): (3:9)
. L L, Ip(')
p
Ip(') = < p; (L)' > : (3:10)
5 ,
L = ; 2 D(L): (3:11)
5 .
-
16 . . , . .
=< (L)'; >< '; >
: (3:12)
- , .. (. [6{10]) .. ([38]) () - , [18] .. .. .
D { , H, ' 2 D H,
f ! A' ; 2 D(A')g (3:13) D(A'), D. , , D(A') - D, D(A') D(A') H. ()
'! A'' = A' j ='; ' 2 D; (3:14) D.
1 [18]. fA'; ' 2 Dg fB'; ' 2 Dg , A' B',
A'' = B''; ' 2 D: (3:15)
: A' A', B' A', A' B', A' C', A' B' B' C'. (4.1) .
1.
f ! '0 ; ' 2 Dg; f ! (' )0; ' 2 Dg; f ! '2 0; ' 2 Dg
,'0 (' )0 '
2 0:
L(') { () D(L), H. - L' L(').
2 [18]. L' L(') - f ! L' ; ' 2 D(L)g , D(L') L' D(L) - (.(3.2))
< L('); >=< ';L' >; 2 D(L'); ' 2 D(L): (3:16) .
L { , - L L, '.
-
17
L' L(') ( , fL'; ' 2 D(L)g).
L' - L(')
L(') = (L')'; ' 2 D(L); (3:17)
.. L(') () ' ! L' ' - ! L' ' 2 D(L).
, L(')
L(') = A(')'; ' 2 D(L); (3:18) '! A(') { () D(L), H, ' 2 D(L) ! A(') , D(A(')) D(A(')) D(L). - L' A
('),
L' A('): (3:19), . [18]. L(') -
L', , (3.18); (3.19).
[6{10, 38] - L(') L(').
A0('); A0(') =Z 10L0(t')dt; ' 2 D(L) (3:20)
, L(') L(0) = 0. [38]
L(') =Z 10L0(t')t'dt = A0(')'; ' 2 D(L);
L(') (3.18) , (.(3.19)), - A0(') fL'; ' 2 D(L)g,
L' A0('): (3:21) 2(. 1). L(') = ''0 (3.19) (3.20)
L' = '
2 0:
3. L(') = 'k(@')m,
L' =k
k +m'k1(@)m +
m
k +m(1)jj@'k@'m1:
-
18 . . , . .
(3.18) [11{13]
L(') =Xjjm
A(x; ')@'(x); (3:22)
'! A(x; ') { () - D(L), H.
A(x; ') =
Xjjm
A(x; ')@;
(3.22) (3.18)
L(') = A(x; ')':
' 2 D(L) ( )
~A(x; ') =Xjjm
(1)jj@[A(x; ') ]; ' 2 D(L); 2 D(A(')): (3:23)
A(')' = 0; ' 2 D(L); (3:24)
~A(x; ') = 0; 2 D(A(')); (3:25)
' (3.24). J('; ;x); = 1; 2; : : : ; n, (3.24) m 1
J('; ;x) =m1Xq+k=0
Xi;j
(1)k@1 : : : @kA1:::k1:::q (x; ') @1 : : : @q'
; (3:26)
{ (3.25). 4. m = 1. (3.24) (3.25)
A0(x; ')' +nX=1
A(x; ')@'
@x= 0;
A0(x; ') nX=1
@
@x
A(x; ')
= 0;
(3.26) {J('; ;x) = A(x; ') '; = 1; 2; : : : ; n: (3:27)
-
19
~A(x; ') fA(x; '); ' 2 D(L)g. ~A(x; ') [10{12] A(x; '), L(').
[10{12] (3.24), A; '(x) (x) q s, s r r q-. (3.26) r r- .
3 [18]. L('), ' 2 D(L) (3.18) - (-, ), - (--, ) .
5.
@nA(x; ')@
'o; ' 2 C2jj( G)
, A(x; ') G C , G{ Rn.
6. ZGK'(x); '(y);x; y
'(y)dy
L2(G), G { K { -
K(X;Y ;x; y) = K(Y;X ; y; x); jK(X;Y ;x; y)j M:
7 [10].
L(') = sin' =sin''
'
, ' 2 L2(a; b) ! sin'' L2(a; b).
8. - z- [44]
i@
@t= H(x; ) ; (x) = 1(x1) 2(x2) : : : z(xz);
H(x; ) { L2(R3z),
H(x; ) =zXk=1
hh2
2mk ze
2
jxkj + e2Xj 6=k
Z j j(xj)j2jxk xj jdxj
i:
k; k = 1; 2; : : : ; z R3, H(x; ) { -
, , H( ; x) = H( ; x).
-
20 . . , . .
- ( , , ..), - : , , , -, , , - , , . .. (. [6{10]).
4.
50 , , , - (2.1) , , , , - . , (2.1). , - - [4, 37]. , -. - , (2.12), (2.13), (2.15),(2.17).
(2.27){(2.29) ( jxj = r < R)
[14]. - m(r) = r'0(r)
(Dm0)0 + m = q(r); 0 < r < R; (4:1)
m(0) = 0; m(R) + hm0(R) = 0; (4:2)
D > 0; 0 h 0.
(Dm0)0 + m = ( ddx
+ g=D)(Dd
dx g)m; (4:3)
g0 + g2=D = (4:4)
g, (4.1)
m01 + g=Dm1 = q(r); (4:5)
-
21
Dm0 gm = m1(r): (4:6) (4.2),
g(0) = 0; m1(0) = 0; m(R) =hm1(R)
D(R) + hg(R): (4:7)
(4.4) g(0) = 0 (0; R]. (4.5) m1(0) = 0 r = 0 r = R, (4.6) (4.7) (m(R) ) r = R r = 0. (4.1), (4.2)c : (4.4) (4.5) (4.6) (4.7).
1953{1955 . - (, , ) - ({16, {37 .. ).
-, : --.
. , - ,
nXk=0
akgk(x)hk(): (4:8)
(4.8) hk() { , - , Pn- { - . (2.15) n - [16]; , (2.21).
gk(x) -, gk(x); hk(x) { ; (2.16) - (2.21) [4].
- . - - . - , - , - ( ) .
, - , [15] - (. 1) [15] .
-
22 . . , . . y
y1
y2
y3
y4
0
a
r1
a
r2
a
r3
a
r4
a
r5
a
r6
x
. 1.
-, - , -. - , - . (2.17). 1; 2; 3, - x() = x0+ G. 32 = 21 = h > 0, l(x) 1 3 .
2#2 = #1 2#2 + #3; !#2 = 56#2 +112(#1 + #3)
, #i #i xi =x0 +i .
l2 2u2h2
= !(u+ Su+ q=)2 (4:9)
(2.17) 0(h4) [4].
. , - , - , , [21, 22, 27, 28]. - 131- , . -, { -
-
23
. . 2 131- -; 5810 17424 .
. 2. 131-
- - : Q ( Dn) c Dn Q '(xi;k), - (. 1).
. -. - , , .. - 'k(x;). 60- - - , x [4]. 1963 . - - . S'k. (2.12) (2.27). S'k S'k+1 : K-
-
24 . . , . .
L'k+1=2 = (x)S'k + q; (4:10)
S'k+1=2, P - wk+1=2 - (2.27)
gkdivDrwk+1=2 + 1(x)wk+1=2 = (x)S('k+1=2 S'k); (4:11)
S'k+1=2 = S'k+1 + wk+1=2; k = 0; N 1: (4:12)
(4.11) gk { - . - [4]: gk = (1 + yk)=2, yk { P (1=2;2N)N (y).
. . . . .. - .. [29, 30] , - , - , . kef (2.24) (2.27), (2.28) T- [4, 25, 29], .
(). -, , . , - , . [25, 26] - - - . , ,{. , - . , - , . - - x .
-
25
, , . 3- 4- ( ) - . , - - .
. . - - . " ", , .
. - S' - (2.27) (.. 5). . - , ( , -).
[19] , , , . ui 2 Vi -
ai(ui; vi) = fi(vi); 8vi 2 Vi; i = 1; 2; (4:13)
ai(ui; vi) { , - ui; vi Vi, fi(vi) { Vi . u = (u1; u2) - V 2 V1 V2,
a1(u1; v1) + a2(u2; v2) = f1(v1) + f2(v2); 8v = (v1; v2) 2 V ; (4:14)
U11u1 = U22u2;
(4.15)
J11u+ J22u2 = ;
i; i = 1; 2; { , , i { - , ai(ui; vi). Ui; Ji; i = 1; 2; V , - (4.14), (4.15) .
-
26 . . , . .
, -.
Siiui = iui; i = 1; 2; (4:16)
ui { ( ) (4.13),i = 1; 2. "" "" .
, - { . - {
Sie = e; i = 1; 2; (4:17)
. - [31, 32].
5. , .
. : , - - - , -.
(2.1), (2.4), (2.8), (2.9),(2.25) x , - t (- ).
, @u@t (2.9), (2.25) (2.1), (2.8) 0 t T n-
du
dt= Au+ q; ujt=0 = u0; (5:1)
u = (u1; : : : ; un), u0 = (u01; : : : ; u0n), q = (q1; : : : ; qn), A { (n n)-. (i; 'i) { A, Sp(A) = fig, f'ig Rn, - z fig B, Kr r > 0
Kr = fz : jz + rj rg: (5:2)
-
27
(5.2) ,
rT 1: (5:3) h , ,
h cou = r1; (5:4) . k+1
uk+1 = uk + k(Auk + qk); k = 0; 1; : : : ; N 1; (5:5) -
[23, 24]. lN =NPi=1
i = P 0N (0), PN (z)
PN (z) =NYi=1
(1 + iz): (5:6)
-
maxz2B
jPN (z)j 1 (5:7)
PN (z) = arg supRN
(R0N (0)); (5:8)
lN = supRN
(R0N (0)) (5.6),
(5.7). zi; i = 1; ; N , (5.8), (5.5) - : z1i ; i = 1; ; N . , B = [M; 0]; M 1, (5.8) , N (5.5)
lN = N2cou: (5:9)
, (5.8) . . N = 2n. PN (z) (5.8) - , (5.1):
yk+1=2 = uk + hk+1f(uk; tk);tk+1=2 = tk + hk+1;yk+1 = yk+1=2 + hk+1f(yk+1=2; tk+1=2);tk+1 = tk+1=2 + hk+1;uk+1 = yk+1 + k+1hk+1(f(uk; tk) f(yk+1=2; tk+1=2));
(5:10)
-
28 . . , . .
k = 0; 1; : : : ; n 1; f(u; t) = Au+ q; . hi; i PN (z). (5.10), - , DUMKA; 2 N 216 36 [23, 24]. - .
. (- ) (2.1), (2.8), (2.25) - . -: , , 1, . . [35, 36] - . - , [42], : - , . - , .
. 3.
. 3, 4 , , , (2.26) -
-
29
, 1986 . - (. 4 "" . 3). , .
. 4. ""
? (2.1),(2.8), (2.25), ,
du
dt= f(u; v; t); (5:11)
u N ', ' = vN; (5:12)
v . (5.11) [0; T ] ujt=0 = u0.
, - - x ( ) . , (5.11) { - . ., [43] - ,
u(t) = u0 +
tZ0
f(u(s); v; s) ds: (5:13)
-
30 . . , . .
- (5.11).
. 5. 1-2-
. 6. 1-
{ 1- { 1- { 2- { 2-
-
31
(2.8), (2.9) v - x . . , , N :
[N ] = 0; (5:14)
( [f ] f ), (5.12), '. :
['] = 0; (5:15)
N .
= S' N0 = SN ,
= vN0: (5:16)
, (5.14)
[N0] = 0;D@N0@n
= 0; (5:17)
(5.15):
[] = 0;D@@n
= 0; (5:18)
n { . . 5
(2.25), (2.26) - ; (5.15) (5.18) ( - (5.14), (5.17)) . . 6 N0, (5.16).
: ? kef (2.25), . , (5.17)
[N0] = 0;vD
@N0@n
= 0;
, ( 3) [33, 34].
-, , - , - .
-
32 . . , . .
1. .. // - ,1{5 1955 {.: - , 1955. . 371{374.
2. .. . {.: -, 1958.
3. .. . {.: ,1961.
4. .., .. -. {.: , 1971; 1981.
5. .., .. // . {.: , 1961. . 30{45.
6. .. .{.: , 1974.
7. Marchuk G.I. Adjoint Equations and analysis of Complex Systems. {KluverAcad. Publ. 1995.
8. Marchuk G.I., Agoshkov V.I. Conjugate operators and algorithms of perturba-tion in non-linear problems // Soviet J. Numer. Anal. and Math. Modell.1988. 1. P. 21{46; 2. P. 115{136.
9. .., .., .. - . {.: -, 1993.
10. .. - // - . .. -, 1994. . 203. . 126{134.
11. .., .. - // . 1984. . 279, 4. . 843{847.
12. .., .. - // . 1985. . 62, 1. . 3{29.
13. .., .. -// - . { : , 1985, 1. . 147{162.
-
33
14. .. - // . 1955. . 19, . 3. . 315{324.
15. .. // . 1958. . 3. . 3{33.
16. .., // - . .. . 1961. . 61.
17. .. // .1968. . 8, 4. . 842{852.
18. .., .. // ( ).
19. .. . {.: , 1986.
20. Lebedev V.I. The composition method and unconventional problems // Sov. J.Num. An. and Math. Mod. 1991. V. 6, 6. P. 485{496,
21. .. // . 1976. . 16, 2.. 293{306.
22. .. - // . 1976. . 231, 1. . 32{34.
23. Lebedev V.I. How to solve sti systems of dierential equations by explicitmethod // Numerical Method and Applications. CRC Press, Boca Raton, 1994.P. 45{80.
24. Lebedev V.I. Explicit Dierence Schemes with Variable Time Steps for SolvingSti Systems of Equation // Numerical Analysis and its Applications: Proc.Lecture Notes in Computer Science 1196. {Springer, 1997. P. 274{283.
25. Lebedev V.I. An Introduction to Functional Analysis and Computational Math-ematics. {Birkhauser, Boston, Basel, Berlin, 1996.
26. Lebedev V.I. Extremal polynomials with restrictions and optimal algorithms //Advanced Mathematics: Computation and Applications / A.S. Alekseev andN.S. Bakhvalov (Editors). NCC Publisher. 1995. P. 491{502.
27. Kazakov A.., Lebedev V.I. Gauss-type Quadrature Formulas for the Sphere,invariant with respect to the Dihedral Group // Proc. of the Steklov Inst. ofMath. 1995. Issue 3. P. 89{99,
-
34 . . , . .
28. .., .. 131- // . . 1999. . 366, 6.
29. Lebedev V.I., Finogenov S.A. On the Order of Parameter Specication inChebyshev Cyclic Iterative Method // Zh. Vychisl. Mat. Mat. Phys. 1971.Vol. 11, . 2. P. 425{439 (in Russian).
30. Lebedev V.I., Finogenov S.A. On the utilization of ordered Chebyshev parame-ters in iterative methods // Zh. Vychisl. Mat. Mat. Phys. 1976. Vol. 16, 4.P. 895{907 (in Russian).
31. .., .. { - . {. , 1983.
32. Agoshkov V.I., Lebedev V.I. Generalized Schartz algorithm with variable pa-rameters // Sov. J. Num. An. and Math. Mod. 1990. V. 5, 1. P. 1{26.
33. .., .., .. // . . . . . 1999. . 1.. 3{15.
34. .., .., .. - // - . -98. . 1999.. 135{142.
35. .., .., .. - // - - -. -98. . 1999. . 143{150.
36. Dementiev V.G., Kosarev A.I., Lebedev V.I., Nechepurenko Yu.M., ShishkovL.K. Spectral Analysis of VVER-1000 Reactor Model at high negative Reactiv-ities // Proceding of the ninth Symposium of AER, 1999. P. 453{468.
37. ., . . {.: , 1974.
38. .. . { .: , 1979.
39. .. . {.:. 1986.
40. .., .. . 2-. {.: , 1985.
-
35
41. .-. , - . { .: , 1972.
42. Nechepurenko Yu.M. A new spectral analysis technology based on the Schurdecomposition // Russ. J. Num. An. and Math. Mod. 1999. V. 14. 3.
43. .. .{.: , 1985.
44. . . { .: , 1950.
-
..
E-mail: [email protected]
- -. , - - , - . -.
- . , - , , [5{14].
-, , . -, ( , ..) - . - ( [12, 17], [4], [13] .).
.. , - [2, 5{8,
-
37
13, 14]. , [14] - 'p ( )
A' = f:
Ip(') = ('; p);
p { , . - 'p
A'p = p;
p, Ip('). - A A0 = A+ A, {
Ip = ('p; A') = ('; A'p); . , - [7, 9].
[2,10], - fkg; fkg
A =mXk=1
(kAk +Bk(kCk))
(Ak ; Bk; Ck { ) - () Ipi ; i = 1; : : : ; n. ..
mXk=1
[('pi ; kAk') + (Bk'
pi ; kCk')] = Ipi ;
'pi { A'pi = pi, '
{ A' = q. fkg; fkg - fkg; fkg . 0k = k + k;
0k = k + k; k = 1; : : : ;m,
. , - . - [13] - , (- ) , - .
-
38 ..
[10] , [11].
.. - A(') A1(')', -
(A('); ) = (';A1(') );
A1(') { ' A(') = f . 'p ( p) A1(')'
p = p.
- , [6].
( -) [7].
, (" "," ", " "). , . - . { - - { , - , .
, "-" , - . . , , -, . - - ( ).
1. 1.1. D R3 - -
@D C(2), x = (x1; x2; x3) 2 D = D[@D, t { -, t 2 [0; T ]; T < 1. U(x; t) = (u1(x; t); u2(x; t); u3(x; t)) -, div U = 0 QT D (0; T ), '(x; t)
-
39
L' @'@t +A' @'@t + div(
eU') 3Pi=1
@@xi
(i@'@xi
) + a' = f QT ;
@'@N +
eU ()n (' '(s)) + ' = 0 @D (0; T );' = '(0)(x); x 2 D; t = 0:
(1)
: eU = (u1; u2; u3 ug), ug(x; t) { -; n = (n1; n2; n3) { @D; eU ()n =(j(eU; n)j (eU; n))=2; eU (+)n = (j(eU; n)j + (eU; n))=2; (eU; n) = 3P
i=1uini; a = a(x; t)
0; 1 = 2 > 0; 3 > 0; (x; t) 0 ; f = f(x; t){ , '(s) = '(s)(x; t) { , -
@D (0; T ); @'=@N 3Pi=1
i cos (n; xi)@'@xi
. ,
, a; ; U; eU; 1; 2; 3; '(s) . , , , .
1.2. (1). W 2;12 (QT ) L2(0; T ;W 22 (D)) \ W 12 (0; T ;L2(D)), H L2(QT ) - (u; v)H (u; v)
RQT
uvdxdt; kukH kuk (u; u)1=2. - LT :
LT @ @t
+AT @ @t
(eU;r) 3Xi=1
@
@xi(i
@
@xi) + a ; 8 2W 2;12 (QT ):
LT = g QT (g 2 L2(QT ));
@ @N + (
eU (+)n + ) = 0 @D (0; T ); = 0; x 2 D; t = T:
(2)
(LT )1g 2 W 2;12 (QT )(c. [14]), k kW 2;1
2(QT )
Ckgk.
Y = f 2 W 2;12 (QT ) : = (LT )1g 8g 2 H; k kY k kW 2;12
(QT )g:
Y W 2;12 (QT ). (1) : f 2 Y ; '(0) 2 L2(D) ' 2 H ,
-
40 ..
,
(';LT ) (f; )ZD
'(0) (x; 0)dx TZ0
dt
Z
eU ()n '(s) d = 0 8 2 Y; (3) '(x; t) (1).
1. f 2 Y ; '(0) 2 L2(D); '(s) 2 L2( (0; T )), (1) ' 2 H,
k'k C(kfkY + k'(0)kL2(D) + k'(s)kL2((0;T )));
C =const. ' 2W 2;12 (QT ), ' (1).
( (3), LT - g 2 H . , ' { " "(, W 2;12 (QT )), - Y QT ( H),, ' (1).)
. , - ' 2 H (1) '(0); '(s). 2
1.3. (1), f(x; t) Q(t) (t 2 (0; T )), X(t) = (X1(t); X2(t); X3(t)), , ..
f(x; t) = Q(t)(x X(t)); (4)
(x) = (x1)(x2)(x3), () { "- ". , X(t) D X(1) X0(t1); X(2) X0(t2), (t1; t2) [0; T ] (0 t1 < t2 T ), X0(t) = (X0;1; X0;2; X0;3)(t) 2 D 8t 2 (0; T ) (W 12 (0; T ))
3. t1 > 0 (t2 < T ), X(t) X0(t) t 2 [0; t1) (t 2 (t2; T ]). Q(t) [0; T ] ( ) L1(0; T ).
f(x; t) (4) (3)
L(X ;'; ) (';LT )TR0Q(t) (X(t); t)dt R
D
'(0)(x) (x; 0)dx
TR0
dtR
eU ()n '(s) (x(); t)d = 0 8 2 Y:(5)
-
41
, f(x; t) Q(t)(x X(t)) 2 Y Q(t) 2 L1(0; T ) 1 (5) ' 2 H .
'
' ('; p) =ZQT
'(x; t)p(x; t)dxdt; (6)
p(x; t) 2 Lp(QT ) (p 2), 'obs =const . SA :
SA 12
t2Zt1
A(t)j ddt(X X0)j2dt+ 12(' 'obs)
2; (7)
A(t) 2 C[t1; t2] { , , 0 < A0 A(t) A1
-
42 ..
(10)., (10) ' 2 H (1). - q(x; t) '. , q(x; t) (10) X(t)( , X(t)) ' .
, (10), , - .
2. 2.1. q;X (8), (10). (" "):
a(X0; 'obs;X; eX) = 0 8 eX 2 (W 12 (t1; t2))3X(t1) = X(1); X(t2) = X(2);
(11)
a(X0; 'obs;X; eX) t2Zt1
A(t)d(X X0)
dt
d eXdtdt+B(X;'obs)
t2Zt1
Q(t)( eX;r)q(X(t); t)dt;
B(X;'obs) TZ0
Q(t)q(X(t); t)dt+ZD
'(0)(x)q(x; 0)dx+
+
TZ0
dt
Z
eU ()n '(s)q(x(); t)d 'obs;( eX;r)q = eX1 @q
@x1+ eX2 @q
@x2+ eX3 @q
@x3; eXi = 0 t 2 ([0; t1)[ (t2; T ]); i = 1; 2; 3:
(11)
ddtAddt (X X0) +B(X;'obs)Q(t)rq(X(t); t) = 0; t 2 (t1; t2);
X(t1) = X(1); X(t2) = X(2):(12)
, (11) ((12)) - X(t) q(x; t) x. -, " " p(x; t), ( '(x; t)) '. ,
-
43
, (12) ( ) - . (12).
2.2. p(x; t)
p(x; t) = f1=(mes ( eD) (et2 et1)) eD (et1;et2); 0 g; (13) eD D; (et1;et2) (0; T ). ' = ('; p) '(x; t) eQ eD (et1;et2) (12) X(t), e' = 'obs( A(t) ), 'obs "" ' eQ.
, eU; a; ; 1; 2; 3; p t, f'ig, f jg - ( (1), (2)):
A'i = i'i; AT j = j j ; (14) H , ('i; j) = ij . (), (14)
p(x; t) =1Xj=1
(p; 'j) j 1Xj=1
pj j : (15)
,
q(x; t) =1Pj=1
pj1 ej(Tt)
j j ;
rq(X(t); t) =1Pj=1
pj1 ej(Tt)
jr j(X(t));
(16)
B(X;'obs) =1Pj=1
pjjfTR0
Q(t)(1 ej(Tt)) j(X(t))dt+
+(1 ejT ) RD
'(0)(x) j(x)dx+
+TR0
(1 ej (Tt))dt R
eU ()n '(s)(x(); t) j(x())dg 'obs:(17)
(16), (17), - (12) . , ,eQ QT ; p = j0 ; (18).. ' ('; j0 ) j0- ' =1Pj=1
('; j)'j X(t),
-
44 ..
('; j0 ) = 'obs 'obs;j0 ('obs; j0) A! +0, 'obs { . (12)
ddtAddt (X X0) + Fj0 (X(t); t) = 0; t 2 (t1; t2);
X(0) = X(1); X(T ) = X(2);(19)
Fj0 (X(t); t) = Q(t)B(X;'obs;j0)
pj0j0
(1 ej0 (Tt))r j0 (X(t)):
B(X;'obs;j0) =pj0j0
8
-
45
pi; i = 0; 1; 2; 3; { t, .. "" p(x; t), - (14).
q(x; t) = q0(t) +3Xi=1
qi(t) cos(
Lxi); (23)
q0(t); : : : ; q3(t) . pi =const, i = 0; 1; 2; 3,
q0(t) = a(T t)p0; qi(t) = pii(1 ei(Tt)); i = a+ i
L
2 (19)
ddtAddt (Xi X0;i)
LQ(t)B(X;'obs)qi(t) sin(
LXi(t)) = 0; t 2 (t1; t2);
Xi(t1) = X(1)i ; Xi(t2) = X
(2)i ; i = 1; 2; 3:
(24)
, ( , ).
2.4. sin () , (24):
ddtAd
dt(Xi X0;i) (
L)2Q(t)B(X;'obs)qi(t)(Xi(t)
(L)213!X3i (t) +O((
L)4) = 0; i = 1; 2; 3:
L, , (24) :
ddtAddt (Xi X0;i) (
L )
2Q(t)B(X;'obs)qi(t)(Xi(t) 13!(L )
2X3i (t)) = 0; t 2 (t1; t2);
Xi(t1) = X(1)i ; Xi(t2) = X
(2)i ; i = 1; 2; 3;
(25) ( ) :
ddtAddt (Xi X0;i) (
L )
2Q(t)B(X)qi(t)Xi(t) = 0; t 2 (t1; t2);
Xi(t1) = X(1)i ; Xi(t2) = X
(2)i ; i = 1; 2; 3;
(26)
B(X) B(X;'obs) (11) q(X(t); t) q(0) =q0(t) +
3Pi=1
qi(t).
, ( D, , p(x; t) q(x; t)
-
46 ..
) (12) ( ) - . , (12) - q(x; t) , .
3. 3.1. (12) .
'obs 'obs;0 + "'obs;1 8 " 2 ["0; "0] { , '0 - (1) f(x; t) = Q(t)(xX0(t)), 'obs;1 =const , 'obs;0 '0. SA(X0; 'obs;0; q;X0) = 0, (10) - "" X0(t), ""'obs;1". .
, a(; ;X; eX) = 0 X X =X0; " = 0
aX(X0; 'obs;0; eX; eeX) [ eX; eeX] ==
t2Zt1
A(t)d eXdt
deeXdtdt+B1( eX)B1( eeX) 8 eX; eeX 2 (W 12 (t1; t2))3;
B1( eX) = t2Zt1
Q(t)( eX;r)q(X0(t); t)dt:
[X;X ] =
t2Zt1
A(t)dXdt
2 dt+ (B1(X))2 A0 t2 t12kXk2(L2(t1;t2))3 + (B1(X))2;
[ eX; eeX] . HA - - X 2 (W 12 (t1; t2))3 [; ] [] = [; ]1=2.
[X1; eX] = F1( eX) 8 eX 2 HA; (27) X1 2 HA { -,
F1( eX) = 'obs;1B1( eX) = @@"a(X0; 'obs;0 + "'obs;1;X; eX)"=0;X=X0 : .
-
47
2. (27) X1 2 HA,
[X1] j'obs;1j;
dX1dt
(L2(t1;t2))3 j'obs;1j=pA0;
kX1k(L2(t1;t2))3 j'obs;1j(t2 t1)=(pA0):
(28)
-, (12) X = X0; " = 0, q(x; t) ( X @a=@"). q(x; t) X(t) (12) ", - (12) { " ".
3.2. , :
(i) supp(p) QT ;
(ii) dist(X0(t), supp(p)) C0 =const 8t 2 (t1; t2);
(iii) - (1) xi; i = 1; 2; 3; 8t 2 (0; T ) QT nsupp(p).
([1], [3]) X0(t)(8 t 2 (t1; t2)) q(x; t) xi; i = 1; 2; 3. , a(X0; 'obs;0+"'obs;1;X; eX) X = X0(t); " = 0; (8 t 2 (t1; t2)),
a(X0; 'obs;0;X; eX) = 0 8 eX 2 (W 12 (t1; t2))3
a(X0; 'obs;0 + "'obs;1;X; eX) = Xi+j0
aij( eX; (X X0)i)"j ; i = (i1; i2; i3); ik 0; k = 1; 2; 3; i+ j = i1 + i2 + i3 + j
aij( eX; (X X0)i) = 1i!j!
@i+j
@X i@"ja(X0; 'obs;0 + "'obs;1; eX; (X X0)i)
a1;0( eX;X X0) = aX(X0; 'obs;0; eX;X X0) = [ eX;X X0] (., , [16]).
-
48 ..
(27) HA F1 2 (HA), [16] (12) - (X0(t); " = 0) -X = X(t; "), X(t; 0) = X0 X(t; ") " " = 0:
X(t; ") = X0(t) + "X1(t) + "2X2(t) + 8 " 2 ["0; "0]; (29) "0 { ( C0; T .), .. (12) " X(t) (29). - (3) X = X(t; ") (10), SA X = X(t; "). - (i){(iii) q(x; t) x, .
. , : 1) X0 2 (W 12 (t1; t2))3; X0(t1) = X(1); X0(t2) =X(2); 2) 'obs = 'obs;0+"'obs;1, 'obs;0 '0, '0 2 H (3) f(x; t) = Q(t)(x X0(t)); 3) (1) p(x; t) , q(x; t) x X0(t) 8 t 2 (t1; t2) (., , (i){(iii)). "0 > 0, , 8" 2 ["0; "0] (10) X(t; ") (29), ". Xj(t); j 1, (29) - (" "). , X1 (27) Xj ; j 1, .
. , , '(x; t) "! 2
3.3. (10)
jX(t)X(+)(t)j < ; (30) =const> 0; X(+)(t) 2 (W 12 (t1; t2))3 { , , X(+)(t) 2 D; 8 t 2 (t1; t2). , X0(t), (30), jX0(t) X(+)(t)j < =2 X0(t1) = X(1); X0(t2) = X(2). - , " (10) (29). " 2 [e"0; e"0], 0 e"0 < "0; e"0 { , (30). ,
3. , X0(t) (30) jX0(t)X(+)(t)j < =2; 8 t. e"0 > 0 , 8 " 2 [e"0; e"0] X(t; ") (30).
(10) - , q(x; t) x. - q(x; t) (
-
49
(19)) - q(x; t), p(x; t), f j0g (. (19)) ..
4. -
.
4.1. -, - . , X0 X1
ddtAdX1dt + (B1(X1) 'obs;1)Q(t)rq(X0(t); t) = 0; t 2 (t1; t2);
X1(t1) = X2(t2) = 0;(31)
B1(X1) =
t2Zt1
Q(t)(X1;r)q(X0(t); t)dt;
X = X(1) X0 + "X1
X(1) . X(1) ( (3)) '(1) (1) f(x; t) = Q(t)(x X(1)).
n- X(n) = X0 + "X1 +: : :+ "nXn, X2; : : : ; Xn ( (31)) [9].
4.2. (12), (. . 5.1).
'obs, X(t) .
1. X0(t), '0 f = Q(t)(x X0) 'obs;0 '0.
'obs 6= 'obs;0, .
2. q(x; t).
3. "'obs;1 , 'obs = 'obs;0 +"'obs;1 (, j"'obs;1j) , - ), X1, .. (31), X = X0 + "X1 = X(1).
-
50 ..
X(1) , X(1) X0 3. , - .
(12) - ( , - .). X(n) , ( ) (3) X = X(n). (-, q(x; t) !).
, . , - q(x; t) x. , -, - q Y . "" q , . , , "- ".
4.3. X1 ( , X(1) =X0 + "X1 = X) , q(x; t). .
A =const.
d2X
dt2= f(t); t 2 (t1; t2); X(t1) = X(t2) = 0
X(t) =
t2Zt1
G(t; t0)f(t0)dt0;
G(t; t0) :
G(t; t0) =
8>>>>>:(t t1)(t2 t0)
(t2 t1) t1 t t0
(t2 t)(t0 t1)(t2 t1) t
0 t t2;
(31) B(X1) X1; X(1):
B1(X1) =
'obs;1B1
0@ t2Zt1
G(t; t0)Q(t0)(rq)(X0(t0); t0)dt01A
A+B1
0@ t2Zt1
G(t; t0)Q(t0)(rq)(X0(t0); t0)dt01A ;
-
51
X1(t) =
'obs;1
t2Zt1
G(t; t0)Q(t0)(rq)(X0(t0); t0)dt0
A+B1
0@ t2Zt1
G(t; t0)Q(t0)(rq)(X0(t0); t0)dt01A ;
( eX;r)q = 3Xi=1
eXi(t) @q@xi
:
, q(x; t) , (rq)(x; t) - . Q(t):
Q(t) = Q0(t t0); (32) Q0 =const> 0; t0 2 (t1; t2). X1(t) :
X1(t) ='obs;1G(t; t0)(rq)(X0(t0); t0)Q0A+Q20G(t0; t0)jrqj2(X0(t0); t0)
: (33)
, jrq(X0(t0); t0)j 6= 0, (33) A = 0, " " X1(t); X(1)(t):
X1(t) ='obs;1Q0
G(t; t0)G(t0; t0)
(rq)(X0(t0); t0)jrqj2(X0(t0); t0)
(34)
X(1)(t) = X0(t) + "'obs;1(t); (35)
(t) =1Q0
G(t; t0)G(t0; t0)
(rq)(X0(t0); t0)jrqj2(X0(t0); t0)
X0(t), - 'obs;0 = '0, ""'obs;1" .
X(t).
.
. , - , . - '(s) .
- - ( ,
-
52 ..
, , - ..). "" .
, - - (, kdX4=dt4k2(L2(t1;t2))3 , - .).
.. , .. .. .
[1] .., .. - . // -. : , 1982. 30. { .: -.
[2] // . -. . 16 { .: , 1985.
[3] .., .., .. - . { .: , 1967 { 736 .
[4] .-. , - . { .: , 1972.
[5] .. . { .: -, 1958.
[6] .. . {.: , 1974.
[7] .. . { .: , 1982.
[8] .. - // . 1981. 4, . 1. . 3{27.
[9] .. . { .: , 1989.
[10] .. // . 1964. 156, 3.. 503{506.
[11] .. - // . . 1964. 2, . 3.. 462{477.
-
53
[12] .., .., .. - . { .:, 1993.
[13] .., .. - // . { -, 1981. C. 3{18.
[14] .., .. // . { .: , 1961. C. 30{45.
[15] .. . { .:, 1961 { 400 .
[16] .. . { .: , 1993. { 440 .
[17] .., .. -. { . 1984. 279, 4.
-
c 1
. . , . .
E-mail: [email protected]
- . . - -, . , - .
1. [3], , -
, , - , . -. [3], , " d L . L, " { . xj yj = xj=". C , .
Lu = x1";x2";x3"
@2u@t2
+ L0u = 0;
L0u =@
@xi
Aij(
x1";x2";x3")@u
@xj
;
(1.1)
1 ( 99{01{01146, 99{01{01153) ( 96{1061).
-
c 55
(y1; y2; y3) Aij(y1; y2; y3) { , -, 1- ( c 1) yj ;Aij = ATji, [1; 3]. ,
0 < (y1; y2; y3); 0 < a(zj ; zj) (Aij(y1; y2; y3)zj ; zi) 8 yj ; zj : (1.2)
(y); A(y) (1.1) .
"
-
56 . . , . .
@N
@yj
= 0 1- N ,
hql1l2l3 =
*Aij
@Nql1i1;l2i2;l3i3@yj
+AijNql1i1j1;l2i2j2;l3i3j3
+: (1.6)
Lv X
q+l1+l2+l32
"q+l1+l2+l32hql1l2l3@q+l1+l2+l3v
@tq@xl11 @xl22 @x
l33
0: (1.7)
,
Nql1l2l3 = 0; hql1l2l3
= 0 q : (1.8)
Nql1l2l3(y1; y2; y3) (1.5) ( -).
Nql1l2l3
=
0. [1] .1. (1.5) ,
Nql1l2l3
= 0 q + l1 +
l2 + l3 > 0, (1.7) , , , hql1l2l3
(hql1l2l3)T = (1)l1+l2+l3hql1l2l3 ; (1.9)
l1 + l2 + l3 l1 + l2 + l3 .
2. w:
v X
q;l1;l2;l30
"q+l1+l2+l3dql1l2l3@q+l1+l2+l3w
@tq@xl11 @xl22 @x
l33
;
dql1l2l3 { , d0000 = E,
w:
LBw b@2w@t2
+1X
l1+l2+l32
"l1+l2+l32bh0l1l2l3 @l1+l2+l3w@xl11 @xl22 @xl33 0; (1.10) b =< >, bh0l1l2l3
(bh0l1l2l3)T = (1)l1+l2+l3bh0l1l2l3 : (1.11) (1.10) (1.7) , , . - ( u ) (1.9), (1.11) ,
hql1l2l3 = 0 ;bh0l1l2l3 = 0 l1 + l2 + l3 : (1.12)
-
c 57
(1.1) , b =hi = 1, () , = (y); Aij = Aij(y), y = x1=".
(1.7) (1.10) , v x3, (1.7)
@2v
@t2+ h020
@2v
@x21+ h002
@2v
@x22+ "2(h400
@4v
@t4+ h220
@4v
@t2@x21+ h202
@4v
@t2@x22
+h040@4v
@x41+ h022
@4v
@x21@x22
+ h004@4v
@x42) +O("4) = 0:
(1.13)
(1.13) , 2 (c. (2.29)) 3.
v = v1+12"2h400
@2v1@t2
(1.13)
E +12"2h400
@2
@t2:
LAv1 = @2v1@t2
+ h020@2v1@x21
+ h002@2v1@x22
+"2(eh220 @4v1@t2@x21 + eh202 @4v1
@t2@x22+ h040
@4v1@x41
+ h022@4v1@x21@x
22
+ h004@4v1@x42
) +O("4) = 0;
eh220 = h220 + 12 h020h400 + h400h020 ;eh202 = h202 + 12 h002h400 + h400h002 :
v1 = w +12"2eh220 @2w@x21 + eh202 @
2w
@x22
E +12"2eh220 @2@x21 + eh202 @
2
@x22
:
":
@2w
@t2+ h020
@2w
@x21+ h002
@2w
@x22
+"2bh040 @4w@x41 + bh022 @
4w
@x21@x22
+ bh004 @4w@x42= 0;
(1.14)
bh040 = h040 + 12 h020h220 + h220h020+14
(h020)
2h400 + 2h020h
400h
020 + h
400(h
020)
2;
(1.15)
-
58 . . , . .
bh022 = h022 + 12 h020h202 + h202h020 + h002h220 + h220h002+14(2(h002h
400h
020 + h
020h
400h
002)
+h002h020h
400 + h
400h
020h
002 + h
400h
002h
020 + h
020h
002h
400);
(1.16)
bh004 = h004 + 12 h002h202 + h202h002+14
(h002)
2h400 + 2h002h
400h
002 + h
400(h
002)
2:
(1.17)
, 1,2 , , .
R = E bE; bA11 = A111 1 ; Q1 = A111 bA11 E;bA22 = hA22i ; Q2 = A22 bA22; ~Q2 = bA122 Q2bA33 = hA33i ; Q3 = A33 bA33; eQ3 = bA133 Q3:
hg(y)i = 0 J
hJ(g)i = 0; (J(g))0 = g:
hfgi2 f2 g2 (1.18)( f g), - .
[3]. hB(y)i = 0
hAJ(B)i = hJ(A hAi)Bi :
[3]
h0200 = bA11; h0k00 = 0; 8k > 2;h4000 =
J(R)A111 J(R)
A111 J(R) bA11 A111 J(R) ; (1.19)h2200 =
J(QT1 )J(Q1)
hJ(R)J(Q1)i J(QT1 )J(R) ; (1.20)N2000 = J(A
111 (J(R) bA11 A111 J(R))); N0200 = J2(Q1): (1.21)
(1.15)
bh0400 = bA11 ( hJ(QR)J(QR)i hJ(QR))i hJ(QR))i) : (1.22) (1.18), bh0400 0.
[2] ( u 1) , ( )h4000 0.
-
c 59
2. , , u { , -
x2 A12 =AT21 = 0. (1.13) (1.14).
Nql1l2l3 = Nql1l2l3
(y) - (1.4)
Hql1l2l3 = Nq2l1l2l3
+@
@y
A11
@Nql1l2l3@y
+Ai1
@Nql1i1;l2i2;l3i3@y
+@
@y
A1jN
ql1j1;l2j2;l3j3
+AijN
ql1i1j1;l2i2j2;l3i3j3
(2.23)
l1 = l3 = 0:
Hq0l0 = Nq20l0 +@
@y
A11
@Nq0l0@y
+A22N
q0l20 = h
q0l0 = const : (2.24)
2 3 0 Hql1l20; Nql1l20
,Hql1l20, C
ql1l20
. q = 0; l = 1
H001 =@
@y
A11
@N001@y
= h001:
, h001 = 0 N001 = 0. q = 0; l = 2
H002 =@
@y(A11
@N002@y
) +A22 = h002:
h002 = hA22i ; A11@N002@y
= J(Q2) + C002;@N002@y
= (A11)1J(Q2) +A111 C002; C002 = bA11 A111 J(Q2) : (2.25) ,
N002 = J(A111bA11 A111 J(Q2)A111 J(Q2)): (2.26)
h004 =
A22N
002
:
N002,
h004 =
J(Q2)A111 J(Q2)
J(Q2)A111 bA11 A111 J(Q2) : (2.27) (2.24) (1.21) (2.26)
h202 =
A22N200
N002 = (J(R)A111 J(Q2)+ J(Q2)A111 J(R))+(
J(R)A111
bA11 A111 J(Q2)+ J(Q2)A111 bA11 A111 J(R)): (2.28)
-
60 . . , . .
(2.24) , Nq0l = 0; hq0l = 0;
q; l . (1.7)
@2v
@t2+ bA22 @2v
@x22+ "2
h400
@4v
@t4+ h202
@4v
@t2@x22+ h004
@4v
@x42
+O("4) = 0: (2.29)
.1, ":
@2w
@t2+ bA22 @2w
@x22+ "2bh004 @4w@x42 = 0; (2.30)
bh004 (1.17), (1.19), (2.27), (2.28). : bh004 = 1 1,
1 =D(J(Q2) bA22J(R))A111 (J(Q2) bA22J(R))E
+12
J(Q2)A111 J(R) J(R)A111 J(Q2)
bA22 bA22 J(Q2)A111 J(R) J(R)A111 J(Q2)
+14[ bA222 J(R)A111 J(R) 2 bA22 J(R)A111 J(R) bA22 + J(R)A111 J(R) bA222];
1 =D(J(Q2) bA22J(R))A111 E bA11 DA111 (J(Q2) bA22J(R))E
+12bA22[J(R)A111 bA11 A111 J(Q2) J(Q2)A111 bA11 J(R)A111 ]
+12[
J(Q2)A111
bA11 J(R)A111 J(R)A111 bA11 A111 J(Q2)] bA22+14[ bA222 J(R)A111 bA11 J(R)A111 2 bA22 J(R)A111 bA11 J(R)A111 bA22
+
J(R)A111
bA11 J(R)A111 bA222]: A22 A11 , -
bh004 = D(J(Q2) bA22J(R))A111 (J(Q2) bA22J(R))ED(J(Q2) bA22J(R))A111 E bA11 DA111 (J(Q2) bA22J(R))E : (2.31)
(2.30) ei(kx2!t)e,
c2e = ( bA22 ("k)2bh004)e; = !=k - . A11 A22 , , ,
. bh004 - (2.31) (1.18) , . -, , , .
-
c 61
, , - (1.1) ( A21 = (A12)T 6= 0) jjA21jj.
v 2 , bh004 : , { . (y) 6= const, bA22 - d1 6= d2 A22(y) = bA22 + ((y) b) bA22. A111 (y) , Q = jjqij jj =
A111 (J(R))
2
q12 6= 0. , - bh004 ; , - d3 = q12(d1 d2)2=4.
(2.30) ei(kx2!t)e;
(c2 d1)(c2 d2) = (d3"k)2. ,
- "k.
, (1.1) (A21 = (A12)T 6= 0) - .
3.
, u 1, - A12 = A21 = 0, 2. , , -., (1.7) (1.13). - (2.23).
Hql1l2 = Nq2l1l2
+@
@y
A11
@Nql1l2@y
+A11
@Nql11;l2@y
+
@
@y
A11N
ql11;l2
+ A11N
ql12;l2
+A22Nql1;l22
:
Nql1l2 = 0 q l2 . Nq0l2 = 0, l2 ,Nql1l2 = 0, q; l1; l2 , , :
Hql1l2 = 0 q l2 ,
, ,
hql1l2 = 0 q l2 :
-
62 . . , . .
, ,
h011 = 0; h013 = 0; h
031 = 0; h
015 = 0; h
033 = 0; h
051 = 0; h
211 = 0; h
213 = 0; h
231 = 0:
(1.9) ,
h003 = 0; h012 = 0; h
021 = 0; h
030 = 0; h
005 = 0; h
014 = 0; h
023 = 0
h032 = 0; h041 = 0; h
050 = 0; h
203 = 0; h
212 = 0; h
221 = 0; h
230 = 0:
(1.8), , (1.7) (1.13). , - (x1; x2) - bh004, bh022 bh040, (1.7) (1.15) { (1.17).
bh004, bh040 ((1.22), 2.31)). - bh022 (1.16) h022.
(1.6)
h022 =
A22N
020
+A11(
@N012@y
+N002):
(1.21)
A22N020
= A22J2(Q1) = hJ(Q2)J(Q1)i.
H012 =@
@y(A11
@N012@y
+N002) +A11@N002@y
+A22N010 = h012:
,
A11(@N012@y
+N002) = C002 + J(h
012 (A11
@N002@y
+A22N010)):
A111 h i
0 =
(A11)1
C002 +
(A11)1J(h012 (A11
@N002@y
+A22N010)):
C002 = bA11(A11)1J(h012 (A11 @N002@y +A22N010))
=
QT1 J(h
012 (A11
@N002@y
+A22N010))
= J(QT1 )(A11
@N002@y
+A22N010))=
J(QT1 )A11
@N002@y
J(QT1 )A11J(Q1) :
(2.25) A11@N002@y
= 002 J(Q2);
-
c 63
J(QT1 )A11
@N002@y
=
J(QT1 )J(Q2)
.
h022 =
J(QT1 )A22J(Q1)
+
J(QT1 )J(Q2)
+ hJ(Q2)J(Q1)i : (3.32)
. h022 =
A22(J(Q1))2
+ 2 hJ(Q2)J(Q1)i.
hJ(A)i = 0 8 A,
(Q1 R)(J(Q1 R))k
=
*1
k + 1J(Q1 R))k+1
0+= 0 8 k 0; (3.33)
D( bA11A111 E)(J(Q1))kE = Q1(J(Q1))k =
*1
k + 1(J(Q1))k+1
0+= 0 8 k 0:
h022 (3.32) :
A22(J(Q1))2 = bA22 (J(Q1))2 J 0(Q2)(J(Q1))2 = bA22 (J(Q1))2+ 2 hJ(Q2)J(Q1)Q1i :
h022 = bA22 (J(Q1))2+ 2 bA11 (A11)1J(Q2)J(Q1). h220 h
202. (1.20)
h220 =
(J(Q1))2
2 hJ(R)J(Q1)i. h220 =
(J(Q1))2
+
J 0(R)(J(Q1))2
2 hJ(R)J(Q1)i=
(J(Q1))2
2 hQ1J(R)J(Q1)i 2 hJ(R)J(Q1)i=
(J(Q1))2
2 bA11 (A11)1J(R)J(Q1) :, (2.28)
h202 = 2
J(R)A111 J(Q2)
+
J(Q2)A111 J(R)
) + 2
J(R)A111
bA11 A111 J(Q2) ; , h020 = bA11, h002 = bA22, h400 = A111 (J(R))2 bA11 A111 J(R)2 - (1.19). (1.16) , h400 (1.19), bh022 = 2 2;
2 = 2 bA11 A111 J(Q1)J(Q2) 2 bA11 bA22 DA111 J( eQ2)J(R)E2 bA11 bA22 (A11)1J(R)J(Q1)+ 2 bA11 bA22 (A11)1(J(R))2
= 2 bA11 bA22 D(A11)1(J(Q1 R))(J( eQ2 R))E ;2 = 2 bA211 bA22 (A11)1J(R) D(A11)1J( eQ2)E
+2 bA211 bA22 (A11)1(J(R))2 = 2 bA211 bA22 (A11)1(J(Q1 R)) D(A11)1(J( eQ2 R))E :
-
64 . . , . .
,
bh022 = 2[ bA11 bA22 D(A11)1J(Q1 R)J( eQ2 R)E( bA11)2 bA22 (A11)1J(Q1 R) DA111 J( eQ2 R)E]:
(3.33), (1.22)
bh040 = bA211 A111 (J(Q1 R))2 bA311 A111 J(Q1 R)2 : , (2.31)
bh004 = bA222 DA111 (J( eQ2 R))2E bA11 bA222 DA111 J( eQ2 R)E2 : (1.14) w =ei(k(1x1+2x2)!t).
c2 =!2
k2= (21 bA11 + 22 bA22) ("k)2 bh04041 + bh0222122 + bh00442 :
S(1; 2) = bh04041 + bh0222122 + bh00442: S(1; 2) bh040 ;bh022 ;bh004. ,
S(1; 2) =
A111 Z
2 bA11 A111 Z2 ;
Z = 21 bA11(J(Q1 R)) + 22 bA22(J( eQ2 R): (1.18) f =A1=211 ; g = A
1=211 Z, , S(1; 2) 0. ,
1. , w { , - { (1.14) { ! - .
4. , -
(1; 2; 3) -
@2u
@t2+
@
@x1
A11
@u
@x1
+
@
@x2
A22
@u
@x2
+
@
@x2
A23
@u
@x3
+
@
@x3
A32
@u
@x2
+
@
@x3
A33
@u
@x3
= 0:
(4.34)
-
c 65
, (1; 2; 0), h
ql1;l2;0
. - (2.23) l3 = 0 (4.34).
Hql1;l2;0 = Nq2l1;l2;0
+@
@y
A11
@Nql1;l2;0@y
!+A11
@Nql11;l2;0@y
+
@
@y
A11N
ql11;l2;0
+A11N
ql12;l2;0
+ A22Nql1;l22;0
+
A23(Nql1;l21;1
+Nql1;l21;1) +A33Nql1;l2;2
= hql1;l2;0:
(4.35)
Nql1;l2;l3 = 0, -, (4.35)
Hql1;l2;0 = Nq2l1;l2;0
+@
@y
A11
@Nql1;l2;0@y
!+A11
@Nql11;l2;0@y
+
@
@y
A11N
ql11;l2;0
+A11N
ql12;l2;0
+A22Nql1;l22;0
= hql1;l2;0:
, A12 = A21 = 0, 2, 3. O("4), - x1; x2, (1.14) , , 3.
, - (1; 2; 3) 2 6= 0, 3 6= 0 -
xn1 = x1; xn2 =
2x2 + 3x3p22 +
23
; xn3 =3x2 + 2x3p
22 + 23
: (4.36)
x1 .
@2u
@t2+
@
@x1
A11
@u
@x1
+
@
@xn2
An22
@u
@xn2
+
@
@xn2
An23
@u
@xn3
+
@
@xn3
An32
@u
@xn2
+
@
@xn3
An33
@u
@xn3
= 0;
(4.37)
An22 =22A22 + 223A23 +
23A33
22 + 23
; An23 = An32 =
23A22 + (22 23)A23 + 23A3322 +
23
;
An33 =23A22 223A23 + 22A33
22 + 23
:
(1.2) a An22, a An33.
-
66 . . , . .
(1; n2 ; 0),
n2 =
p22 +
23 .
ei(k(1x1+2x2+3x3)!t)
ei(k(1x1+n2 x
n2 )!t). -
-
c2 =!2
k2= 21 bA11 + (n2 )2 bAn22 ("k)2Sn(1; n2 );
Sn(1; n2 ) =
A111 Z
2 bA11 A111 Z2 ; Z = 21 bA11J(Q1R)+(n2 )2 bAn22J( eQn2 R);
bAn22 = 22 hA22i+ 223 hA23i+ 23 hA33i22 + 23 ; Qn2 = 22A22 + 223A23 +
23A33bAn22 :
(1.18), ,, Sn(1; n2 ) 0.
, 2. ,
(4.34), { ", (1.10) - , .
, , - , .
[1] Bakhvalov N.S., Eglit M.E. Variational properties of averaged equations for pe-riodic media // Proc. of the Steklov Institute of Mathematics. 1992. Issue 3.P. 3{18.
[2] Bakhvalov N.S., Eglit M.E. Eective equations with dispersion for waves propa-gation in periodic media // Doklady Math. 2000. Vol. 370, 1. P. 1{4.
[3] .., .. Long-waves asymptotics with dispersion for thewaves propagation in stratied media. Part 1. Waves orthogonal to the layers// Russian J. Numer. Analys. and Math. Modelling. 2000. 15, 1. P. 3{14.
[4] Bakhvalov N.S., Panasenko G.P. Homogenization. Averaging processes in peri-odic media. {Dordrecht: Kluwer. 1989. {366 P.
-
- Rn
. .
E-mail:[email protected]
{ . { . :(i) , (ii) (iii) , -, Rn. , (ii) - , n .
The classical interpretation of matrix is representation of the operator inxed coordinate system. For a symmetric matrix this is the representation of thequadratic form as well. In the paper the new concept is presented: (i) a stronglynonsingular matrix, (ii) irreducible low and upper Hessenberg matrices and (iii)two system of special polynomials, taken together represents the dierent aspectsof the some object exterior to the linear vector space Rn. For instance, if element(ii) is Jacobi matrix with nonsingular spectrum, then this object is the oscillationsystem with n degrees of freedom.
{ - . { - . :(i) , (ii) - (iii) , , Rn. -, ii , n [2].
[3-6] , - , , A. - ( , , [7]) .
-
68 . .
, , ((i),(ii) (iii)), Rn-.
: Bk = Bek, Bl = eTl B, ek k- E, k- l- B. Bk k- B, k. b , b, .
R = (rij) { . - [4], T
T =
26666664
11 12 : : : 1n21 22 23 : : : 2n
32. . .
. . ....
. . .. . . n1;n
n;n1 nn
37777775 . T - R:
Rk+1 = T Rk = r11T ke1; k = 0(1)n (1)( k = n R, Rn+1 Rn+1), :
lk =rl;k+1rkk
k1Xj=m
rjkrkk
lj ; (2)
l = 1(1)k + 1; k = 1(1)n; m = max(l 1; 1): ,
k+1;k =rk+1;k+1rkk
;
k;k =rk;k+1rkk
rk1;krk1;k1
;
k;k1k1;k =rk1;k+1rk1;k1
rk1;krk1;k1
k1;k1 rk2;krk2;k2
; (3)
k = 1(1)n 1: n 1 T . - , T . .
n(x) = xn nXj=1
pnj+1xj1
-
- Rn 69
T , P = (pn; : : : ; p1)T . (1) k = n Rn, R, (2) k = n. { [4]:
P = R1T Rn; (4)
Tn = 1rnn
(RP n1Xj=1
rjnTj): (5)
L R - A = LR. A (LT = DR, D { ,), .
L R - A,
Ak(m; l) =Ak1 AlAm aml
k, Ak1 - k 1 Ak(k; k) = Ak. [1],
A = LDU; (6)
L = (lij), U = (uij), i; j = 1(1)n, D = diag(d1; : : : ; dn) , D - ,
lij = jAj(i; j)j;di = 1=(jAi1jjAij); jA0j = 1; (7)uij = jAi(i; j)j;
(6) ,
lkkdkukk = jAkj=jAk1j; k = 1(1)n: (8) , A , (LT = U), "k = sign(jAkj=jAk1j),E = diag("1; : : : ; "n) R = (ED)1=2U
A = RTER: (9) ,
Ak1[i; j] =
26666664a11 : : : a1;j1 a1;j+1 : : : a1k: : : : : : : : : : : :
ai1;1 : : : ai1;j1 ai1;j+1 : : : ai1;kai+1;1 : : : ai+1;j1 ai+1;j+1 : : : ai+1;k: : : : : : : : : : : :ak1 : : : ak;j1 ak;j+1 : : : akk
37777775 ;i; j = 1(1)k:
-
70 . .
, aij Ak. - (6)
A1 = UDL; (10)
U = U1D1 = (uij); L
= D1L1 = (lij):
[3] Ak1[i; j]:
1. (10) - A1
lij = (1)i+j jAi1[j; i]j; i j;uij = (1)i+j jAj1[j; i]j; j i: (11)
, , A = RTR , T , , [4].
.
1. A -
, (11)
2. k- U k- L
k 1- Uk1(x) Lk1(x) ,
Uk1(x) =kPi=1
uikxi1 =
=kPi=1
(1)i+kjAk1[k; i]jxi1 =
= Ak1 Ak1 : : : xk2 xk1
;Lk1(x) =
kPi=1
lkixi1 =
=kPi=1
(1)i+kjAk1[i; k]jxi1 =
1
Ak1 : : :xk2
Ak xk1
:
-
- Rn 71
jAk1j. - Uk1(x), L
k1(x) .
, k- U = U1D1 (k- L = D1L1) Uk1(x) (L
k1(x)) x.
V =
266641 x1 : : : xn111 x2 : : : xn12...
......
1 xn : : : xn1n
37775 (i) Rn-.
k(x) =kYi=1
(x xi) =kXi=1
ck;kixi;
'(k)l (x) = k(x)=(x xl); l = 1(1)k; k = 1(1)n;
V V 1 [3], [6]:
V = ZW;
Z = (zij), W = (wij), i; j = 1(1)n, { ,
wij =iXl=1
xj1l
'(i)l (xl)
;
zji = i1(xj)
V 1 =W1Z1;
Z1 = (z?ij) W1 = (w?ij), i; j = 1(1)n, -
z?jl =
1
'(j)l (xl)
; l = 1(1)j;
w?ij = cj1;ji; i = 1(1)j;
. W ( wij
(i 1)- xj1 x1; : : : ; xi) - [3] ( )
T =
26666664
x11 x2
1. . .. . .
. . .1 xn
37777775 ;
-
72 . .
W1 k(x), k = 0(1)n 1. 2 A. 1.
(7). (10) P = U(ED)1=2 E = diag("1; : : : ; "n), -
A1 = PEP T : (12) P = (pj1i1 ), i = 1(1)j, j = 1(1)n,
pj1i1 = (1)i+j1p
"j jAj1jjAj jjAj1[j; i]j; j i; (13)
"j = sign(jAj j=jAj1j):
2. k- P - pk1(x) k 1,
pk1(x) =k1Xi=0
pk1i xi =
=1p
"kjAk1jjAkjkXi=1
(1)i+kjAk1[k; i]jxi1 =
=1p
"kjAk1jjAkj
Ak1 Ak1 : : : xk2 xk1 : (14)
2. A -
-. , -
3. R T - (1). D Rk+1 = T Rk, k = 1(1)n 1, T = DT D1 = (tij) { R = DR { -. ( D ):
tkk = kk ;tk;k1tk1;k = k;k1k1;k: (15)
. R , - T ,
Rk+1 = T Rk; k = 1(1)n 1:
-
- Rn 73
k Rk+1 = T Rk . Rl = DRl, Rk+1 = D1 T DRk = T Rk, U = D1 T DT ,
URk =kXj=1
rjkUj = 0; k = 1(1)n 1:
k = 1 , U1 = 0. Uj = 0, j = 1(1)k 1, Uk = 0, k = 1(1)n 1. T , Un = 0. , U , . D1 T D. 2
3. T R, (1)
R =n1Xk=1
T k1R1eTk +RneTn ;
T , (4):
T R =n1Xk=1
T kR1eTk +RPeTn :
{ , :
T R = RF; (16)
F {
F =n1Xk=1
ek+1eTk + Pen:
, , , (xi) = 0, xi , v(xi) = (1; xi; : : : ; xn1i ) { . P = R1.
1. :
PT = FP; (17)
v(xi)PT = xiv(xi)P: (18)
-
74 . .
A, A = LDU , D, (7), - :
A An+1An+1 an+1;n+1
=LD 0lD d
U u0
=
=LDU LDulDU d+ lDu
:
U TU , L TL, (1) k = n : u = TUUn = UP ,l = LnTL = P TL, (4).
An+1 = LDU = AP; An+1 = P TA:
4. A - TU TL, - ,
(x) =1jAj
A APvT (x) xn
=
1jAj
A v(x)P TA xn
; (19)
v(x) = (1; x; : : : ; xn1)T . , -, : vT (xi)PU TU PLv(xi) TL, PUU = PLL = E.
. (19) ( ) ( ). , :
AP =nPi=1
pjAnj+1:
, (18). 2
, , .
5. Rn- .
. A, A =LR TL, TR, p
(L)k (x),
p(R)k (x), k = 0(1)n 1. T (L), T (R), ,
L = l11nXk=1
ekeT1 T
k1L ;
R = r11nXk=1
T k1R e1eTk
-
- Rn 75
A. , p(L)k (x) p(R)k (x), k = 0(1)n1, L1 R1, A.
A .
4. ,
, A = RTR. , (9), R E . (9) :
A = ~RT ~D ~R; (20)
~D = diag( ~d1; : : : ; ~dn), ~R = (~rij), i; j = 1(1)n, { ,
~di = jAij=jAi1j; ~rij = jAi(i; j)j=jAij (21) R ~R
R = ( ~DE)1=2 ~R; (22)E = diag("1; : : : ; "n), "i = sign ~di, 3 -
T = (DE)1=2 ~T ( ~DE)1=2 (23) ~T = (~tij), i; j = 1(1)n, (9) (20) -.
, T , R, ~T , ~R , - , (1) - (3). , (22), (23) - ~T ~R :
~Rk+1 = ~T ~Rk = ~T k ~R1; k = 0(1)n 1; (24)
~tjk = ~rj;k+1 k1Pi=m
~rik~tji; (25)
j = 1(1)k + 1; k = 1(1)n 1; m = max(j 1; 1);, ,
~tk+1;k = 1;~tk;k = ~rk;k+1 ~rk1;k ; (26)
~tk1;k = ~rk1;k+1 ~rk2;k ~rk1;k~tk1;k1; k = 1(1)n 1: { -
~R A, ~D+ = diag( ~d+1 ; : : : ; ~d
+n )
~D = ~D+E :
-
76 . .
E ( 2n) -, A+ =~RT ~D+ ~R 2n A = ~RT ~D ~R. - A+ A .
2. A - (9) (20) :
~D = E ~D+; ~R = ~R+; R = R+:
. ~D+. , (20) ~rii = 1,i = 1(1)n,
jAi(i; j)j = j ~RTi jj ~Dijj ~Ri(ij)j = jEijj ~RTi jj ~D+i jj ~Ri(ij)j == jEijjA+i (i; j)j:
, (6) L = UT
uij = jAi(ij)j = jEiju+ij ;,
~rij =uijqjAji
=ju+ij jjA+i j
= r+ij :
. , A = RTER = ~RT ~D ~R R = ( ~D+)1=2 ~R =( ~D+)1=2 ~R+ = R+: 2
5. Rn-cRn- A = (aij), aij = ai+j2,
Rn-. , [3-4].
6. Rn- ~T , - (20), :
~T =
266664b1 g1
1 b2. . .
. . .. . . gn11 bn
377775
bj = rj;j+1 rj1;j ; j = 1(1)n;gj =
jAj1jjAj+1jjAj j2 ; j = 1(1)n 1: (27)
-
- Rn 77
. [3-4]:
jAl1(l; k)j = ul1;k+1 + jAl1jul2;l1ul1;k ul2;kjAl1jjAl2jjAl1j (28)
jAl2julk = jAl1jjAl1(l; k)j ul;l1ul1;k (29)( , jAlj l- k-, ulk). jAl1(l; k)j - A, :
ul1;k+1jAl1j =
ul2;kjAl2j +
ul1;kjAl1j
ul1;ljAl1j
ul2;l1jAl2j
+jAl2julkjAl1j2 ;
, (16):
~rl1;k+1 = ~rl2;k + ~rl1;k~rl1;l ~rl2;l1
+jAl2jjAljjAl1j2 ~rlk : (30)
l = k - (26) ( ), :
~tk1;k =jAk2jjAkjjAk1j2 :
(26) (27). , ~tj1;k = 0 j < k, ..
T . : (30),
~rl1;k+1 =lX
i=l2
~rik~tl1;i;
(25)
~rl1;k+1 =kX
i=l2
~rik~tl1;i;
kXi=l+1
~rik~tl1;i = 0: (31)
l = k 1 ~tk2;k = 0 k = 3(1)n. , (28) l+2. , , k, , ~tkj;k = 0 ~tkj1;k = 0. , ~T . 2
-
78 . .
T , (9).. B ,
jbij j = jbjij.. A -
(9) T ,
T =
266664b1 a1"1"2
a1 b2. . .
. . .. . . an1"n1"nan1 bn
377775 ;
aj =
vuutabsjAj1jjAj+1jjAj j2 :
,
T T = ETE :
1. .. . { .: , 1966.
2. .., .. . { .-.: , 1950.
3. .., .. .{ : , 1986.
4. .. - // . . . 1986. XXVII, 2. . 84{90.
5. .. . // . . .1987. XXVIII, 6. C. 148.
6. .. . { -: , 1994.
7. .. . 1068, 1996.
-
Sn{
..
- Sn{. .
1. 1955 - Sn{ ( ). . , - .. .. [1] : \... , , . , Sn{ ". H , - , , - [2{6]. - .
H , - , Sn{, , , - .
2. Sn{ q(r), [1, 4, 6]
1r2
@
@r(r2'(n)) +
@
@
1 2r
'(n)+ '(n) = f (n)(r); (1)
-
80 ..
'(n)(R; ) = S() 0: (2) f (n)(r) = 12
sR 11 '
(n1)(r; )d + q(r)-
"n" ; 0 r R; 1 1; (r) s(r) > 0. 1 1 2n (j ; j+1),
0 = 1, 0 r R N (rk; rk+1) , rk. Mkj , rk r rk+1; j j+1. (1) Mkj \" dM = r2drd ( n) :
r2k+1
j+1Zj
'(rk+1; )d r2kj+1Zj
'(rk ; )d+ (1 2j+1)rk+1Zrk
r'(r; j+1)dr
(1 2j )rk+1Zrk
r'(r; j)dr +
rk+1Zrk
(r)r2dr
j+1Zj
'(r; )d =
rk+1Zrk
f(r)r2dr; (3)
= j+1 j . "" Sn{ '(r; )
Mkj . - f(r) .
Akj'kj +Bkj'k+1;j + Ckj'k;j+1 +Dkj'k+1;j+1 = Ekfk + Fkfk+1; (4)
k = 0; 1; :::; N1; j = 0; 1; :::; n1. Akj ; Bkj ; :::; Fk, [1,6].
(4) :
'Nj = S(j); j = 0; 1; :::; n; (5)
'0j = '0;2nj ; j > n: (6)
(2), | .
- . - -, 'k0, k = 0; 1; :::; N: (1) = 1, -
@'@r
+ '(r;1) = f(r) (7)
-
Sn - 81
'(R;1) = S(1). - . - [1, 6] . . 0 j < n Ckj (4), j n - Dkj > 0. , , , . , M - [7], - [8]. (1), (2) '(r; ) f(r) 0 S() 0, fk = f(rk) 0 S(j 0) 'kj k j . .
3. "" Sn- . , , , .. . , (1),(2) ( [4,6], [1, 6] .), , .
H (7)
(1 + (rk)rk)'k0 'k+1;0 =rk+1Zrk
f(r)dr; 'N0 = S(1); (8)
rk = rk+1 rk . , Mkj 0; (0 j n), (3) :
8>>>>:r2kj+1Zj
d+ (1 2j+1)rk+1Zrk
rdr +
rk+1Zrk
r2dr
j+1Zj
d
9>>>>;'k;j+1+
r2k+1
j+1Zj
d 'k+1;j+1 (1 2j )rk+1Zrk
rdr 'kj = rk+1Zrk
f(r)r2dr: (9)
(5) 'k0 (8) - 0 j < n 'k;j+1 (9). , 'k;j+1 . - (9) .
{ n j 2n 1
-
82 ..
'k+1;j+1 :8>>>>:r2k+1j+1Zj
d+ (1 2j+1)rk+1Zrk
rdr +
rk+1Zrk
r2dr
j+1Zj
d
9>>>>;'k+1;j+1
r2k
j+1Zj
d 'k;j+1 (1 2j )rk+1Zrk
rdr 'k+1;j = rk+1Zrk
f(r)r2dr: (10)
k = 0 '0;j+1 '0;2n(j+1) (6).
(10) 'k+1;j+1 - , . - : (8){(10) - . - (8) ; (9)
r2kj+1Zj
d+ (1 2j+1)rk+1Zrk
rdr + r2k+1
j+1Zj
d+ (1 2j )rk+1Zrk
rdr 0;
(10). , -
M - [7], .. A1 0, f(r) S() -. , f(r) 0; S() 0 [8]
0 maxk;j
'k;j maxn
max0kN1
rk+1Zrk
f(r)dr=(rk)rk ;
max0kN1
rk+1Zrk
f(r)r2dr=
rk+1Zrk
r2dr; max0jn
S(j)o:
f S, : j 'kj j; j f(r) j; j S(j). - (8){(10).
, , -, ..
[1] .., .. .{ .: , 1981. { 454 .
-
Sn - 83
[2] Carlson B.G. Solution of the transport equation Sn{approximations / Los AlamosScientic Laboratory Report LA{1891, 1955.
[3] ., . Sn{ // - . { .: , 1959. {408 .
[4] .. . { .: , 1961.{ 667 .
[5] .H. Sn- // . {.: , 1962. {91.
[6] .. . { .: , 1978. {216 .
[7] .., .. { . { .: H, 1984.{ 218 .
[8] .. . { H-, 1998. { 9 . (/H. . -, ; 1118)
[9] .. . { .: H, 1988. { 263 .
-
1
. . , . .
E-mail: [email protected]
- .
- - [1]{[20] . ..[3, 4].
- , - . , - - . - . . -, [13], [15], [21].
1. :(
d'dt + A(t)'+ F (') = f; t 2 (0; T )
't=0
= u;(1.1)
1 ( 00{01{00611)
-
85
A { ,
A(t)' = 2X
i;j=1
@
@xiai;j
@'
@xj+
2Xi=1
ai@'
@xi+ a':
x = (x1; x2) 2 R2, { - @, ai;j , ai, a - :
a(x; t) 0;2Xi=1
@ai@xi
= 0; ai;j = aj;i;
2Xi;j=1
ai;jij 2Xi=1
2i 8i 2 R; = const > 0:
A Y = L2( (0; T )) Y D(A) =f' : ' 2 Y ;A' 2 Y; '=@ = 0g. f 2 Y , u 2 H = L2(), F (') - ,
F (') =Qe='; ' > 0;
0; ' 0;Q; ; > 0 - . (1.1) , , [23], [24].
S(') =
2k'jt=0k2 + 12
Z T0kb' 'k2dt; (1.2)
=const 0; b' = b'(t) 2 Y { , kk { H . b'(t) , , -. [13]. , .
(1.1) u 2 H (.. u) . [6],[13]: '='(t) - u , (1.1), - (1.1) (1.2) .
:8>:@'@t +A(t)'+ F (') = f; t 2 (0; T )
't=0
= uS(') = min
uS('):
(1.3)
(1.3) [6], [13], [15], [25] . (1.2) , , [15].
-
86 . . , . .
[6], [13], (1.3) - ' = '(t), ' = '(t) u :(
@'@t +A(t)'+ F (') = f; t 2 (0; T )
't=0
= u;(1.4)
(@'
@t +A(t)' + (F 0('))' = (b' '); t 2 (0; T )
't=T
= 0;(1.5)
u 't=0
= 0; (1.6)
A(t) { , (t), F 0 F .
(1.4){(1.6) - [13], [25] . , , ', ', u (1.4){(1.6) , .
2. (1.4){(1.6) -
- . (1.4), (1.5) . ,
(1.4) . o (1.4) '(x; t), t 2 (0; T ) HA [26], - @'@t 2 L2( (0; T )) (0; T )
(@'
@t; )(t)+['; ](t)+
2Xk=1
ak
@'
@xk;
+(F ('); )(t) = (f; )(t); ('
t=0
; ) = (u; );
(2.1) 2 HA. (; ) { H , HA { , A(t),
['; ] =Z
(2X
i;j=1
ai;j@'
@xj
@
@xi+ a' )dx
['] = ['; ']1=2. , -
- , [26]. , Pi i [26]. - (2.1)
'h(x; t) =NXi=1
bi(t) i(x);
-
87
N { . bi(t) :
(@'h@t
; i)(t) + ['h; i](t) +2Xk=1
ak@'h@xk
; i
+ (F ('h); i)(t) = (f; i)(t); (2.2)
('h(x; 0) '0; i) = 0 i = 1; N: (2.3) (1.4){(1.6) :8>>>>>:
@'h@t ; i
+ ['h; i] +
2Xk=1
ak@'h@xk
; i
+ (F ('h); i) = (f; i)
('ht=0uh; i) = 0;
(2.4)
8>>>>>:@'h@t ; i
+ ['h; i]
2Xk=1
ak@'h@xk
; i
+ (F 0('h)'h; i) = (b' 'h; i)
('ht=T
; i) = 0;(2.5)
(uh 'ht=0
; i) = 0; i = 1; N: (2.6)
: e'h = 'h 'h, e'h = 'h 'h, euh = uh uh, 'h, '
h, uh { :8>>>>>:
@'h@t ; i
+ ['h; i] +
2Xk=1
ak@'h@xk
; i
= (f; i)
('ht=0uh; i) = 0;
(2.7)
8>>>>>:@'h@t ; i
+ ['h; i]
2Xk=1
ak@'h@xk
; i
= (b' 'h; i)
('ht=T
; i) = 0;
(2.8)
(uh 'ht=0
; i) = 0; i = 1; N: (2.9)
e'h, e'h, euh :8>>>>>:@ e'h@t ; i
+ [e'h; i] + 2X
k=1
ak@ e'h@xk
; i
+ (F ('h + e'h); i) = 0
(e'ht=0euh; i) = 0;(2.10)
-
88 . . , . .
8>>>>>>>>>:@ e'h@t ; i
+ [e'h; i] 2X
k=1
ak@ e'h@xk
; i
+
(F 0('h + e'h)('h + e'h); i) = (e'h; i)(e'ht=T ; i) = 0;
(2.11)
(euh e'ht=0; i) = 0; i = 1; N: (2.12) :
8>>>>>: @ e'(n+1)h@t ; i
!+ [e'(n+1)h ; i] + 2X
k=1
ak@ e'(n+1)h@xk
; i
!+ (F ('h + e'(n)h ); i) = 0;
(e'(n+1)h t=0eu(n+1)h ; i) = 0;(2.13)
8>>>>>>>>>>>>>>>:
@ e'(n+1)h
@t ; i
!+ [e'(n+1)h ; i] 2X
k=1
ak@ e'(n+1)h@xk
; i
!+
+(F 0('h + e'(n)h )('h + e'(n)h ); i) = (e'(n+1)h ; i)(e'(n+1)h t=T ; i) = 0;
(2.14)
(eu(n+1)h e'(n+1)h t=0; i) = 0; i = 1; N: (2.15) (2.13){(2.15)
(2.7){(2.9), .
3. (2.7){(2.9) f; b' 2 Y =
L2( (0; T )).
3.1. f; b' 2 Y . > 0 (2.7){(2.9) 'h, '
h, uh,
k'hkY + k'hkY + kuhk c(kfkY + kb'kY ); c = const > 0: (3.1) [6], [9], [13] -
. (3.1) c { .
.
-
89
3.1. 'h (2.7)
k'ht=T
k2 + k'hk2Y 1kfk2Y + k'h
t=0k2; (3.2)
= 94mes, { , A.
. (2.7) [13] -
@'h@t
; 'h
+ ['h; 'h] = (f; 'h);
t 0 T ,
12k'hk2(T ) +
TZ0
['h]2(t0)dt0 =
TZ0
(f; 'h)dt0 +
12k'hk2(0):
['h]2
Z
2Xk=1
@'h@xk
2dx C2()k'hk2;
C() = 32(mes)1=2 { -x [27],
" > 0
12k'hk2(T ) + k'hk2Y
14"kfk2Y + "k'hk2Y +
12k'hk2(0):
" = =2, (3.2). 'h
(2.8). 3.1. 3.1. 3.1 (2.7){(2.8)
'h = G0uh +G1f; 'h = G
(T )1 (b' 'h);
G0 : H ! Y; G1 : Y ! Y; G(T )1 : Y ! Y { . 'h; '
h (2.7), (2.8) (2.9),
uh:
Luh = P; (3.3)
L = T0G(T )1 G0 + E; P = T0G
(T )1 (b'G1f), T0' = 't=0; E' = '.
L HN { f igi=1;N . 0 , ..
(Luh; uh) = kuhk2 +TZ0
kG1uhk2dt > 0:
-
90 . . , . .
Luh = P , -
kuhk dkPk; (3.4) d = 1=min(L), min(L) { L., min(L) .
P 3.1. :
kPk 1p(kb'G1fkY ) 1p
kb'kY + 3=2kfkY ;
(3.4)
kuhk d1=2kb'kY + d3=2kfkY : (3.5) 'h { (2.7), 3.1
k'hkY 1kfkY + 1=2kuk;
(3.5)
k'hkY 1+
d
2
kfkY + d
kb'kY : (3.6)
(2.8) (3.6):
k'hkY 1kb' 'hkY 1kb'kY +
12
+d
3
kfkY + d
2kb'kY : (3.7)
(3.5){(3.7), (3.1),
c = max
dp+d
+1+
d
2;
d
3=2+1+
d
2+
12
+d
3
:
.
4. .
(x) = Qe=x; x 2 (0;1), F (').
4.1. 0; 00
j(x)j Q; j0(x)j q1Q; j00(x)j q2Q; (4.1)
q1 =4 e
2; q2 = max(1;2), i = j00(i)j; i = 1; 2, 1;2 = 2(12 +p36 ),
00(x) = expf x( 2x)g=x4, x 2 (0;1).
4.1
-
91
4.2. F Y Y ,
kF (')kY QpTmes; kF 0(') kY q1Qk kY ; 8 '; 2 Y; (4.2)
q1 (4.1).
(2.13){(2.15) - .
4.1. e'(0)h ; e'(0)h ; eu(0)h R:
ke'(0)h kY + ke'(0)h kY + keu(0)h k R; R > 0: Q 1=(cq1 + g0=R), g0 = cq1(k'hkY + k'hkY + kuhk), - (2.13){(2.15) e'(n)h ; e'(n)h ; eu(n)h .
. (2.13){(2.14) ke'(0)h kY+ke'(0)h kY +keu(0)h k R. ( n) e'(n+1)h ; e'(n+1)h ; eu(n+1)h{ , 3.1 4.2
ke'(n+1)h kY+ke'(n+1)h kY+keu(n+1)h k c(kF ('h+e'(n)h )kY+kF 0('h+e'(n)h )('h+e'(n)h )kY ) cq1Q(k'h+e'(n)h kY+k'(n)h +e'hkY ) cq1Q(ke'(n)h kY+ke'(n)h kY )+cq1Q(k'hkY+k'hkY ): ,
ke'(n)h kY + ke'(n)h kY + keu(n)h k (cq1Q)n(ke'(0)h kY + ke'(0)h kY )+ 1 (cq1Q)n1 cq1Q g0Q R; Q 1=(cq1 + g0=R). , R.
4.1
maxtke'(n)h kL1() +maxt ke'(n)h kL1() + keu(n)h kL1() qR (4.3)
q, n R.
HN - L2() vh =NPi=1
bi i
L1() [26]. , -
Q .
4.2. e'(0)h ; e'(0)h ; eu(0)h R = r0 = k'hkY + k'hkY + kuhk. Q < Q0, Q0 = [2c(q1 +qq2r0)]1, (2.13){(2.15) .
-
92 . . , . .
. R = r0 4.1, - 4.1 Q 1=(2cq1). (2.13){(2.15) n = e'(n+1)h e'(n)h , n = e'(n+1)h e'(n)h , vn = eu(n+1)h eu(n)h :8>>>>>:
@n@t ; i
+ [n; i] +
2Xk=1
(ak@n@xk
; i) = (F ('h + e'(n1)h ) F ('h + e'(n)h ); i)(nt=0vn; i) = 0;
(4.4)
8>>>>>>>>>>>>>:
@n@t ; i
+ [n; i]
2Xk=1
(ak@n@xk
; i) = (F 0('h + e'(n1)h )('h + e'(n1)h ); i)(F 0('h + e'(n)h )('h + e'(n)h ); i) (n; i)
(nt=T
; i) = 0(4.5)
(vn nt=0
; i) = 0; i = 1; N: (4.6)
3.1 4.1, 4.2, (4.4){(4.6) (4.3)
knkY + knkY + kvnk c(kF ('h + e'(n1)h ) F ('h + e'(n)h )kY+kF 0('h+e'(n1)h )(e'(n)h e'(n1)h )kY +k(F 0('h+e'(n)h )F 0('h+e'(n1)h )('h+e'(n)h )kY )
cQ(q1ke'(n)h e'(n1)h kY + q1ke'(n)h e'(n1)h kY + 2qq2r0ke'(n)h e'(n1)h kY )= (ke'(n)h e'(n1)h kY + ke'(n)h e'(n1)h kY );
= cQ(q1 + 2qq2r0). , m n -
:
ke'(n+m)h e'(n)h kY+ke'(n+m)h e'(n)h kY+keu(n+m)h eu(n)h k n n+m1 ; k = const > 0:(4.7)
< 1, (4.7) , - .
5. -
.
-
93
(2.13){(2.14) - :8>:
d'dt +A
h(t)' = ef;'t=0
= u;(5.1)
8>:d'
dt +Ah(t)' = '+ g;
't=T
= 0;(5.2)
u 't=0
= 0; (5.3)
' = '(t), ' = '(t), u { N , Ah(t); Ah { N , ef = ef(t); g = g(t) { -.
(5.1){(5.3) , [21], [22]. ( - -) - [22]:8>:
'k+1(j) 'k(j) +A
k+1=2'k+1(j) + 'k(j)
2 =efk+1=2;
'0(j) = uj(5.4)
8>:'
k+1(j) 'k(j) +A
k+1=2'k+1(j) + 'k(j)
2 = 'k+1=2(j) + gk+1=2;
'M(j) = 0;(5.5)
uj+1 = uj + j+1('0(j) uj) + j+1(uj uj1); (5.6) j+1; j+1 { , 'k(j), 'k(j), uj {
, 'k+1=2(j) = 'k+1(j) + 'k(j)
2 , { , k =0;M 1; = T=M .
(5.4){(5.6) [22]. - (5.4), (5.5) (5.4){(5.6) BiCGSTAB- [28], [29].
, - (2.13){(2.15), (5.4){(5.6).
. - - h=0:05, =0:05.
-
94 . . , . .
( , )
kuj+1 ujkkujk < ";
" = h2 = 2:5 103, kk { L2(). -
: aij = 1; i = j; aij =0; i 6= j; ai = 0; a = 0, Q = = = 1. ' (1.1) , b' = b'(x1; x2; t), t 2 (0; 1); x1 2 (0; 1); x2 2 (0; 1). , u , (2.4){(2.6) , , u0 = x21x
22(1 x1)(1 x2), [21]
j+1 =
8>:2
+ ; j = 0
4
Tj()Tj+1()
; j > 0; j+1 =
8>:0; j = 0
Tj1()Tj+1()
; j > 0;(5.7)
=+ ; = + (2max)
1[1 1 max
2
1 + max2
!2T=];
= + (2min)1[1 1 min
2
1 + min2
!2T=];
max =8h2
cos2h
2; min =
8h2
sin2h
2:
. 5.1 , - ", . . 5.1, .
5.1. n(")
1 0.5 0.2 0.1 0.05 0.01 0.001 0.0005 n 8 8 8 9 11 16 42 54
, [18], - u (5.1){(5.3), u, (5.1){(5.3) = 0, O(). R R = kuj uk=kuk, uj { , . . 5.2 R . , R , - .
-
95
5.2. R
1 0.5 0.2 0.1 0.05 0.01 0.001 0.0005
R 0.9765 0.9541 0.8931 0.8078 0.6804 0.3167 0.0586 0.0351
(1.2). . 5.3 - S(') . . 5.4 S(') - .
5.3. S(') 1 0.1 0.05
0 0.12732 0.11895 0.118481 0.02019 0.01666 0.013562 0.02009 0.01687 0.014193 0.02006 0.01649 0.013284 0.02006 0.01654 0.013525 0.02006 0.01653 0.013416 * 0.01652 0.013457 0.01653 0.013428 0.01653 0.013439 * 0.01343 17 0.01343
5.4. S(')
1 0.5 0.2 0.1 0.05
S(') 0.02006 0.01963 0.01839 0.01653 0.01343
(2.13){(2.15), -
(2.10){(2.12), { 2-3 - . , - . 0:05 1, , { -. ,
-
96 . . , . .
. -, .
- - , .
[1] .. . {.: , 1998.
[2] .. . {.: , 1969.
[3] .. // . 1964..156, N3. C.503{506.
[4] .. // . . 1964. .2. .3..462{477.
[5] .. . {.: , 1969.
[6] .-. , - . {.: , 1972.
[7] Marchuk G.I., Penenko V.V. Application of optimization methods to the prob-lem of mathematical simulation of atmospheric processes and environment //Modelling and Optimization of Complex Systems: Proc. of the IFIP-TC7 Work.conf. New York: Springer, 1978. P. 240-252.
[8] Le Dimet F.X., Talagrand O. Variational algorithms for analysis and assimilationof meteorological observations: theoretical aspects // Tellus. 1986. 38A. P. 97-110.
[9] . . - // Research ReportDNM 91/2, {M.: , 1991.
[10] Kurzhanskii A.B., Khapalov A.Yu. An observation theory for distributed-parameter systems // J. Math. Syst. Estimat. Control. 1991. V.1, . 4, P.389-440
[11] .. - . {M.: , 1992.{ . 17.07.92, 2333{1392.
-
97
[12] Zou X., Navon I.M., Le Dimet F.X. Incomplete observations and control of grav-ity waves in variational data assimilation // Tellus A. 1992. V. 44A. P.273{296.
[13] Agoshkov V.I., Marchuk G.I. On solvability and numerical solution of data as-similation problems // Russ. J. Numer. Analys. Math. Modelling. 1993. V. 8. .1. P. 1{16.
[14] .., .., .. - . {M.: -, 1993.
[15] Marchuk G.I., Zalesny V.B. A numerical technique for geophysical data assimi-lation problem using Pontryagin's principle and splitting-up method // RussianJ. Numer. Analys. Math. Modelling. 1993. V. 8. 4. P. 311{326.
[16] Agoshkov V.I. Control theory approaches in data assimilation processes, inverseproblems and hydrodynamics // Computer Mathematics and its Applications.1994. V.1. P. 21.
[17] Glowinski R., Lions J.L. Exact and approximate controllability for distributedparameter systems // Acta Numerica. 1994. V.1. P. 269.
[18] Marchuk G.I., Shutyaev V.P. Iteration methods for solving a data assimilationproblem //Russ. J. Numer. Analys. Math. Modelling. 1994. V. 9. 3. P. 265{279.
[19] ., .. - // . . 1996. .32, 5. C. 613{629.
[20] .., .., .. - // .. . . . 1997. .2. 12. C. 1449{1458.
[21] .., .. - // . 1997. T.37, 7. C. 816{827.
[22] Parmuzin E.I., Shutyaev V.P. Numerical analysis of iterative methods for solvingevolution data assimilation problems // Russ. J. Numer. Anal. Math. Modelling.Vol. 14. 3. 1999. P. 265{274.
[23] .. // . . 1959. T. XIV. B. 2(86). C. 87{158.
[24] . -. {.: , 1985.
-
98 . . , . .
[25] .. // . . 1998. .34, 3.C. 383{389.
[26] .., .. - . {.:, 1981.
[27] .. . {.:, 1994.
[28] Freund R.W., Golub G.H., N.M. Nachtigal. Iterative solution of linear systems// Acta Numerica. 1992. P. 57{100.
[29] H. van der Vorst. Bi-CGSTAB: A fast and smoothly converging variant of Bi-CGfor the solution of nonsymmetric linear systems // SIAM J. Sci. Statist. Comput.1992. V. 13. P. 631{644.
-
1
. .
E-mail: [email protected]
-, -. , : . , - - . , - , , .
1.
, , .
, - , . , - . :
@'
@t+K(') ' = S'+ f; 'jt=0 = '0; (1)
' 2 ; f 2 F;
1 , 99{05{64038.
-
100 . .
{ - (), K(') { , ':
(K(')'; ') = 0; K = K;S { - ( - ):
(S'; ') 2('; '); S = S: (1) , ,
S, f :
12@
@t('; ') 1
2@
@tk'k2 = 0: (2)
, . , (1) ,.. - .
, (1) , .. c [1]. - , , - [1]. , , .. , '0 2 A, A { (1). , , -, . , (1), '0 2 A, . , (1) - , , - . , , , , , , .
, '(t) , , - '(t) (1) - .
M' K(') '+ S': M '(t) :
A'0 K('(t)) '0 +K('0) '(t) + S'0:(, K(') '.) ' = '(t) + '0, - '(t) '0 :
@'0
@t+A('(t))'0 = 0; '0jt=0 = '00: (3)
-
101
, A M K('0) '(t), , , - '(t). ( .)
2.
-. , , (1) ( f S - ), -, '0 '0 - . , - . , , , , , . - [2]. , - .
:
q H + @@p
p2
m2@H
@p;
H { , { , { , m2 > 0. , - , -.
q
@q
@t+ 2J(H; q) = ~": (4)
~" { , ,J { , 2 =const > 0.
p = p0; p1
@
@t
@H
@p= f1(H); p = p0; p2
@H
@t= 0; p = p1:
-
102 . .
L :
L
8>>>>>:+
@
@p
p2
m2@
@pp 2 (p1; p0);
@
@pp = p0;
p2 p = p1:
L@H
@t= F (H; ~"):
L { , :
@H
@t= L1F (H; ~"):
L1 { . ( ) - , . - { . , - , : -, , , - - L.
- , - , - , . .
- (1) - [3].
(1):
@'
@t+K(') ' = S'+ f; 'jt=0 = '0; t 2 [0; T ]
@'
@t+K(') ' = S + f; 'jt=T = 'T : (5)
(') S :
K(') = K('); S = S;
-
103
(5) :
@'
@t+K(')' = S'+ f; 'jt=0 = '0;
(6)
@'
@tK(') ' = S'+ f; 'jt=T = 'T :
(6) ', ' .
@
@t('; ') + (K(')'; ') + (K(')'; ') = (f; ') (f; '):
(K';') = (';K') = (K'; ')
('; ')T ('; ')0 =TZ0
(f; ') dtTZ0
(f; ') dt: (7)
, , ' . 'T ; f
, - . , , .
(6) , '(t) '. . , : - ( ).
, - '(t) (0;1) t =1. , .
t1 = T t, ' :
@'
@t1K(') ' = S' + f1; 'jt1=0 = '0:
'0
0 { .
@'0
@tK(')'0 = S'0 : (8)
-
104 . .
(8) '0
,
12@k'0k2@t
= (S'1; '0) 2k'k2;
k'0k e2tk'00 k:
- .
@'0
@t1K(')'0 = S'0 + f; '0 jt1=0 = 0:
, f , f , . :
12@k'0k2@t
2k'k2 + (f; '0)
2k'0k2 + kfk k'0k 2k'0k2 + 12"kfk2 + "
2k'0k2 =
="2 2
k'0k2 + 1
2"kfk2:
, , " = 2, :
@
@tk'0k2 2k'0k2 + 1
2kfk2;
k'0k2 (1 e2t) 12kfk2 1
2kfk2:
, , - , . , , - , , , - (7) . , - .
3.
. . (1) . - { f . ,
-
105
- ( ) . (1).
@'
@t+K(') ' = S'+ f
, - .
- , 1959 . [4] , . - . - ; , , (1) S = 0; f = 0, - (') ( ). , :
duidt
= Qi(u); i = 1; : : : ; N; u 2 RN ;(9)
ui(0) = ui0:
(9)
NXi1
@Qi@ui
= 0
NXi1
u2i = C:
, -
(u) = AeP
u2i
B ;
; { .
du0idt
= Qi(u0) + fi; u0i(0) = ui0: (10)
ui = u0i ui. , kfk , (0; t) kuik , f 6= 0 t 0. (10)
-
106 . .
u. ui
ui =
tZ0
Xj
gij(t; t0)fj dt0; (11)
u =
tZ0
G(t; t0)f dt0;
G(t; t0) . : -
(11) , (u), kuk:
< G(t; t0) >=< G() >= C()C1(0); = t0 t; (12) () =< u(t+)u0(t) > { . - (12) , , . ,
< u >=
tZ0
C()C1(0)f d: (13)
(13) t ! 1 - f :
< u >=
1Z0
C()C1(0)f d: (14)
( [5].) - : (1), ""? "-" , .. : -, . "", , - . :
kfkfk = " 1:
-
107
, (1) :
1. ' =< ' > { (< > { , .. ).
2. E = ('; ') { .3. E
= ('; ') { .
4. E E = (' '; ' ') { .
5. D(E) = [('; ') ('; ')]2 { . '0 = ' '.
k'0kk 'k =
(E E)1=2E1=2
:
:
"1 =D(E)E2 ; "2 =
k'0kk 'k :
"1 1 , 1="1 . , 1="1 [6], '
N =1"1
=(Pi)2P2i
;
i { C(0). , - : 1 N N . , , - ( .. - - ) , . , - ( ), , " " . , "1 1 (13) (1) ( , ) .
, , - , (13) .
, "2 1. B , , - "2 '. , [7]:
@'0
@t+A1'0 = 0; (15)
-
108 . .
1 (1), ' .
, 0 - - F < 0 0T >. - '0 : C0 < '0 '0T >
A1C0 + C0AT1 = F
( , Re(A1) > 0), : C() < '0(t+ ) '0(t) >
C() = eA1C0 ( > 0):
(15) f , -
< '0 >= A11 f;
11 , ,
A11 =
1Z0
C()C10 d;
.. - .
, , , .
- (, ). "" , , ( - ). , -, "" - . , . , - - .
, -
,
-
109
, ... , -, . , (, , ) - , .
1. .., .. .{.: , 1994. {252 .
2. .., .. // . 1958. . 2. . 66{104.
3. .. .{.: , 1974. {303 .
4. Kraichnan R.H. Classical uctuation-relaxation theorem // Phys. Rev. 1959.V. 109. P. 1407{1422.
5. Leith C.E. Climate response and uctuation dissipation // J. Atmos. Sci. 1975.V. 32, 1. P. 2022{2026.
6. - .. , "-" // -. 1969. 2. . 24{36.
7. .. // .. . 1998. . 34, 6. . 741{751.
-
1
. . , . .
E-mail: [email protected]
- . { - , , { - ; - , - .
. - . . - . , , . - : A, B, C. - . . . - , .
The work deals with the numerical modelling of the marine dynamics andan analysis of the sensitivity of the obtained solutions.Our goal is threefold:
1 , 00-05-64051
-
111
{ to develop numerical model of the marine general circulation that simulatesthe large-scale structure of the hydrological elds, its time-space variability andlocal peculiarities{ to construct a cost-eective exible computational algorithms for the modeland retrospective data analysis of the solution{ to develop methods for assessment of the solution on the base of adjoint equa-tion technique.The model is based on the primitive equation system of thermohaline sea dy-namics. The governing equations are written in the bottom following system ofcoordinates (-system).The special symmetrized form of notation of the dierential equations is usedin the model. The operator of the problem prior to its space approximation isrepresented as the sum of suboperators of a simpler structure. The numericalalgorithm of the model is based on implicit splitting schemes and the grid ap-proximations of the split problems with respect to space variables, which obeythe main conservation law that holds for the original system.The space approximation of the model equations is realized on staggered com-bined grids: the combinations of grids A, B, and C can be used. The model isimplemented on a personal computer with rather cost-eective code.The sensitivity of the solution is considered. Adjoint method is used for this pur-pose. In order to characterize state of the system a special functional dependingon the solution of the adjoint problem is introduc