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Economical schemes for the multidimensional heat conduction equation with discontinuous coefficients
ECONOMICAL SCHEMES FOR THE ~ULT~D~~ENSIONAL HEAT CONDUCTION EQUATION WITH DISCONTINUOUSCOEFFICIENTS* I.V. FRYAZINOV Moscow (Received
26 June 1971)
ECONOMICAL schemes with equations on graphs for the multidimensional conduction equation with discontinuous
coefficients
are constructed
heat
for a class
of non-stepped surfaces (lines) of discontinuity of the coefficients. The scheme considered is a generalization of a locally on~dimensional scheme. The transition from the time layer tj_t of the grid function
yj-l
to the layer tj = rj_i + r
(7 is the step of the time grid) of the function ,j is executed by the successive solution of difference
problems on graphs.
In this paper schemes for the multidimensional heat conduction equation with variable coefficients are constructed which are economical (absolutely stable and requiring O(1) arithmetical operations per grid node). The schemes are suitable for the solution of problems in the case of non-stepped surfaces (lines)
of discontinuity
(see Figs.
of the coefficients,
often encountered in technology
1, 2 for examples of lines of discontinuity).
The method considered below is one of the realizations of the method of total approximation and is a natural generalization of a number of economical methods [l-41. The transition from the time layer tj_1 of the grid function yj- I to the layer tj = tj_l + T (7 is the grid time-step) of the function yj is executed by the successive solution of problems at the grid nodes situated on graphs, and not on segments parallel to the coordinate axes as in a locally one-dimensional scheme.
The scheme proposed belongs to a class
and is regarded as additive,
of compound schemes
its approximation is understood in the over-all
sense (see [3, 41). Additive schemes with equations on graphs were previously considered
in k-71.
To prove the convergence *Z/L u>hisl.
Mat. mat. Fiz.,
of the scheme prior estimates
13, 1, 8081, 1973.
100
obtained in [6, 71
Schemes
for the multidimensional
FIG.
heat conduction
equation
FIG.
1.
2.
are used. The principles of the choice of graphs were given in [S, 71, and methods of solving problems on graphs were given in [8, 101.
1. Formulation of the problem 1. Formulation of the initial problem. In order not to encumber the treatment, we construct the scheme and study its properties by a specific example we confine ourselves to the case of two dimensions - where the spatial region is the rectangle Q = lo < xo < &, a = 1, 21, and a boundary first kind is specified on its boundary I?. Let I’r be a curve possessing equation
x, = [(x,), hb,)
continuous
where for definiteness
= 0,
hb,)
divides
left) with boundaries
sl,~Wt
r
the rectangle and rl,
8=HP~(O~tg9,
SO
and defined
~=div(~(~,t)grada)+i(r,t),
by the
that
dt/dx, > 0, d2@dxla > 0, d3(‘/dxiS are
fi into two parts C$, and f21 (right and
that I-‘, = I’ .rl l?l.
H=O
Let C$, = sl\r,,
uF,ilp=~~
u
We consider the following problem: find the function x,) continuous in Q which satisfies the equation
(1.1)
of the
0 < b, C b, < m,,
= %,
and that for b, < xl ,( b, the derivatives continuous (see Fig. 1). The curve r,
curvature we assume
condition
r,,&n,
Q, =
u rl.
u = U(X, t), x = (x1,
(5, t) E
Qo,
1. V. Fryazinou
102
the initial
and boundary u
(x, 0) =
0.2) u (x,
conditions
uO(&
t) = u cc,t),
625;
x
r,
x =
O
and the conditions of conjugacy on the curve TX, where the heat conduction coefficient x = x (z, t> becomes discontinuous:
(1.3) Here the notation [vf = u - u1 has been introduced; u , u1 are the right and left limits of the values Ef the function u on the curve P r; n is the normal to r,
inward with respect
to the domain fi,;
f, ZZ’,Y, X
are specified
functions.
Let
x(5, 2) >, ci = const > 0.
(1.4) We suppose up, TX,Y, f
that the function
f(~, t) also has a discontinuity
and ITI be such that a unique
solution
possessing
in Ql = El x (0 < t< T) and Qp = a,
continuous
fourth derivatives
with respect
for x ‘es rl.
of problem (L&(1.4)
Let exists,
x (0 < t\< T) (but not in Q) to x,, x, and second derivatives
with respect to t, the derivatives af/axa, a = 1, 2, continuous in Gl and Qp. In the case considered the conditions imposed on ri, f, V, 1c, U” and ensuring the smoothness of the solution of problem (l.l)-(1.4) indicated above, Nevertheless, in order to simplify are of course extremely rigorous. ment we confine ourselves to the particular case given above.
the treat-
2. The grid. points
We introduce in 3 the grid w, the nodes of which are the fir) f&f f i,) of the straight lines x, = x, , i, = Xi = (X, t X2 )E il of intersection (NJ
0, 1, . * I, N,, r’H’= 0, x a
=m,,x
curve IT,: the straight
passing
lines
(i,) (&-If o: >x a , a = 1, 2, matched
coordinate axes ox, and ox?, intersect grid 0 (see Fig. 1).
through the nodes zi E &, the curve r,
with the
parallel
only at the nodes
to the
of the
In technology, problems involving the heat conduction equation are often encountered where the surfaces (curves) lr‘, of discontinuity of the heat-conduction coefficients are such that it is possible to construct a sequence as h + 0 (h is the maximum spacing of the grid :I of grids 0, matched with lTl. For the solution
of such problems
schemes
are constructed
in the same way
Schemes
for the multidimensional
heat conduction
as is done below for the particular case considered convergence
of the schemes
103
equation
(see also L.5, 71). The
is studied on a sequence (as h + 0) of grids matched
with rl. We will use a system of notation without subscripts, 5 = 5i = @)
_&(‘&0.5)
-&%*m)))
0.5
=
Z(mI) = (Zi(il**) ) zz(iz) ))
) &@z) ))
Z(m*) = (Q’),
(kO.5)
= Ix
a
(i,*l)
of the grid o,
a
-
m = 0.5,
772= 1,
(&(‘cL*‘) + Z,%))~
We call the points x(ko-5 a ) intermediate Let h
putting
xza’I,
fia
=
nodes in the direction OX,(U = 1, 2).
0.s(h(&0*5)
+
h(iom5)),
;L =
1, 2 be steps
h = max max A, (5). XEWa=i,z We introduce the sets of nodes of the grid o:
10
=
a r-l sl,
and we associate
00
(f0.5,)
(kO.5) =4,-$, Ha
ri,
y=anr,
(kO.5) ,
a, P = 1, 2,
(*0.5a)
a + @
nodes x’ = x
and instead of x
(f0.5,)
(f0.5,)
E 0’
.
We
will often
we will write
We introduce the time grid wT=
3.
fl
0
the area
the set of all intermediate
omit the superscripts x’ E 0’.
yi =
with each node x E fl the area H(x) = ti;ti,, with each inter
mediate node x’ = x
We denote by o’
w n i-20,
=
Graphs.
Itj = jr, j = 0, 1, . . ., jo, T Grid functions.
=
T/j,].
Operators.
Let xIcP1) = b,, xICpz) = b,.
enumerate from left to right the segments parallel to the coordinate
We
axis ox,,
belonging to fi, passing through the grid nodes x = (x,(~‘), xdi2)) with coordinates o< x~(~I)\
the system of graphs G = 1G,, G,, . . . , Gm 1. Let an arc of
1. V. Fryazinov
104
each of the graphs G, for k = 1, 2, ...,m' < m be one of the segments indicated above (graphs of “segment” type), the vertices segment:
the points (x,‘~‘,
C, be the ends of the corresponding
O), (x<~~, &) for k = 1, 2, . . . , /3, and the points
(x,(k+P2*P1-‘), 0), (~j~+~~-~~-i), m,)for k = /3r + 1, 8, + 2, ..., m' . We enumerate in the direction in which the coordinate x, (or x2) increases (i,f, x(i,$ the nodes x = (LX, E yr beginning with the number k = m + 1. We construct four segments emerging from the node x E yl, parallel to the coordinate aces ox,, ox, and belonging to n.
type.
We consider the graphs C,, k = m' + 1,m' + 2, .*.,m, graphs of ‘cross” fi,) The vertices of the graph G (m' + I,
xy), (p,
O),(p,
6 &,>,(0,x,'I?,(%,xJi2fj where xii%) = [(x,f ‘I)), i, = k -
N, + pz,p, < i,< p2,the arcs G, are the segments indicated above, emerging from the node x E yI. The number of graphs of “cross” type m - m' = N, - 1. The total number of graphs m = N, + N, - l++ @, - &. In Fig. 1 (Fig. 2) the vertices
of different graphs are labelled
by different symbols.
three graphs of “segment” type and four graphs of “cross”
In Fig. 1
type (p, = 2, & = 7,
N, = 8, N, = 5, m' = 3, m = 7) are labelled. We denote by gk the set of nodes of the grid o situated on the graph Gk G,, a =
(gk = w fl G&, by gi the set of iatermediate nodes x’ = x i*“*5a), 1, 2. We introduce the set of nodes 00, k = WO n gk, vk = Y n Gkr (in this case only one node x E rl belongs to each set yi ,$. finctions
defined on 0 we will use the subscript-free
Y=Ycq WYJ
(z.5) YX
t),
= y(x
system of notation: EO.5)
et,) I t),
Yi n gk
Y,
a
(+I,) = (Y
-.YVh,
([email protected]
= (y _ y(-la)j,h~~*5),
The grid function y, considered
@la) Y
Yi. k =
For the grid
(f0.5)
, the difference
analogue of a derivative,
to be def&ed at the intermediate
introduce the difference
nodes x’ = x
analogue of the derivative
(t0.5,)
will be E
0’.
We
for the net functions
defined at the nodes x E o:
At the intermediate nodes of the net w’ we also define the function a’“oJa) a x (z(*oJ$ 2)) which is the difference also k31). By (1.4)
analogue of the coefficient
x
(see
*
Schemes for the multidimensional (1.7)
a
UO.5,)
>/c, >o. A,, n corresponding
to the differential
We introduce
the operators
Ld, L, where
L,U = a /a~,(xau/ax,), Lu = L,u + Ls (+1a) _
Ly
-
105
heat conduction equation
(“‘“W y jp.5~ y -
;,
(-la) Y
Y-
f-0.5,) fJ
g-0.5) (I
Q
operators
’ )
hy = AlY t by. Because
of the notation
tors Afk’, 1, 2, *.
of (IS),
h = 1, 2, . . . , m. At the nodes
* , m’ > we
At the nodes
situated
x f g,,
axis OX,, we put
(1.9)
Py
(1.10)
,A
ayt
we introduce
type (k =
.
type parallel
to the
k = m’ + 1, . . . , m.
a = 1, 2,
Formulation
x = y~,k’
the functions
Here we have introduced right limits of the value 4.
k = 1, 2, . . . , m'
’ = %,k’
&Y +bY*
4(k) = Q/2 t
(1.11)
,*f Usegment”
the opera-
G,, the node x c yr k we put ,
A’k’ Y =
Finally,
on a gra;h
on the arcs of a graph of ‘cross”
coordinate
At the vertex
. We introduce
);
put
A’k.’Y = &Yt
(1.8)
(1.6), A,y = (ay,
.’ = Uo,k,
k = m’ + 1, m’ + 2, . . . , pt.
(k = 1, 2, . . . , m> $JCk) = (fi
+ f/)/2,
x E y, ,k’
the notation xj = X(‘j), where x = f; fP, fl are the of f for x E rr.
of the difference
problem.
We will now construct
the
scheme. At the nodes x er &?kon,each graph G,, k = 1, 2, . . . , m we write down the difference equation of the balance of heat, taking into account the heat flows in the direction of the arcs G,, arriving at the node x, e&erging from the node x (for x E y, ,k ) and passing through the node x (for x es w~,$. Then from these equations we construct s scheme in which the transition from the time layer tj_I of the function y P1 to the layer tj of the function yJ is executed by the successive solution of the difference problems at the nodes on the graphs G,,
k = 1, 2, . . ., m.
At the nodes on the arcs of graphs G, of Usegment” type parallel as in the case coordinate axis ox,, w$ write down the same equations locally one-dimensional scheme:
to the of a
1.V. Fryazinov
106
Here the later Fk is the unknown below.
function,
the function
yk_r will be defined
on the arcs of the graphs G, of *cross”
At the nodes
m' +2,..., m>,parallel
type (k = m’ + 1,
axis OX,, we also consider
to the coordinate
the
equations (1.13)
(& - y&/r
= (CzJ&
a
);
a
+ /j/2,
Here either a = 1, or a = 2. At the vertex down the difference
equation
heat flows in the direction
x EWu
of G,, the node x E yr a we write
of the balance of the coordinate
of heat,
(1.1) over the area H(x) (we will denote
segment
and its length, (1.3),
approximately
(1.14)
(?b - y&/r the notation
condition
into account
the
by the same letters),
the shape
taking
and its area, the
into account
the conjugacy
we have
We replace
Using
taking
axes ox1 and OX,. Integrating
equation condition
k’
the last equation = (a’;&,
(I.L+o.
Il),
by the difference
+ (a&&,
equation:
+ (r,’ + Q/2,
we write equations
(1.12)-(1.14)
XE
Y, k’
and the boundary
(1.2) in the form
At the nodes
on the graphs
ordinary
pivotal
Kcrossn
type, equation
condensation
ends of the Ucrossn
of “segment”
type, equation
yk_r is known).
(if the function
(1.15) can, for example,
we make direct
pivotal
(1.15) is solved
be solved
condensations
by an
On a graph of
as follows:
from the
along an arc to its
centre, the node x E yr k. From the four pixotal relations for x E yr ,k and the equation(l.14) the value of the function yk at the node x E. Yr k is found. The inverse pivotal condensations are performed and the values of’the function yk on the arcs of Gk are found. We establish original problem (l.l)-(1.4) and the following on graphs:
a correspondence between the difference problem with equations
Schemes for the multiddimensional
JR- YA--1 = A(k)gj& + p, t
(1.16)
Yk--i,
yi= We introduce the scalar
product
x=
yk;
k = I, 2,. . . , m,
xi-gk,
j = 1,2,....jo, y"= 220, XEii).
XEi3,
Ym,
the space
107
equation
vj,
ijk =
x E g,,
ZE &
F
yk =
heat conduction
K of grid functions
defined
at the nodes x e
W, with
and norm
0)of grids o such that the difference between of arcs (s(xp), B = 1, 2,...,NJ, into which the grid nodes
We consider
the sequence
adjacent lengths divide the curve r,
(as h +
is a quantity
Here and below we denote
of the order of the square
by the single
letter
of their half sum, or
M all the constants
independent
of h, 7, m, t.
Theorem The difference
scheme
the grid norm L,(u) (1.18) holds,
(1.16) with the equations on graphs converges in Subject to condition (1.17) the estimate
as h + 0, r + 0.
I\rj - u (x, $1 \I ,( M (h2 + 44. where y is the solution
of the original (1.19) Before
problem (l.l)-
IlYj -
passing
write it in a rather
ub,
ti> 11,<
of the difference (1.4).
(1.17)
u.is the solution
is not satisfied,
M (h+ ~'6.
to a study of the scheme different
problem (1.16),
If the condition
(1.16) (see section
2), we will
form. UVa-l)
We call the nodes and right, respectively) We denote
with coordinates in the direction
boundary
rL1), x, ox,.
the set of left and right boundary
nodes
nodes (left
. At
along ox, by 0: a
1. V. Fryazinou
108
the boundary
nodes
we write
where = __t. (7a(~o*5a)y~~*5)
Kay
:1,2(j)
_ a(?10*5a)y~h~~o~5a)),
xEw+.
Ya
“a
At these (1.21)
nodes
we put
x,
We introduce function
the operators
y defined
fJ$ +. a
x p
= - I&y,
A, establishing
a correspondence
at the nodes x E gk, and the grid function
nodesxEgk,k=l,2
,...,
At the nodes
between
the grid
defined
at the
m.
on the arcs of G, parallel
to the coordinate
axis oxa we put
Td
(1.22)
A,Y
at the node xez (1.23)
‘- -AaY,
x =
a=lora=2,
wO,ks
y, ,k
A, y = - 8,
We also introduce
+ XJY,
x =
Yl,k’
the function
(1~24) The expressions AQI, Tk differ from -Ack’y, $fk) only at the boundary nodes. Comparing (1.2fQo.24) and (l.lS), we notice that the equation and boundary
(i*25)
condition
(Fk- Yk,l)fr + AkFk= qks
Then the scheme
(1.26)
(1.15) can be written
(1.16) assumes
the form
” -Tyk-‘ + AA& - a)k, !$=,I,2
)...(
7%
as
x= g;t.
Schemes
for the maltidi~ns~onal
2. Convergence 1. The approximation is defined
by the value
tion of problem We introduce
(1.16)
heat conduction
and accuracy of the scheme
error.
The accuracy
of the difference
of the scheme
(1.16) (or (1.26))
zj = yj - u(x, tj), where y is the solu-
and u is the solution
the functions
109
equation
uk G akj, k = 0,
of the original
problem
(l.Ml.4).
1, . . . , m:
and the functions .zk= yk - r+, k = 1, 2, . , . , m. For x ~3 gk. we denote function zk by ik = yk ._ ui. Since either UK E $-I, or Uk p_ul, 20 also _ g. Substituting yk = ,ek + Zk = yk - u j-iorzkzyk we obtain for Zk and gk the problem zOE sj-* 7 5 E 0, i& - Zk-1 + d&h = $k, x E gk, Zk = (2.1)
uk,
T
yk
=
zK
+
d
the i*
&26),
x c+z 8%
&, Zk--it
x
z
gh,
k = 1,2, . . . , m, zj= where pk i skj
z,,
5 E 0,
j = I,
2, . . . . 10,
is the approximationerror
z*=o,
of the kth
XEW,
equation
of (1.26)
(or (I. 16)), J;K =&
(2-2)
We associate G, (the sets (2.3)
-A&-
fuj-
U&_,1/T,
x
E &”
with each node x E w the set r(x) of numbers
of the graphs
gk), to which the node x belongs: r(X) = ta,CX), 2,t.A X f
In the first case
001,
r(x) = II,(x),
I E G,, ll Gt,, in the second
16 Is ,< m, E,(x) < E,(x).
Using
Taylor’s
3;,, k = 1, 2, . . . , m can be written
where on arcs parallel
x. E GI1;
formula,
in the form
to the caordinate
axis OX,,
X E y* f. I,(X), Z&Y) are integers
the approximation
error
I. V. Fryazinov
110
At the vertices
x E y, K 5 Es Yl,k,
~+O(ftiz+A22+r),
(2.7)
$k” = 0,
(2.8)
T;pk = - (/4&l +
2
@kbz,
E
Yl,k(+0.5,)
The
is determined at the intermediate to the conditions (1.17)
function
Subjeci
P,
If condition
= 0023,
E
x
g;.
g’,.
(1.17) is not satisfied,
x(f0.5a)g’&
p$ko*5a) =O(h),
(2.10)
E
@0.5,)
(kO.5,) (2.9
nodes x’ = x
CC~
E
Since Gk = O(1) for x E o,, k (i;I,” = O(l)), equations (1.16) do not appraximate the original equation (1.1) anywhere except at the nodes LZE yi, k (z E I’%), We will regard the compound
scheme
approximation in the over-all of the nodes x E o, belongs
sense (see [3, 41, and also [S-71). Since each to two graphs, the arc of one of them being parallel
to the coordinate O(l), nevertheless
understanding
axis oxI, the other to the axis ox,, although on solving the original problem we have
qJl,O + q$ = This
(1.16) as additive,
and (2.2)-(2.10)
$
its
tily = o(l),
I,J?~,O =
+div(xgradu)+f=O$
imply that
(2.11)
It follows
from (2.11) that the scheme
in the over-all
(2.12)
sense,
(1.16) approximates
that is, on solving
11 ~q+o
the original
IIPH h-+0,
the original
problem (1.1)_(1.4)
problem we have
T-+0.
Q-=r(X) Here the subscript
(2.13)
q
assumes
1~ II(C+t)“II+
2.
0 as
Is(x) of r (xl.
h * 0,
T+ 0
Since $‘k = 0( 0,
(like b/7).
We introduce the spaces H,, k = 1, 2, . . . , m of net at the nodes x: E gk, k = 1, 2, . . . , m with the scalar products
Prior estimates.
functions defined and norms
all the values
Schemes
for the ~~‘~~t~d~~ensio~l heat conduction
By means
of Green’s difference
operators
A,, mapping the spaces Kk onto &
formula (see [31) it can be shown that the are non-negative,
that is, for any
*
YEJ-$
k = 1, 2, . . . , m.
(2.141 Subject
(2.14) the prior estimates
to condition
for the solution
of problem
imply the convergence
nodes
111
equation
(2.1) were obtained
of the scheme
Let H k ’ &$I 2,.*-, x’ =x ’ a = g,‘, @,q)k’
=
in [6, 71. This
m be the space
of grid functions
with the scalar
products
c
and (2.121, (2.13)
(1.16) (at the rate O(h + dr)). defined
at the
and norms
~(~‘)~(~‘)~k(~‘~,
llgll,’
=
l(E,
c)kl,
X’Egk’ (+0.5)
where Hk($ ) = H a We finally
introduce
(M.5,) =
x’ =x
forx’
w’,
(k,rl)‘=
a =
=x
M3.5,)
the space
e
gk’.
HI of grid functions
1, 2, with the scalar
Y.E(0+W(+
Wl’=
where H(x’> = Hk(x’ ) for X’ E gk’ , k = 1, 2, Let
Tk,
on x,’ formulas:
‘k’
Sk,
Tk* be operators
(+Oe6Q,) = y(+O*‘) %a
. . . , m,
if
( Tky)(+0.6a) = _ y,/hi+O.@, (skE),)‘+O*5,z) = -@Wewill
x
consider
a(+.5a)
the functions
gffO.5a)
?‘,y,
if
gk’,yatthenodesx-
W
Q’,
the spaces & on and defined by the following x.(+la) c x e
r ,
r,
;c(%J E r,
.
5; S,C$ to be defined
f+0*5,) E
and norm
. . . , m.
xczr,
if 9
(Tky)(f0*6a) = y(+la) /h$+0.5),
(2.15)
at the nodes
rnapp~~g~ respectively,
, and Hk’ on Hk, k = 1, 2,
(TkY)
product
defined
E&*
at the nodes
xi =
xk’ ,
112
Fryazinou
1. V.
We define follows
the operator
from (2.15),
definite.
Comparing
(2.17)
where gk en N,’ . It
S, are self-conjugate,
positive-
formula (see 131), that the operators
(2.8)) are conjugate
(2.6), (2.8) and (1.22),
(1.23) can be written A,
It follows conjugate.
and (2.6),
(2-S),
A, of (1.22),
(2.6), (2.8),
(1.7) that the operators
It can be shown by Green’s
and T,* ((2.25)
(2.18)
Tk* by formulas
to one another,
(1.23),
we notice
T,
that is,
that the operators
in the form
= T,*S,T,.
from (2.16), (2.17) that the operators A, are non-negative and selfAt the nodes X’ E o’ we introduce the function I, putting
&(X1)=/J
(+_0.5,f k ,
x’ =x
EO.5,)
E g;,
k-l,
2, . . . .
m.
In 16, 71 the estimate
was obtained 3.
for the solution
The accuracy
of the problem (2. I), (2. I7), (2.16),
of the scheme.
(I. 16) or (1.26) we use the estimate
From (2.9), (2.18) (2.21)
MI
If condition (2.22)
to condition
It follows
the accuracy
of the scheme
from (2.4)-(2.7)
that
(1. f7) we have
,< Mb’. (1.17) is not satisfied,
(2. lo), (2.18) imply that
ilutl’ i Mh.
From (2.19)-(2.22) proved.
subject
TV investigate (2.19).
(2.4).
The scheme
we obtain
the estimates
for the equation
(1.18),
(I. 19).
The theorem
is
Schemes
is constructed condition
tinuity
similarly.
(1.17))
Figure
for the mul~idimene~onu~
follows
2 shows
Its convergence
another
example
to
of 16, 71.
of a domain fi and a boundary
coefficients,
and the system
113
equation
at the rate O(h’ + dr) (subject
from the estimates
of the heat conduction
constructed
heat conduction
the grid Zi matched
rr of discon-
withr,
of graphs G = (G,, G1, . . . , G,,j indicated.
was Here
15 of them ate graphs of ‘segment” type, and two graphs (G,, G,) each have 12 vertices (the vertices ate fabelled by the symbols A,01.At the vertices of the graphs GI, G, situated on rl the equations (1.14) ate written down, and at the nodes on the arcs G,, G,, . . . , G, equations (1.13) ate written the scheme (1.16) is constructed from equations (1.15).
down.
As before,
The problems on the graphs G,, G, ate solved by means of a combination of ordinary and cyclic (see [llI> pivotal condensations. In the general
method the system
for the construction in the locally
of the scheme
one-dimensional
of graphs
instead
G = {G,, G,, . . . , G,f
of the system
of segments
is chosen considered
method.
The nodes of the grid 0 matched with the surface (curve) rl of discontinuity of the coefficients, ate the points of intersection with one another of arcs of the graphs
and their vertices.
(curves)
of discontinuity
conditions equations
connect
At the vertices
of the graphs
of the heat-conduction
the heat flows along several
of the balance
of heat ate written
situated
coefficients, coordinate
down, connecting
on surfaces
where the conjugacy directions,
difference
the ‘difference”
heat flows along the same directions. ate written
down as in the case
At the nodes on arcs the same equations of locally one-dimensional schemes. Therefore,
at the nodes on each graph G, problem ‘,1.25) is posed. Then the scheme is constructed from equations (1.25). The conditions (2. I2)_(2.14) ensure gence of the scheme 16, 71).
(weaker
In the construction of vertices
conditions
of a scheme
and arcs must be chosen,
and (2.13) ate satisfied In conclusion
of stability
graphs
of the scheme
with the smallest
ace given in
possible
but in such a way that conditions
for the corresponding
I thank A. A. Samatskii
scheme
(1.26) convet-
number (2.12)
(see ES, 71).
for discussions,
advice
and continued
interest. Tru~s~u~e~ by J. Betty
USSR
CMMP
131-H
114
I. V. Fryuz~nov REFERENCES
1,
YANENKO, N. N. A difference method of calculating the multidimensional equations of heat conduction. Dokl. Akad. Nauk SSSR, 125.6, 1207-1210, 1959.
2.
YANENKO, N. N. The method of fractional steps for the solution of multidimens ional problems of mathematic phys its (Meted drobnykh shagov resheniya mnogomernykh zadach matematicheskoi fizikif, ‘Nauka”, SO, Novosibirsk, 1967.
3.
SAMARSKII, A. A. Introduction to the theory of difference teoriyu raznostnykh skhem), *Nauka*, Moscow, 1971.
4.
S~A~~I, A. A. Difference schemes for the multidimensional differential equations of mathematical physics. App. Math. Svazek, 10,2, 146-164, 1965.
5.
FRYAZINOV. I. V. Economical schemes for the equations of heat conduction with a boundary condition of the third kind. Zh. vychisl. Mat. mat. Fiz., 12, 3, 6 12-626, 1972.
6.
FRYAZINOV, I. V. The convergence of additive schemes with equations on graphs. Zh. vychisl. Mat. mat. Fiz., 12, 5, 1208-1219. 1972.
7.
FRYAZINOV, I. V. Economical additive schemes with equations on graphs. I. The third boundary value problem for the heat conduction equation. Preprint IPM Akad. Nauk SSSR. No. 10, 1971; II, Means of choosing graphs for the construction of economical schemes. No. 11; III. The heat conduction equation with discontinuous coefficients. No. 12.
8.
VOEVODIN, A. F. Gas-&ermodynamical calculation of the flows in simple and complex turboducts. Izv. SO Akad. Nauk SSSR, Ser. ‘Bkhn. Nauk, 8, 2, 45-55, 1969.
9.
RYABEN’KII, V. S. Calculation of the heat conduction on a system of rids. Zh. vFchisi. Mat. mat. Fiz., 10,1, 236-239, 1970.
schemes
Wvedenie v
10. FRYAZINOV, I. V. Algorithm for the solution of difference problems on graphs, Zh. vychisl. Mat. mat. Fiz., 10, 2, 474-477, 1970.
11. ABRAMOV. A. A. and ANDREEV, V. B. On the application of the method of successive substitution to the determination of periodic solutions of differential and difference equations. Zh. @chist. Mat. mat. Fiz., 3, 2, 377380, 1963.
26 June 1971)
ECONOMICAL schemes with equations on graphs for the multidimensional conduction equation with discontinuous
coefficients
are constructed
heat
for a class
of non-stepped surfaces (lines) of discontinuity of the coefficients. The scheme considered is a generalization of a locally on~dimensional scheme. The transition from the time layer tj_t of the grid function
yj-l
to the layer tj = rj_i + r
(7 is the step of the time grid) of the function ,j is executed by the successive solution of difference
problems on graphs.
In this paper schemes for the multidimensional heat conduction equation with variable coefficients are constructed which are economical (absolutely stable and requiring O(1) arithmetical operations per grid node). The schemes are suitable for the solution of problems in the case of non-stepped surfaces (lines)
of discontinuity
(see Figs.
of the coefficients,
often encountered in technology
1, 2 for examples of lines of discontinuity).
The method considered below is one of the realizations of the method of total approximation and is a natural generalization of a number of economical methods [l-41. The transition from the time layer tj_1 of the grid function yj- I to the layer tj = tj_l + T (7 is the grid time-step) of the function yj is executed by the successive solution of problems at the grid nodes situated on graphs, and not on segments parallel to the coordinate axes as in a locally one-dimensional scheme.
The scheme proposed belongs to a class
and is regarded as additive,
of compound schemes
its approximation is understood in the over-all
sense (see [3, 41). Additive schemes with equations on graphs were previously considered
in k-71.
To prove the convergence *Z/L u>hisl.
Mat. mat. Fiz.,
of the scheme prior estimates
13, 1, 8081, 1973.
100
obtained in [6, 71
Schemes
for the multidimensional
FIG.
heat conduction
equation
FIG.
1.
2.
are used. The principles of the choice of graphs were given in [S, 71, and methods of solving problems on graphs were given in [8, 101.
1. Formulation of the problem 1. Formulation of the initial problem. In order not to encumber the treatment, we construct the scheme and study its properties by a specific example we confine ourselves to the case of two dimensions - where the spatial region is the rectangle Q = lo < xo < &, a = 1, 21, and a boundary first kind is specified on its boundary I?. Let I’r be a curve possessing equation
x, = [(x,), hb,)
continuous
where for definiteness
= 0,
hb,)
divides
left) with boundaries
sl,~Wt
r
the rectangle and rl,
8=HP~(O~tg9,
SO
and defined
~=div(~(~,t)grada)+i(r,t),
by the
that
dt/dx, > 0, d2@dxla > 0, d3(‘/dxiS are
fi into two parts C$, and f21 (right and
that I-‘, = I’ .rl l?l.
H=O
Let C$, = sl\r,,
uF,ilp=~~
u
We consider the following problem: find the function x,) continuous in Q which satisfies the equation
(1.1)
of the
0 < b, C b, < m,,
= %,
and that for b, < xl ,( b, the derivatives continuous (see Fig. 1). The curve r,
curvature we assume
condition
r,,&n,
Q, =
u rl.
u = U(X, t), x = (x1,
(5, t) E
Qo,
1. V. Fryazinou
102
the initial
and boundary u
(x, 0) =
0.2) u (x,
conditions
uO(&
t) = u cc,t),
625;
x
r,
x =
O
and the conditions of conjugacy on the curve TX, where the heat conduction coefficient x = x (z, t> becomes discontinuous:
(1.3) Here the notation [vf = u - u1 has been introduced; u , u1 are the right and left limits of the values Ef the function u on the curve P r; n is the normal to r,
inward with respect
to the domain fi,;
f, ZZ’,Y, X
are specified
functions.
Let
x(5, 2) >, ci = const > 0.
(1.4) We suppose up, TX,Y, f
that the function
f(~, t) also has a discontinuity
and ITI be such that a unique
solution
possessing
in Ql = El x (0 < t< T) and Qp = a,
continuous
fourth derivatives
with respect
for x ‘es rl.
of problem (L&(1.4)
Let exists,
x (0 < t\< T) (but not in Q) to x,, x, and second derivatives
with respect to t, the derivatives af/axa, a = 1, 2, continuous in Gl and Qp. In the case considered the conditions imposed on ri, f, V, 1c, U” and ensuring the smoothness of the solution of problem (l.l)-(1.4) indicated above, Nevertheless, in order to simplify are of course extremely rigorous. ment we confine ourselves to the particular case given above.
the treat-
2. The grid. points
We introduce in 3 the grid w, the nodes of which are the fir) f&f f i,) of the straight lines x, = x, , i, = Xi = (X, t X2 )E il of intersection (NJ
0, 1, . * I, N,, r’H’= 0, x a
=m,,x
curve IT,: the straight
passing
lines
(i,) (&-If o: >x a , a = 1, 2, matched
coordinate axes ox, and ox?, intersect grid 0 (see Fig. 1).
through the nodes zi E &, the curve r,
with the
parallel
only at the nodes
to the
of the
In technology, problems involving the heat conduction equation are often encountered where the surfaces (curves) lr‘, of discontinuity of the heat-conduction coefficients are such that it is possible to construct a sequence as h + 0 (h is the maximum spacing of the grid :I of grids 0, matched with lTl. For the solution
of such problems
schemes
are constructed
in the same way
Schemes
for the multidimensional
heat conduction
as is done below for the particular case considered convergence
of the schemes
103
equation
(see also L.5, 71). The
is studied on a sequence (as h + 0) of grids matched
with rl. We will use a system of notation without subscripts, 5 = 5i = @)
_&(‘&0.5)
-&%*m)))
0.5
=
Z(mI) = (Zi(il**) ) zz(iz) ))
) &@z) ))
Z(m*) = (Q’),
(kO.5)
= Ix
a
(i,*l)
of the grid o,
a
-
m = 0.5,
772= 1,
(&(‘cL*‘) + Z,%))~
We call the points x(ko-5 a ) intermediate Let h
putting
xza’I,
fia
=
nodes in the direction OX,(U = 1, 2).
0.s(h(&0*5)
+
h(iom5)),
;L =
1, 2 be steps
h = max max A, (5). XEWa=i,z We introduce the sets of nodes of the grid o:
10
=
a r-l sl,
and we associate
00
(f0.5,)
(kO.5) =4,-$, Ha
ri,
y=anr,
(kO.5) ,
a, P = 1, 2,
(*0.5a)
a + @
nodes x’ = x
and instead of x
(f0.5,)
(f0.5,)
E 0’
.
We
will often
we will write
We introduce the time grid wT=
3.
fl
0
the area
the set of all intermediate
omit the superscripts x’ E 0’.
yi =
with each node x E fl the area H(x) = ti;ti,, with each inter
mediate node x’ = x
We denote by o’
w n i-20,
=
Graphs.
Itj = jr, j = 0, 1, . . ., jo, T Grid functions.
=
T/j,].
Operators.
Let xIcP1) = b,, xICpz) = b,.
enumerate from left to right the segments parallel to the coordinate
We
axis ox,,
belonging to fi, passing through the grid nodes x = (x,(~‘), xdi2)) with coordinates o< x~(~I)\
the system of graphs G = 1G,, G,, . . . , Gm 1. Let an arc of
1. V. Fryazinov
104
each of the graphs G, for k = 1, 2, ...,m' < m be one of the segments indicated above (graphs of “segment” type), the vertices segment:
the points (x,‘~‘,
C, be the ends of the corresponding
O), (x<~~, &) for k = 1, 2, . . . , /3, and the points
(x,(k+P2*P1-‘), 0), (~j~+~~-~~-i), m,)for k = /3r + 1, 8, + 2, ..., m' . We enumerate in the direction in which the coordinate x, (or x2) increases (i,f, x(i,$ the nodes x = (LX, E yr beginning with the number k = m + 1. We construct four segments emerging from the node x E yl, parallel to the coordinate aces ox,, ox, and belonging to n.
type.
We consider the graphs C,, k = m' + 1,m' + 2, .*.,m, graphs of ‘cross” fi,) The vertices of the graph G (m' + I,
xy), (p,
O),(p,
6 &,>,(0,x,'I?,(%,xJi2fj where xii%) = [(x,f ‘I)), i, = k -
N, + pz,p, < i,< p2,the arcs G, are the segments indicated above, emerging from the node x E yI. The number of graphs of “cross” type m - m' = N, - 1. The total number of graphs m = N, + N, - l++ @, - &. In Fig. 1 (Fig. 2) the vertices
of different graphs are labelled
by different symbols.
three graphs of “segment” type and four graphs of “cross”
In Fig. 1
type (p, = 2, & = 7,
N, = 8, N, = 5, m' = 3, m = 7) are labelled. We denote by gk the set of nodes of the grid o situated on the graph Gk G,, a =
(gk = w fl G&, by gi the set of iatermediate nodes x’ = x i*“*5a), 1, 2. We introduce the set of nodes 00, k = WO n gk, vk = Y n Gkr (in this case only one node x E rl belongs to each set yi ,$. finctions
defined on 0 we will use the subscript-free
Y=Ycq WYJ
(z.5) YX
t),
= y(x
system of notation: EO.5)
et,) I t),
Yi n gk
Y,
a
(+I,) = (Y
-.YVh,
([email protected]
= (y _ y(-la)j,h~~*5),
The grid function y, considered
@la) Y
Yi. k =
For the grid
(f0.5)
, the difference
analogue of a derivative,
to be def&ed at the intermediate
introduce the difference
nodes x’ = x
analogue of the derivative
(t0.5,)
will be E
0’.
We
for the net functions
defined at the nodes x E o:
At the intermediate nodes of the net w’ we also define the function a’“oJa) a x (z(*oJ$ 2)) which is the difference also k31). By (1.4)
analogue of the coefficient
x
(see
*
Schemes for the multidimensional (1.7)
a
UO.5,)
>/c, >o. A,, n corresponding
to the differential
We introduce
the operators
Ld, L, where
L,U = a /a~,(xau/ax,), Lu = L,u + Ls (+1a) _
Ly
-
105
heat conduction equation
(“‘“W y jp.5~ y -
;,
(-la) Y
Y-
f-0.5,) fJ
g-0.5) (I
Q
operators
’ )
hy = AlY t by. Because
of the notation
tors Afk’, 1, 2, *.
of (IS),
h = 1, 2, . . . , m. At the nodes
* , m’ > we
At the nodes
situated
x f g,,
axis OX,, we put
(1.9)
Py
(1.10)
,A
ayt
we introduce
type (k =
.
type parallel
to the
k = m’ + 1, . . . , m.
a = 1, 2,
Formulation
x = y~,k’
the functions
Here we have introduced right limits of the value 4.
k = 1, 2, . . . , m'
’ = %,k’
&Y +bY*
4(k) = Q/2 t
(1.11)
,*f Usegment”
the opera-
G,, the node x c yr k we put ,
A’k’ Y =
Finally,
on a gra;h
on the arcs of a graph of ‘cross”
coordinate
At the vertex
. We introduce
);
put
A’k.’Y = &Yt
(1.8)
(1.6), A,y = (ay,
.’ = Uo,k,
k = m’ + 1, m’ + 2, . . . , pt.
(k = 1, 2, . . . , m> $JCk) = (fi
+ f/)/2,
x E y, ,k’
the notation xj = X(‘j), where x = f; fP, fl are the of f for x E rr.
of the difference
problem.
We will now construct
the
scheme. At the nodes x er &?kon,each graph G,, k = 1, 2, . . . , m we write down the difference equation of the balance of heat, taking into account the heat flows in the direction of the arcs G,, arriving at the node x, e&erging from the node x (for x E y, ,k ) and passing through the node x (for x es w~,$. Then from these equations we construct s scheme in which the transition from the time layer tj_I of the function y P1 to the layer tj of the function yJ is executed by the successive solution of the difference problems at the nodes on the graphs G,,
k = 1, 2, . . ., m.
At the nodes on the arcs of graphs G, of Usegment” type parallel as in the case coordinate axis ox,, w$ write down the same equations locally one-dimensional scheme:
to the of a
1.V. Fryazinov
106
Here the later Fk is the unknown below.
function,
the function
yk_r will be defined
on the arcs of the graphs G, of *cross”
At the nodes
m' +2,..., m>,parallel
type (k = m’ + 1,
axis OX,, we also consider
to the coordinate
the
equations (1.13)
(& - y&/r
= (CzJ&
a
);
a
+ /j/2,
Here either a = 1, or a = 2. At the vertex down the difference
equation
heat flows in the direction
x EWu
of G,, the node x E yr a we write
of the balance of the coordinate
of heat,
(1.1) over the area H(x) (we will denote
segment
and its length, (1.3),
approximately
(1.14)
(?b - y&/r the notation
condition
into account
the
by the same letters),
the shape
taking
and its area, the
into account
the conjugacy
we have
We replace
Using
taking
axes ox1 and OX,. Integrating
equation condition
k’
the last equation = (a’;&,
(I.L+o.
Il),
by the difference
+ (a&&,
equation:
+ (r,’ + Q/2,
we write equations
(1.12)-(1.14)
XE
Y, k’
and the boundary
(1.2) in the form
At the nodes
on the graphs
ordinary
pivotal
Kcrossn
type, equation
condensation
ends of the Ucrossn
of “segment”
type, equation
yk_r is known).
(if the function
(1.15) can, for example,
we make direct
pivotal
(1.15) is solved
be solved
condensations
by an
On a graph of
as follows:
from the
along an arc to its
centre, the node x E yr k. From the four pixotal relations for x E yr ,k and the equation(l.14) the value of the function yk at the node x E. Yr k is found. The inverse pivotal condensations are performed and the values of’the function yk on the arcs of Gk are found. We establish original problem (l.l)-(1.4) and the following on graphs:
a correspondence between the difference problem with equations
Schemes for the multiddimensional
JR- YA--1 = A(k)gj& + p, t
(1.16)
Yk--i,
yi= We introduce the scalar
product
x=
yk;
k = I, 2,. . . , m,
xi-gk,
j = 1,2,....jo, y"= 220, XEii).
XEi3,
Ym,
the space
107
equation
vj,
ijk =
x E g,,
ZE &
F
yk =
heat conduction
K of grid functions
defined
at the nodes x e
W, with
and norm
0)of grids o such that the difference between of arcs (s(xp), B = 1, 2,...,NJ, into which the grid nodes
We consider
the sequence
adjacent lengths divide the curve r,
(as h +
is a quantity
Here and below we denote
of the order of the square
by the single
letter
of their half sum, or
M all the constants
independent
of h, 7, m, t.
Theorem The difference
scheme
the grid norm L,(u) (1.18) holds,
(1.16) with the equations on graphs converges in Subject to condition (1.17) the estimate
as h + 0, r + 0.
I\rj - u (x, $1 \I ,( M (h2 + 44. where y is the solution
of the original (1.19) Before
problem (l.l)-
IlYj -
passing
write it in a rather
ub,
ti> 11,<
of the difference (1.4).
(1.17)
u.is the solution
is not satisfied,
M (h+ ~'6.
to a study of the scheme different
problem (1.16),
If the condition
(1.16) (see section
2), we will
form. UVa-l)
We call the nodes and right, respectively) We denote
with coordinates in the direction
boundary
rL1), x, ox,.
the set of left and right boundary
nodes
nodes (left
. At
along ox, by 0: a
1. V. Fryazinou
108
the boundary
nodes
we write
where = __t. (7a(~o*5a)y~~*5)
Kay
:1,2(j)
_ a(?10*5a)y~h~~o~5a)),
xEw+.
Ya
“a
At these (1.21)
nodes
we put
x,
We introduce function
the operators
y defined
fJ$ +. a
x p
= - I&y,
A, establishing
a correspondence
at the nodes x E gk, and the grid function
nodesxEgk,k=l,2
,...,
At the nodes
between
the grid
defined
at the
m.
on the arcs of G, parallel
to the coordinate
axis oxa we put
Td
(1.22)
A,Y
at the node xez (1.23)
‘- -AaY,
x =
a=lora=2,
wO,ks
y, ,k
A, y = - 8,
We also introduce
+ XJY,
x =
Yl,k’
the function
(1~24) The expressions AQI, Tk differ from -Ack’y, $fk) only at the boundary nodes. Comparing (1.2fQo.24) and (l.lS), we notice that the equation and boundary
(i*25)
condition
(Fk- Yk,l)fr + AkFk= qks
Then the scheme
(1.26)
(1.15) can be written
(1.16) assumes
the form
” -Tyk-‘ + AA& - a)k, !$=,I,2
)...(
7%
as
x= g;t.
Schemes
for the maltidi~ns~onal
2. Convergence 1. The approximation is defined
by the value
tion of problem We introduce
(1.16)
heat conduction
and accuracy of the scheme
error.
The accuracy
of the difference
of the scheme
(1.16) (or (1.26))
zj = yj - u(x, tj), where y is the solu-
and u is the solution
the functions
109
equation
uk G akj, k = 0,
of the original
problem
(l.Ml.4).
1, . . . , m:
and the functions .zk= yk - r+, k = 1, 2, . , . , m. For x ~3 gk. we denote function zk by ik = yk ._ ui. Since either UK E $-I, or Uk p_ul, 20 also _ g. Substituting yk = ,ek + Zk = yk - u j-iorzkzyk we obtain for Zk and gk the problem zOE sj-* 7 5 E 0, i& - Zk-1 + d&h = $k, x E gk, Zk = (2.1)
uk,
T
yk
=
zK
+
d
the i*
&26),
x c+z 8%
&, Zk--it
x
z
gh,
k = 1,2, . . . , m, zj= where pk i skj
z,,
5 E 0,
j = I,
2, . . . . 10,
is the approximationerror
z*=o,
of the kth
XEW,
equation
of (1.26)
(or (I. 16)), J;K =&
(2-2)
We associate G, (the sets (2.3)
-A&-
fuj-
U&_,1/T,
x
E &”
with each node x E w the set r(x) of numbers
of the graphs
gk), to which the node x belongs: r(X) = ta,CX), 2,t.A X f
In the first case
001,
r(x) = II,(x),
I E G,, ll Gt,, in the second
16 Is ,< m, E,(x) < E,(x).
Using
Taylor’s
3;,, k = 1, 2, . . . , m can be written
where on arcs parallel
x. E GI1;
formula,
in the form
to the caordinate
axis OX,,
X E y* f. I,(X), Z&Y) are integers
the approximation
error
I. V. Fryazinov
110
At the vertices
x E y, K 5 Es Yl,k,
~+O(ftiz+A22+r),
(2.7)
$k” = 0,
(2.8)
T;pk = - (/4&l +
2
@kbz,
E
Yl,k(+0.5,)
The
is determined at the intermediate to the conditions (1.17)
function
Subjeci
P,
If condition
= 0023,
E
x
g;.
g’,.
(1.17) is not satisfied,
x(f0.5a)g’&
p$ko*5a) =O(h),
(2.10)
E
@0.5,)
(kO.5,) (2.9
nodes x’ = x
CC~
E
Since Gk = O(1) for x E o,, k (i;I,” = O(l)), equations (1.16) do not appraximate the original equation (1.1) anywhere except at the nodes LZE yi, k (z E I’%), We will regard the compound
scheme
approximation in the over-all of the nodes x E o, belongs
sense (see [3, 41, and also [S-71). Since each to two graphs, the arc of one of them being parallel
to the coordinate O(l), nevertheless
understanding
axis oxI, the other to the axis ox,, although on solving the original problem we have
qJl,O + q$ = This
(1.16) as additive,
and (2.2)-(2.10)
$
its
tily = o(l),
I,J?~,O =
+div(xgradu)+f=O$
imply that
(2.11)
It follows
from (2.11) that the scheme
in the over-all
(2.12)
sense,
(1.16) approximates
that is, on solving
11 ~q+o
the original
IIPH h-+0,
the original
problem (1.1)_(1.4)
problem we have
T-+0.
Q-=r(X) Here the subscript
(2.13)
q
assumes
1~ II(C+t)“II+
2.
0 as
Is(x) of r (xl.
h * 0,
T+ 0
Since $‘k = 0( 0,
(like b/7).
We introduce the spaces H,, k = 1, 2, . . . , m of net at the nodes x: E gk, k = 1, 2, . . . , m with the scalar products
Prior estimates.
functions defined and norms
all the values
Schemes
for the ~~‘~~t~d~~ensio~l heat conduction
By means
of Green’s difference
operators
A,, mapping the spaces Kk onto &
formula (see [31) it can be shown that the are non-negative,
that is, for any
*
YEJ-$
k = 1, 2, . . . , m.
(2.141 Subject
(2.14) the prior estimates
to condition
for the solution
of problem
imply the convergence
nodes
111
equation
(2.1) were obtained
of the scheme
Let H k ’ &$I 2,.*-, x’ =x ’ a = g,‘, @,q)k’
=
in [6, 71. This
m be the space
of grid functions
with the scalar
products
c
and (2.121, (2.13)
(1.16) (at the rate O(h + dr)). defined
at the
and norms
~(~‘)~(~‘)~k(~‘~,
llgll,’
=
l(E,
c)kl,
X’Egk’ (+0.5)
where Hk($ ) = H a We finally
introduce
(M.5,) =
x’ =x
forx’
w’,
(k,rl)‘=
a =
=x
M3.5,)
the space
e
gk’.
HI of grid functions
1, 2, with the scalar
Y.E(0+W(+
Wl’=
where H(x’> = Hk(x’ ) for X’ E gk’ , k = 1, 2, Let
Tk,
on x,’ formulas:
‘k’
Sk,
Tk* be operators
(+Oe6Q,) = y(+O*‘) %a
. . . , m,
if
( Tky)(+0.6a) = _ y,/hi+O.@, (skE),)‘+O*5,z) = -@Wewill
x
consider
a(+.5a)
the functions
gffO.5a)
?‘,y,
if
gk’,yatthenodesx-
W
Q’,
the spaces & on and defined by the following x.(+la) c x e
r ,
r,
;c(%J E r,
.
5; S,C$ to be defined
f+0*5,) E
and norm
. . . , m.
xczr,
if 9
(Tky)(f0*6a) = y(+la) /h$+0.5),
(2.15)
at the nodes
rnapp~~g~ respectively,
, and Hk’ on Hk, k = 1, 2,
(TkY)
product
defined
E&*
at the nodes
xi =
xk’ ,
112
Fryazinou
1. V.
We define follows
the operator
from (2.15),
definite.
Comparing
(2.17)
where gk en N,’ . It
S, are self-conjugate,
positive-
formula (see 131), that the operators
(2.8)) are conjugate
(2.6), (2.8) and (1.22),
(1.23) can be written A,
It follows conjugate.
and (2.6),
(2-S),
A, of (1.22),
(2.6), (2.8),
(1.7) that the operators
It can be shown by Green’s
and T,* ((2.25)
(2.18)
Tk* by formulas
to one another,
(1.23),
we notice
T,
that is,
that the operators
in the form
= T,*S,T,.
from (2.16), (2.17) that the operators A, are non-negative and selfAt the nodes X’ E o’ we introduce the function I, putting
&(X1)=/J
(+_0.5,f k ,
x’ =x
EO.5,)
E g;,
k-l,
2, . . . .
m.
In 16, 71 the estimate
was obtained 3.
for the solution
The accuracy
of the problem (2. I), (2. I7), (2.16),
of the scheme.
(I. 16) or (1.26) we use the estimate
From (2.9), (2.18) (2.21)
MI
If condition (2.22)
to condition
It follows
the accuracy
of the scheme
from (2.4)-(2.7)
that
(1. f7) we have
,< Mb’. (1.17) is not satisfied,
(2. lo), (2.18) imply that
ilutl’ i Mh.
From (2.19)-(2.22) proved.
subject
TV investigate (2.19).
(2.4).
The scheme
we obtain
the estimates
for the equation
(1.18),
(I. 19).
The theorem
is
Schemes
is constructed condition
tinuity
similarly.
(1.17))
Figure
for the mul~idimene~onu~
follows
2 shows
Its convergence
another
example
to
of 16, 71.
of a domain fi and a boundary
coefficients,
and the system
113
equation
at the rate O(h’ + dr) (subject
from the estimates
of the heat conduction
constructed
heat conduction
the grid Zi matched
rr of discon-
withr,
of graphs G = (G,, G1, . . . , G,,j indicated.
was Here
15 of them ate graphs of ‘segment” type, and two graphs (G,, G,) each have 12 vertices (the vertices ate fabelled by the symbols A,01.At the vertices of the graphs GI, G, situated on rl the equations (1.14) ate written down, and at the nodes on the arcs G,, G,, . . . , G, equations (1.13) ate written the scheme (1.16) is constructed from equations (1.15).
down.
As before,
The problems on the graphs G,, G, ate solved by means of a combination of ordinary and cyclic (see [llI> pivotal condensations. In the general
method the system
for the construction in the locally
of the scheme
one-dimensional
of graphs
instead
G = {G,, G,, . . . , G,f
of the system
of segments
is chosen considered
method.
The nodes of the grid 0 matched with the surface (curve) rl of discontinuity of the coefficients, ate the points of intersection with one another of arcs of the graphs
and their vertices.
(curves)
of discontinuity
conditions equations
connect
At the vertices
of the graphs
of the heat-conduction
the heat flows along several
of the balance
of heat ate written
situated
coefficients, coordinate
down, connecting
on surfaces
where the conjugacy directions,
difference
the ‘difference”
heat flows along the same directions. ate written
down as in the case
At the nodes on arcs the same equations of locally one-dimensional schemes. Therefore,
at the nodes on each graph G, problem ‘,1.25) is posed. Then the scheme is constructed from equations (1.25). The conditions (2. I2)_(2.14) ensure gence of the scheme 16, 71).
(weaker
In the construction of vertices
conditions
of a scheme
and arcs must be chosen,
and (2.13) ate satisfied In conclusion
of stability
graphs
of the scheme
with the smallest
ace given in
possible
but in such a way that conditions
for the corresponding
I thank A. A. Samatskii
scheme
(1.26) convet-
number (2.12)
(see ES, 71).
for discussions,
advice
and continued
interest. Tru~s~u~e~ by J. Betty
USSR
CMMP
131-H
114
I. V. Fryuz~nov REFERENCES
1,
YANENKO, N. N. A difference method of calculating the multidimensional equations of heat conduction. Dokl. Akad. Nauk SSSR, 125.6, 1207-1210, 1959.
2.
YANENKO, N. N. The method of fractional steps for the solution of multidimens ional problems of mathematic phys its (Meted drobnykh shagov resheniya mnogomernykh zadach matematicheskoi fizikif, ‘Nauka”, SO, Novosibirsk, 1967.
3.
SAMARSKII, A. A. Introduction to the theory of difference teoriyu raznostnykh skhem), *Nauka*, Moscow, 1971.
4.
S~A~~I, A. A. Difference schemes for the multidimensional differential equations of mathematical physics. App. Math. Svazek, 10,2, 146-164, 1965.
5.
FRYAZINOV. I. V. Economical schemes for the equations of heat conduction with a boundary condition of the third kind. Zh. vychisl. Mat. mat. Fiz., 12, 3, 6 12-626, 1972.
6.
FRYAZINOV, I. V. The convergence of additive schemes with equations on graphs. Zh. vychisl. Mat. mat. Fiz., 12, 5, 1208-1219. 1972.
7.
FRYAZINOV, I. V. Economical additive schemes with equations on graphs. I. The third boundary value problem for the heat conduction equation. Preprint IPM Akad. Nauk SSSR. No. 10, 1971; II, Means of choosing graphs for the construction of economical schemes. No. 11; III. The heat conduction equation with discontinuous coefficients. No. 12.
8.
VOEVODIN, A. F. Gas-&ermodynamical calculation of the flows in simple and complex turboducts. Izv. SO Akad. Nauk SSSR, Ser. ‘Bkhn. Nauk, 8, 2, 45-55, 1969.
9.
RYABEN’KII, V. S. Calculation of the heat conduction on a system of rids. Zh. vFchisi. Mat. mat. Fiz., 10,1, 236-239, 1970.
schemes
Wvedenie v
10. FRYAZINOV, I. V. Algorithm for the solution of difference problems on graphs, Zh. vychisl. Mat. mat. Fiz., 10, 2, 474-477, 1970.
11. ABRAMOV. A. A. and ANDREEV, V. B. On the application of the method of successive substitution to the determination of periodic solutions of differential and difference equations. Zh. @chist. Mat. mat. Fiz., 3, 2, 377380, 1963.