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.. () . . . , . ON STABILITY OF AN NONLOCAL DIFFERENCE SCHEME: NUMERICAL STUDY Morozova V.A. (Moscow) The numerical investigation is represented concerning stability with respect to initial data of the explicit difference scheme for heat conduction equation with nonlocal boundary conditions. The norms of powers of transition matrices are calculated. Stability criterions are checked for various grid spaces. The conditions numbers are obtained for matrices which defines energy norms. . , (. [1])
y
n +1

-y

n



+ Ay n = 0,

n = 0,1,K,

> 0,

y

0

,

(1)

tn = H,

y n = y (t n ) H -- n A - , H . , A n . H 198


.. -- -10, 2002, .198-199

( , )

y = ( y, y ) .

D - H , ( y, v) D = ( Dy, v) y D = ( Dy, y ) . E - -

H E - y = ( y , y ) . [2] (1) H D , y n D n ,
y
n +1 D

y

nD

, n = 0,1,K

(2)

y0 H . .
h = {xi = ih, i = 0,1,K, N , hN = 1}


yin = y ( xi , tn ), y
n +1 i

n xx ,i

=

yin+1 - 2 yin + yin-1 . h2


y
0 i

-y

n i


y = u0 ( xi ), y

=y
n +1 0

n xx ,i

, i = 1, 2,K, N - 1, hN = 1,
n x ,0

= 0, y

-y

n x,N

=

h y
2

n +1 N

-y

n N

(3)
,



. (3) (1) H y = ( y1 y2 L y N )T . (3) (1),
nn yn = y1 y2 L y

(

nT N

)



A
(4)

( Ay )i = - y

, i = 1, 2,K, N - 1, y0 = 0, 2 1 ( Ay ) N = - ( y x ,0 - y x , N ), = . h
x x ,i

199


, | | 1. (3) 0,5h 2 . (3), [3], , H D . . A. (4) (. [4]). | |1 4/h2 . | |<1, , H. =1 , [5], [6]. , 0 =0, 2, , . 1 min(A) A N = 40 . 4 N 2 = 6400 .
min(A) min(A)

1. A .

0 2.47 -0.1 2.79 H [5],
( y, v ) =
N -1

0.1 0.5 0.7 0.99 1. 2.16 1.10 0.63 0.02 0. -0.5 -0.7 -0.99 -1. 4.39 5.50 9.00 9.87 ,
y = ( y, y ) .


i =1

yi vi h + 0,5 y N v N h,

(5)

200


.. -- -10, 2002, .198-201

(4) A* , ( A* v ) i = -v xx ,i , i = 2, 3,K, N - 1, 1 1 ( A* v )1 = 2 ( 2v1 - v 2 - v N ), ( A* v ) N = 2 ( -2v N -1 + 2v N ). h h , A* AT , A . A* = R -1 AT R, R = diag(h, h,K, h, 0.5h ) - , (5). A A0 = 0,5( A* + A) . , N A0 , (4), . (3) H E > 0 . 0 A0 N . . 2. A0 N = 40 0 0.2 0.3 0.5 0.7 1. -0.0003 -0.0040 -0.0124 -0.0410 - - h 2 0
3. A0 N = 100



h 2 0

0 -

0.2 -

0.3 -0.0005

0.5 -0.0035

0.7 -0.0120

1. -0.0410

,
-1 2 N -1 2 1. N +1

. (3)
201


1 . 2 h
2



(6)

[5] , (3) = 1 1 , > 0 h 2 2(1 + )
y n M 1 y0 ,

(7)

M1 M1 0 . (7) L2 ­. (3) . y0 y0 ( x j ) = ( -1) j yn = S n y0 ,
S = E - A - n - . n . n .

. - L2 ­, (5). - y D = ( Dy , y ) , D = ( * ) -1 - , (5) A . , 1 (3) H D (6). y
n

y

nD



n

N = 40 = 0.7 .

= 0,5(1 + )h 2 , . > 0 , < 0 - . = 0 . 4. . N = 40 , = 0.7 , = 0 n 0 30 200 500 1000 0.994 1.004 1.017 0.971 0.881 yn
y
nD

0.182

0.173

0.163
202

0.154

0.139


.. -- -10, 2002, .198-203 5. .

N = 40 , = 0.7 ,

= -0.1

n

y y

n nD

0 0.994 0.182

30 0.017 0.003

200 0.011 0.002

500 0.010 0.002

1000 0.009 0.001

, = 0,5h 2 y n D n , , n . = 0.1 . , n = 30 y n = 238 , y n D = 41 . S n . L2 - S n .
6. . N = 40 ,

n
S
n

0 1

1 1.338

30 1.051

200 1.052

= 0.7 , = 0 500 1000 1.004 0.91 = 0.7 , = -0.1 500 1000 1.004 0.92

7. . N = 40 ,

n
S
n

0 1

1 1.222

30 1.044

200 1.046

, 0,5h 2 (3) L2 ( (7)), H E . . (1) - H D ,
y
nD

y

0D

,

(7), M 1 D : M 1 = max ( D ) / min ( D ) . , (3) 1
203


D D = ( * ) -1 , - , A . (6) H D . M 1 = -1 h . D * = R -1 T R , (5). 8. M 1 . N = 40 0 0.1 0.2 0.3 0.4 0.5 0.6 1.41 1.50 1.61 1.76 1.94 2.18 2.49 M
1


M

1

0.7 2.95

0.8 3.70

0.9 5.36

0.95 7.67

0.99 1.94

0.999 54.79

9. M 1

M N M

1

0.1 1.50

0.2 1.61

0.3 1.76

. N = 60 0.4 0.5 1.94 2.18

0.6 2.49

0.7 2.95

0.8 3.70

0.9 5.36

10. M 1 N .

= 0.7 3 10 20 30 40 50 100 2.32 2.97 2.95 2.95 2.95 2.95 2.95 1 . M 1 ( N ) N .

= 1 . M 1 .
= 0.1 M 1 1.5 . = 0.9 M 1 5.4 . N M 1 . . = 0 A (5) , , 1, 2 ,K, N .

, T E . 204


.. -- -10, 2002, .198-205

T R = E , R = diag(h, h,K, h, 0.5h ) . D 2 M 1 ( 0) = 2 . . . . . 1. .. . .: . 1989, 3- , 616 . 2. .., .. . .: . 1973, 416 . 3. . . . X ". . ." , 2003. 4. . . , . . . IY « ». , , 2002, . 52-55. 5. . . . . . ., . 15, . . ., 1977, N 2, . 20-32. 6. . . () . , Numerikus Modzerek, N 14, 1979, 70 .

205