Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.iki.rssi.ru/earth/pres2006/sushkevich2.pdf Äàòà èçìåíåíèÿ: Mon Feb 19 19:17:04 2007 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 13:42:47 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 64

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(, ) .. , .. , .. , .. , ..
tamaras@keldysh.ru

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.. " " 13-17 2006 , ,

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This is Slide No. 2.

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2005 40 , , . 2004 IBM NASA ( 135 ). , , .
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- . . -- , - .

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- . - " ­ " , , . - - .

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: (1) "", "", .. ; (2) ; (3) ; (4) . : - "", "".

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This is Slide No. 6. : (). , . , , . , , , , , , .
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, , . - , . . - , . - , ( ) .
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This is Slide No. 8. . - . - -- . - -- .

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This is Slide No. 9. , .. . , - , (- < x, y < , r = (x, y )) (0 z h) . c

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t = {z , r, s : z = 0, s +}, b = {z , r, s : z = h, s -}.

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This is Slide No. 10. ( ) (r, s) ^ ( R 0 )
cos + sin cos + sin sin + (z )(x, y , z , , ) = z x y
2

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= s (z )
0

d
0

(z , , ; , ) (x, y , z , , ) sin d + F in (z , , ), (x, y , z = 0, , ) |t = F 0 (x, y , , ), (x, y , z = h, , ) |b = F h (x, y , , ) +

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This is Slide No. 11.
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^ ^ K = F in, |t = F 0, |b = R + F

h

(1)

:

^ ^ ^ ), Dz µ + (z ); D (s, g rad) + (z ) = Dz + (s, r z
,

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^ S s(z )


^ (z , s, s )(z , r, s ) ds , ds = dµ d , S (1) 1.
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^ ^ ^ K D - S ; ^ ^ ^ Kz Dz - S .

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This is Slide No. 12.
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^ [R](h, r, s)

+

q (r, s, s+)(h, r, s+) ds


+

(2)






:

^ R(1) = q (r, s) 1.
( ), (2), : - - ; - - ; - - ; - - .
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F 0 , F h , q r , (1) (x, y , z , , ) , . . .. ( // . 1956. . I. . 3. C. 328­343.); .. ( // . . 1957. . 113. 3. C. 541­543). "" .. , .
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This is Slide No. 14. (1)

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= a + q .
a () ""

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^ K a = F in, a |t = F 0, a |b = F h.

(3)

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q , , --

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^ ^ K q = 0, q |t = 0, q |b = Rq + E ,

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^ E (r , s) Ra -- (,
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^ K = 0, |t = 0, |b = f (sh; r, s)

(5)
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^ (sh; z , r, s) = P (f ) (, f )


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ds

-

- h -

(s ; z , r - r, s)f (s ; r

- h

h

- , sh

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(6) (s-; z , r, s) -- h

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^ K = 0, |t = 0, |b = f (s-; r, s) h
s- - h

(7)

f (s-; r, s) = (r) (s - s-) . h h
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, s- , h z = h x, y .

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f (r ) -- - , (5) -- , :


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^ (z , r, s) = Pr (f ) (r , f )
-

r (z , r-r, s)f (r) dr
(8)



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r (z , r, s) =

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This is Slide No. 18. f (sh ; s) (5) --

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1 ^ (s ; z , s) = Pz (f ) (z , f ) 2
h

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z (s ; z , s)f (s ; s ) ds .

-

- h

h

- h

- h

(11) --


z (s ; z , s) =
-

- h

(s-; z , r, s) dr h



(12)

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^ Kz z = 0, z |t = 0, z |b = (s - s-) h

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^ (z , s) = Pc(f ) = f W (z , s), f = const,


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1 W (z , s ) = 2




ds

-

- h -

(s-; z , r, s) dr h 1 2



=
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-

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=



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z (s-; z , s) ds-, (15) h h

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^ Kz W = 0, W |t = 0, W |b = 1.
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- - (5) r :


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g (p) = F [f (r)]
-

f (r) exp[i(p, r)] dr, B F [],
(17)
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p = (px , py ) (- < px , py < ) , :

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^ L(p)B = 0, B |t = 0, B |b = g (sh; p, s)


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^ ^ ^ L(p) Dz - i(p, s) - S .
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(18)

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1 ^ B (sh; z , p, s) = (g ) (, g ) 2



-

(s-; z , p, s)g (sh; p, s-) ds-. h h h
(19)

(19) -

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(s-; z , p, s) = F [(s-; z , r, s)] h h
c s- - p -- h

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^ L(p) = 0, |t = 0, |b = g (s-; p, s), h g (s-; p, s) F [f (s-; r, s)] = (s - s-) . h h h

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This is Slide No. 23. (20) (5),

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r (z , p, s) F [r (z , r, s)] =


1 2



-

(s-; z , p, s) ds-, h h

^ L(p)r = 0, r |t = 0, r |b = 1,

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gr (p) = g (p) = F [f (r)] .
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^ Br (z , p, s) = F [(z , r, s)] = r (gr ) (r , gr ) r (z , p, s)gr (p).
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This is Slide No. 24.
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Contents

q (sh; z , r, s) =
k =1

k k

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(4) (5)

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^ K k = 0, k |t = 0, k |b = Ek ^ Ek = Rk
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k 2 , E1 = E .
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This is Slide No. 25.
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, :

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^ ^ [Gf ](sh; h, r, s) R(, f ) =

+

q (r, s, s+)(, f ) ds+ = f (sh; r, s- ) dr â

1 = 2 â

+



ds

-

- -

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q (r, s, s+)(s- ; h, r - r, s+)ds+.
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1(z , r, s) = (, E ), ^ ^ k (z , r, s) = (, Rk-1) = (, G
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This is Slide No. 26.
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(4) (6) -- ()

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q = (, Y ),

(23)

"" ()


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Y
k =0

^ Gk E =
k =0

^ ^ Rk , R0 = E ,

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This is Slide No. 27. - (17)


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Bq (sh; z , p, s) F [q (sh; z , r, s)] =
k =1

k Bk , Bk F [k ],
(25)
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(18) (V F [E ]) :

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^ k = 1 : L(p)B1 = 0, B1 |t = 0, B1 |b = V (p, s); ^ ^ k 2 : L(p)Bk = 0, Bk |t = 0, Bk |b = T Bk
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This is Slide No. 28.
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:

^ ^ ^ [Qg ](sh; h, p, s) F [Gf ] = T (, g ) = 1 = (2 )2 1 = 2 â

+
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dp
- -
+

v (p - p , s, s+)(, g ) ds
2 +

+

=

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1 ds (2 )

g (sh; p , s- ) dp â
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This is Slide No. 29.
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(25) (19):

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^ ^ B1 = (, V ), Bk = (, T Bk-1) = (, Qk-1V ) .
C (25) -- - (4) (19)

Bq = (, Z ); q = F -1[Bq ] = F -1[(, Z )].

(26)

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- "" ( -):


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F [Y ] = Z
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^ QV =
k k =0

^ T Bk .

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This is Slide No. 30.
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(4) . - "" , . (23) -- , (4) (-, - ).

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This is Slide No. 31.

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S P - (-) - = (I , Q, U, V )T :

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Q = I p cos 2 cos 2 , U = I p sin 2 cos 2 , V = I p sin 2 ,
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-- , -- , 0 p 1 -- .

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This is Slide No. 32.
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^ ( R 0)

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^ ^ K = F, |t = F0, |b = R + F
:

H

(28)

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^ ^ D (s, g rad)+ (z ) = Dz + s, r



^ , Dz µ + (z ); z
Page 32 of 64

( )

^ S s(z )


^ P (z , s, s )(z , r, s ) ds , ds = dµ d ;

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^ [R](H, r, s)

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This is Slide No. 33.
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(28)­(29)

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= 0 + R .
0 "" ^ () (R 0) :

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^ K 0 = F, 0 |t = F0, 0 |b = FH .

(30)

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R , , -- ^ (R 0 , E 0)

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^ ^ K R = 0, R |t = 0, R |b = RR + E,

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^ E(r , s) R0 -- (, ) , , .
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This is Slide No. 34.
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^ K = 0, |t = 0, |b = f (sH ; r, s).

(32)
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- (sH ; z , r , s) , m , m = 1, ..., M , M = 4 , im :

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= i11 + i22 + i33 + i44,
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This is Slide No. 35. , f fn (sH ; r , s) , (32)
N

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(r, s) =
n=1

n(r, s),
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^ K n = 0, n |t = 0, n |b = tnfn


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tn = {mn}, m = 1, . . . , M , n = 1, . . . , N ,
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This is Slide No. 36. , (33) n :

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n = (n, fn) = 1 = 2


ds

-

- -

n(s-; z , r - r, s)fn(sH ; r, s-) dr.
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n = {mn }, n = 1, . . . , N , mn (s- ; z , r , s), m = 1, ..., M ,

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^ K n = 0, n |t = 0, n |b = tnf , f (s ; r, s) = (r) (s - s ),
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This is Slide No. 37. n = {mn (z , r , s)} :

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^ mn = [P (f )]mn (mn, fn)
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1 2



ds

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- -

mn(s ; z , r - r, s)fn(s ; r, s ) dr.
H

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This is Slide No. 38.
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, N n :

Contents

^ =

11 ... m1 ... M 1

. . . . .

. . . . .

. . . . .

1n ... mn ... M n

. . . . .

. . . . .

. 1N . ... . mN . . ... . M N

(35)
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m = 1, ..., M , M 4, mn ^ n , n = 1, ..., N , N 4, tn (33), n , , , ^ (35).

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This is Slide No. 39.

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^ ^ = P (f ) (, f ) = {m}, m = 1, . . . , M , M 4.
(36) (32),
N
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m

^ = [P (f )]m
n=1

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This is Slide No. 40.
Title Page




Contents

R =
k =1

k k ,
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^ k = 1 : K 1 = 0, ^ k 2 : K k = 0,

1 |t = 0, k |t = 0,

1 |b = E;
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^ k |b = Rk-1.
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This is Slide No. 41.
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, (35):

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^ ^^ [Gf ](s ; H, r, s) R(, f ) =
-
+

^ ^ q (r, s, s )(, f ) ds .
+ +

(37)­(38) (35) :

^ ^^ ^^ 1 = (, E); k = (, Rk-1) = (, G

k -1

E)

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(31) ( = 1) :


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^ ^ R = (, Y); Y Y E
k =0

^ Gk E =
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^ Rk (39)

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This is Slide No. 42. (39), (31), "" z = H X - , -- (31) (39).

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This is Slide No. 43.

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- - r (32) :

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^ L(p)B = 0, B |t = 0, B |b = g(sH ; p, s)


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^ ^ ^ L(p) Dz - i(p, s)-S ; (p, s) = px sin cos +py sin sin .

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This is Slide No. 44. (40)
N

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Contents

B(z , p, s) =
n=1

Bn(z , p, s),
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^ L(p)Bn = 0, Bn |t = 0, Bn |b = tngn,

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gn(sH ; p, s) F [fn(sH ; r, s)] .

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This is Slide No. 45.
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(41) n = 1, ..., N

Contents

Bn(z , p, s) = (n, gn) =

1 2

n(s-; z , p, s) gn(sH ; p, s-) ds-,

-

n = {mn (s- ; z , p, s)}, n = 1, ..., N , N 4 , c s- - p --

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^ L(p)n = 0, n |t = 0, n |b = tng ,


(42)
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g (s ; p, s) F [f (s ; r, s)] = (s - s ) .
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This is Slide No. 46. - Bn = {Bmn} :

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^ Bmn(z , p, s) = [T (g)]mn (mn, gn) =

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=

1 2

mn(s-; z , p, s) gn(sH ; p, s-) ds-.

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(43)

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This is Slide No. 47. , N n :

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^=

11 ... m1 ... M 1

. . . . .

. . . . .

. . . . .

1n ... mn ... M n

. . . . .

. . . . .

. 1N . ... . mN , . ... . M N

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(44)
Contents

(40) :

^ ^ B = T (g) (, g) = {Bm}, m = 1, . . . , M , M 4,
(45) - ,
N

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^ Bm = [T (g)]m
n=1

(mn, gn),

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This is Slide No. 48.
Title Page

- -


Contents

BR (z , p, s) F [R (z , r, s)] =
k =1

k Bk (z , p, s) (46)
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-- (Bk F [k ]) :

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^ k = 1 : L(p)B1 = 0, ^ k 2 : L(p)Bk = 0,

B1 |t = 0, Bk |t = 0,

B1 |b = W;
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^ Bk |b = T Bk-1.
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This is Slide No. 49.
Contents

, (44):

^ ^ ^^ [Qg](s-; H, p, s) F [Gf ] = T (, g) =


=

1 (2 )

2 -

dp

+

^ ^ v (p - p , s, s+) (, g) ds+ ,

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(47)­(48) :

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^ ^^ ^^ B1 = (, W); Bk = (, T Bk-1) = (, Qk-1W).

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This is Slide No. 50.
Title Page

C (46) -- - (31) S -

Contents

^ BR = (, Z),

(49)
Page 50 of 64

, - ""


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^ Z ZW
k =0

^ Qk W =
k =0

^ T Bk

(50)
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-- ( -).

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This is Slide No. 51.
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, . domain decomposition. , , " - - " " - - - - ".
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This is Slide No. 52.
Contents

, , . . , M .

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¡

This is Slide No. 53.
d , 1
F1 1 1

¡ ¡ ¡ ¡ ¡

¡ ¡

¡ ¡ ¡ ¡ ¡ ¡ ¡

¡ ¡ ¡
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m = 1 = top
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d , 2 d , 2

,
F2

F1

1

,

1 m=2
Contents

. . .
d
1

d , m d , m

Fm-

^ s d Tm d d d © d d s d ^ d d Fm T © ^m Rm d d

Fm

^ R

m



m

, m

m

, m m m+1

Page 53 of 64

d , M d , M d , M + 1

Fm+1

. . .
FM FM

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m=M
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M

,

M

^ R
d

b

, M M
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d d

m = M + 1 = bottom
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This is Slide No. 54.

Title Page

Contents

^ ^ K = Fin, |t = F, |b = Rb + F, t b
m = 2 Â M :
Page 54 of 64

|d |d

,m ,m

^ = (R +
m

^ Tm

) + F

m-1

,
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^ ^ = (Rm + Tm) + F ; m
M

:

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F = F ; F t 1

= F ; d , 1 = t ; d , M + 1 = b ; b

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This is Slide No. 55.
Title Page

, :


Contents

=
n=0

n(n

)

2M : n -

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(n) = {1 , 1 , 2 , 2 , . . . , (n), (n), . . . , M , M } ; m m
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(n)

(n)

(n)

(n)

(n)

(n)



= {, , , , . . . , , , . . . , , } ; 1 1 2 2 m m M M

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This is Slide No. 56. n -

Title Page

Contents

F

(n)

= {F

(n) 1

,F

(n) 1

,F

(n) 2

,F

(n) 2

,...,F

(n) m

,F

(n) m

,...,F

(n) M

,F

(n) M

};



= {, , , , . . . , , , . . . , , } ; 1 1 2 2 m m M M


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E = {E , E , E , E , . . . , E , E , . . . , E , E } ;
""
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1

1

2

2

m

m

M

M

Z = {Z, Z, Z, Z, . . . , Z , Z , . . . , Z , Z } . 1 1 2 2 m m M M

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This is Slide No. 57.
Contents

2M -- : m = 1 Â M

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^ K (0) = Fin, (0) |d m m m ^ K (0) = Fin, (0) |d m m m

,m ,m

= F , (0) |d m m = 0, (0) |d m

,m+1

= 0;
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,m+1

= F . m
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This is Slide No. 58.
Title Page

n 1 2M m = 1 Â M :

Contents

^ ( K mn) = 0, (n) |d m

,m ,m

=F

(n-1) m

, (n) |d m
,m+1

,m+1

= 0;
(n-1) m
Page 58 of 64

^ ( K mn) = 0, (n) |d m


= 0, (n) |d m

=F

F

(n) 1

= 0; F

(n) M

^ (n) ^ (n) = Rb M + Rb M ;

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F F

(n) m

^ ^ ^ (n) ^ (n) = Tmm-1 + Tmm-1 + Rm(n) + Rm(n) ; m m
m+1

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(n) m

^ =R

^ (n) + R m

m+1

(n) (n) ( ^ ^ mn) + Tm+1m+1 + Tm+1m+1 .
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This is Slide No. 59. m = 1 Â M :

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Title Page

Contents

(n) = , F m m

(n-1) m

; (n) = , F m m

(n-1) m

.

z [hm, hm+1] :
, fm (z , s) = m , fm (z , s) = m fm(hm, s ) m sm ; fm(hm, s ) m sm .
+
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(s ; z , s)fm(hm, s )ds ; mm m m

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(s ; z , s)fm(hm+1, s )ds ; mm m m
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-- z = hm
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This is Slide No. 60.
Contents

-- m = 1 Â M --

^ K ^ K


m m

= 0, |d m = 0, |d m

,m

= f,m, |d m = 0, |d m

,m+1

= 0;
Page 60 of 64

,m

,m+1

= f,m ,

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f,m = t (s - s ) , s ; m m m f,m = t (s - s ) , s . m m m

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This is Slide No. 61. n -

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Title Page

(n) = , F

(n-1)

,F

(n-1)

^ = Ptr (n

-1)

,

Contents

^ Ptr -- . - , :

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^ ^ = GF = Ptr (, F).
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n -

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(n) =

^ , Ptr (n

-1)

=

^ , G
·

n-1

E.
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This is Slide No. 62.
Title Page

-- () :

Contents

= (, Z) .
"" -- Z

Page 62 of 64

^ Z ZE
n=0

^ Gn E = E +
n=0

^ Ptr (n) = E +
n=0

F

(n)

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This is Slide No. 63. :

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Title Page

Contents

1. .
Page 63 of 64

2. "" - . 3. , "" .
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Contents

This is Slide No. 64. Perl.
Page 64 of 64

....
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