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Difference methods of solving the Cauchy problem for elliptic equations
256
P. N. Vabt~hchevich
Remark 2. At each stage of the algorithm the inverse basis matrix B [No, M] obtained here is used. And f i n 0 n S ~ O, then because of the change of signs of the remaining artificial condition vectors, it is necessary to change the signs of the elements of the matrix B [NO", M]. Translated by J. Berry
REFERENCES 1. ROMANOVSKII, L V. Aigorithnu for solvln8 exrremal problems (A]goritmy reshenii eks~emal'nykh zadach). Nauka, Moscow, 1977. 2. KUNZI, H. P. and CRELLE, V. Non.linear programming (Netlineinoe programmitovanie), Soy. radio, Moscow, 1965. 3. BULAVSKII, V. A. Note on the beginnin8 of the computation in linear programming. In: Optimalplanning (Optimarnoe planirovanie). No. 15, 76-78. Novosibirsk, 1970.
U.S.S.R. Compu~ Maths. Math. Phys. Vol. 21, No. 2, pp. 256-259, 1981. 0041-$553/81/020256-04507.50/0 Printed in Great Britain
©1982. Perpmon Press Ltd.
D I F F E R E N C E METHODS OF SOLVING THE CAUCHY PROBLEM F O R ELLIPTIC EQUATIONS* P. N. VABISHCHEVICH Moscow
(Received 25 May 1979; rev/sed 26 May 1980)
USING the example of a model problem for Laplace's equation in a rectangle, difference methods are considered for solving the iU-posed Cauchy problem for elliptic equations. It is proposed to accomplish an additional regularization with the aid of smoothing sptines. The Catchy problem for elliptic equations belongs to the class of problems that are correct by Tikhonov's definition [1, 2]. In this paper it is solved by difference methods with smoothing of the solution by spline~ 1. We consider the simplest Cauchy problem for the Laplace equation:
u=+u~-0,
(1) (2)
~'Zh. v.VchiR M~t. mat. F ~ , 21, 2, 509-511, 1981.
257
Short communications
(~--1, 2 .... ,M, zt~=0, z,w=l), g2.=(j-l)hu (j=-l, 2,...,N, gt,.=0, g.~=l) be a mesh uniform in x and y with steps hx and hy respectively. We denote by uii the mesh solution of problem (1), (2), and by vi/an auxiliary mesh function. From the known ui, 1- 1, ui,/-2, approximating Eq. (1), we determine Let ~,==(i-t)h,
hv z
v,~=2u, ~-,-u,.~_~
-
~
(u,.~ ~-,-2u,.~-,÷ u,-~ ~_,).
(3)
We then smooth the mesh function vi~ and take it as the solution u i / o f problem (1), (2) on the/-th layer. 2. To smooth l V for constant / we use cubic sphnes [3]. The smoothing problem reduces to finding a cubic spline $1 (x) minimizing the functional 51
.V~_-- ~ , [S;(z,) -~',,]:+~9.[Sj].
where c~ is the regularization parameter, and t
.q[Sd= ~ [S/'(z)]2 d: o
is the stabilizing functional If we write Pij "=S f ' (xi), we obtain a linear system of equations for determining Pij, i = 1, 2 . . . . , M, with constant/: ptj----0, a2..p2;~ aczpz ~ a : . p . : ~ ]2:, a,.,-2p,-: ~+a, ,_ip,_~ ;+a,.p,~+a , ~ p , . ~ i-3,
4 .....
;+a,.,-2p:*2 :~f,:,
M-2,
a ~-i,ts-sp.w-
a,~-~ a . ~ - i . . ~ - 2 p . w - 2,~+ a.~s- t M-- tp M-- L J ~ [ . ~ - J.~.
Here a~. ,+2--a~. ~-~'=3ah~ -3, a~, ~+~--a,. ~_~= 0.5-12¢thZ 3, ~-2vo+v,-~,~), ~-=1, 2,...,M, 1==1, 2. . . . . N.
a,-*2+tScth~ -s,
System (4) is solved by using a five.point pivotal condensation [4]. Then the difference solution of the Cauchy problem u o ~ r'~;-ah~ -1 (P,+ i, j-2p,~+p,_ 1, j),
is determined, where vii is taken from the relation (3). The parameter a was chosen from the remainder J~
(u,~- v,j)-h;=,:l:o (cO ==6:.
:~S$ c 2':~
- O
(4)
P. N. Va~che~ch
258
The value of the remainder/5 was determined on each/-th layer by the error of the f'mite-difference approximation of Eq. (1). When using (3) the error in determining vi] at each i-th point was estimated by the error of approximation of the Laplace operator, multiplied by hy 2, that is, by the expression
This gives the estimate for the remainder M
62~ ~
Avi~Zh~'
which may depend strongly on y. To solve the non-linear equation ¢02 (a) =/5 2 the method of secants was used, and therefore a was determined in terms of the given initial a 1 and a2 by the iterative sequence a,,+ ~-~,,~ (,,',, + (~,-~,,_ ~) [qDoi(a,,) -8 i ] [q~oi (c~.) -to I (c~.-l) ] -l) -l.
We note that the parameter a can also be found by using the tangent method [51. 3. As a test problem (1), (2) with an exact solution ~(z, y) ,-sin (a kz) sh (aky) considered. The error of the difference solution was estimated by the expression M
)-'
M
i~l
was
I" l 1
~t
0
0.5
u
t.O
t,{tE
LgE
-5
-5
- iO
-
FIG. 1
0
k=5
SO FIG. 2
;.0
Short communications
0
259
0.5
Y
I
II
f.g 1
-5
-fO FIG. 3
Figure 1 shows e as a function o f y in the difference solution of (1), (2)without regularization (a = O, k = 1) with a different number of points in the variables x and y. Here curve 1 is for M = 20, N = 20, 2 is for M = 10, N = 20, 3 is for M = 30, N = 20, 4 is for M = 20, N = 10, 5 is for M = 20, N = 30. It is seen that the use of a large number of points gives the best accuracy for small values o f y, and more open meshes give a more exact approximate solution for large values ofy. The effect of the various harmonics (k = 1. . . . . 5) is illustrated in Fig. 2, where M = 20, N = 20. Figure 3 shows the variation of e (3') on smoothing with various regularizing parameters;M = 20, N = 20; curve 1 is for a = 10 -1, curve 2 is for a = 1 0 - 3, curve 3 is for a = 10 - 5, and curve 4 is for ,, = 1 0 - 7. Also given here is an example of the solution of the problem for a given remainder (curve 5 is for = 1.45.10-4, 82 = 1 0 - 1 0 , j = 2). Translated by J. Berry
REFERENCES
1. LAVRENT'EV, M. M. On some ill.posed problems of mathemarioM phy~cs (0 nekotorykh nekorrektnykh zadachakh matematicheskoi fiziki). Izd.vo SO Akad. Nauk SSSR, Novott'birsk, 1962. 2. TIKHONOV, A. N. and ARSENIN, V. Ya. Methods o/solving ill.posed problems (Metody resheniya nekorrektnykh zadach), Nauka, Moscow, 1974. 3. STECHKIN, S. B. and SUBBOTIN, Yu. N. Splines in computational mathematics (Splainy v vychislitel'noi matematike), Nauka, Moscow, 1976. 4. SAMARSKII, A. A. and NIKOLAEV, E. S. Methods of solving mesh equations (Metody reshenJya setochnyk.h mavnenii). Nauka, Moscow, 1978. 5. REINSCH, C. H. Smoothing by spline function. Numer. Math., 10, 3, 177-183, 1967.
P. N. Vabt~hchevich
Remark 2. At each stage of the algorithm the inverse basis matrix B [No, M] obtained here is used. And f i n 0 n S ~ O, then because of the change of signs of the remaining artificial condition vectors, it is necessary to change the signs of the elements of the matrix B [NO", M]. Translated by J. Berry
REFERENCES 1. ROMANOVSKII, L V. Aigorithnu for solvln8 exrremal problems (A]goritmy reshenii eks~emal'nykh zadach). Nauka, Moscow, 1977. 2. KUNZI, H. P. and CRELLE, V. Non.linear programming (Netlineinoe programmitovanie), Soy. radio, Moscow, 1965. 3. BULAVSKII, V. A. Note on the beginnin8 of the computation in linear programming. In: Optimalplanning (Optimarnoe planirovanie). No. 15, 76-78. Novosibirsk, 1970.
U.S.S.R. Compu~ Maths. Math. Phys. Vol. 21, No. 2, pp. 256-259, 1981. 0041-$553/81/020256-04507.50/0 Printed in Great Britain
©1982. Perpmon Press Ltd.
D I F F E R E N C E METHODS OF SOLVING THE CAUCHY PROBLEM F O R ELLIPTIC EQUATIONS* P. N. VABISHCHEVICH Moscow
(Received 25 May 1979; rev/sed 26 May 1980)
USING the example of a model problem for Laplace's equation in a rectangle, difference methods are considered for solving the iU-posed Cauchy problem for elliptic equations. It is proposed to accomplish an additional regularization with the aid of smoothing sptines. The Catchy problem for elliptic equations belongs to the class of problems that are correct by Tikhonov's definition [1, 2]. In this paper it is solved by difference methods with smoothing of the solution by spline~ 1. We consider the simplest Cauchy problem for the Laplace equation:
u=+u~-0,
(1) (2)
~'Zh. v.VchiR M~t. mat. F ~ , 21, 2, 509-511, 1981.
257
Short communications
(~--1, 2 .... ,M, zt~=0, z,w=l), g2.=(j-l)hu (j=-l, 2,...,N, gt,.=0, g.~=l) be a mesh uniform in x and y with steps hx and hy respectively. We denote by uii the mesh solution of problem (1), (2), and by vi/an auxiliary mesh function. From the known ui, 1- 1, ui,/-2, approximating Eq. (1), we determine Let ~,==(i-t)h,
hv z
v,~=2u, ~-,-u,.~_~
-
~
(u,.~ ~-,-2u,.~-,÷ u,-~ ~_,).
(3)
We then smooth the mesh function vi~ and take it as the solution u i / o f problem (1), (2) on the/-th layer. 2. To smooth l V for constant / we use cubic sphnes [3]. The smoothing problem reduces to finding a cubic spline $1 (x) minimizing the functional 51
.V~_-- ~ , [S;(z,) -~',,]:+~9.[Sj].
where c~ is the regularization parameter, and t
.q[Sd= ~ [S/'(z)]2 d: o
is the stabilizing functional If we write Pij "=S f ' (xi), we obtain a linear system of equations for determining Pij, i = 1, 2 . . . . , M, with constant/: ptj----0, a2..p2;~ aczpz ~ a : . p . : ~ ]2:, a,.,-2p,-: ~+a, ,_ip,_~ ;+a,.p,~+a , ~ p , . ~ i-3,
4 .....
;+a,.,-2p:*2 :~f,:,
M-2,
a ~-i,ts-sp.w-
a,~-~ a . ~ - i . . ~ - 2 p . w - 2,~+ a.~s- t M-- tp M-- L J ~ [ . ~ - J.~.
Here a~. ,+2--a~. ~-~'=3ah~ -3, a~, ~+~--a,. ~_~= 0.5-12¢thZ 3, ~-2vo+v,-~,~), ~-=1, 2,...,M, 1==1, 2. . . . . N.
a,-*2+tScth~ -s,
System (4) is solved by using a five.point pivotal condensation [4]. Then the difference solution of the Cauchy problem u o ~ r'~;-ah~ -1 (P,+ i, j-2p,~+p,_ 1, j),
is determined, where vii is taken from the relation (3). The parameter a was chosen from the remainder J~
(u,~- v,j)-h;=,:l:o (cO ==6:.
:~S$ c 2':~
- O
(4)
P. N. Va~che~ch
258
The value of the remainder/5 was determined on each/-th layer by the error of the f'mite-difference approximation of Eq. (1). When using (3) the error in determining vi] at each i-th point was estimated by the error of approximation of the Laplace operator, multiplied by hy 2, that is, by the expression
This gives the estimate for the remainder M
62~ ~
Avi~Zh~'
which may depend strongly on y. To solve the non-linear equation ¢02 (a) =/5 2 the method of secants was used, and therefore a was determined in terms of the given initial a 1 and a2 by the iterative sequence a,,+ ~-~,,~ (,,',, + (~,-~,,_ ~) [qDoi(a,,) -8 i ] [q~oi (c~.) -to I (c~.-l) ] -l) -l.
We note that the parameter a can also be found by using the tangent method [51. 3. As a test problem (1), (2) with an exact solution ~(z, y) ,-sin (a kz) sh (aky) considered. The error of the difference solution was estimated by the expression M
)-'
M
i~l
was
I" l 1
~t
0
0.5
u
t.O
t,{tE
LgE
-5
-5
- iO
-
FIG. 1
0
k=5
SO FIG. 2
;.0
Short communications
0
259
0.5
Y
I
II
f.g 1
-5
-fO FIG. 3
Figure 1 shows e as a function o f y in the difference solution of (1), (2)without regularization (a = O, k = 1) with a different number of points in the variables x and y. Here curve 1 is for M = 20, N = 20, 2 is for M = 10, N = 20, 3 is for M = 30, N = 20, 4 is for M = 20, N = 10, 5 is for M = 20, N = 30. It is seen that the use of a large number of points gives the best accuracy for small values o f y, and more open meshes give a more exact approximate solution for large values ofy. The effect of the various harmonics (k = 1. . . . . 5) is illustrated in Fig. 2, where M = 20, N = 20. Figure 3 shows the variation of e (3') on smoothing with various regularizing parameters;M = 20, N = 20; curve 1 is for a = 10 -1, curve 2 is for a = 1 0 - 3, curve 3 is for a = 10 - 5, and curve 4 is for ,, = 1 0 - 7. Also given here is an example of the solution of the problem for a given remainder (curve 5 is for = 1.45.10-4, 82 = 1 0 - 1 0 , j = 2). Translated by J. Berry
REFERENCES
1. LAVRENT'EV, M. M. On some ill.posed problems of mathemarioM phy~cs (0 nekotorykh nekorrektnykh zadachakh matematicheskoi fiziki). Izd.vo SO Akad. Nauk SSSR, Novott'birsk, 1962. 2. TIKHONOV, A. N. and ARSENIN, V. Ya. Methods o/solving ill.posed problems (Metody resheniya nekorrektnykh zadach), Nauka, Moscow, 1974. 3. STECHKIN, S. B. and SUBBOTIN, Yu. N. Splines in computational mathematics (Splainy v vychislitel'noi matematike), Nauka, Moscow, 1976. 4. SAMARSKII, A. A. and NIKOLAEV, E. S. Methods of solving mesh equations (Metody reshenJya setochnyk.h mavnenii). Nauka, Moscow, 1978. 5. REINSCH, C. H. Smoothing by spline function. Numer. Math., 10, 3, 177-183, 1967.