- Email: [email protected]
Some difference schemes for solving boundary problems
SOME DIFFERENCE SCHEMES FOR SOLVING
BOUNDARY
PROBLEMS*
YE. G. D’YAKONOV Moscow (Received 27 September 1961)
VARIOUSdifference schemes are available for finding approximately the solution of a boundary problem of the equation au/at = Lu, where L is an elliptical operator with constant coefficients. The principal requirements for such schemes are stability and convergence. In explicit difference schemes the time step has to be chosen considerably smaller than the step for space variables, while in implicit schemes (which are free from this restriction) systems of linear algebraic equations have to be solved. If the number of space variables p 3 2, then both explicit and implicit schemes require much computing work. A series of investigations (see [I]-[5]) was therefore undertaken to study problems of constructing absolutely stable difference schemes in which the solution of equation systems with a large number of variables is not necessary. The present paper is devoted to the analysis of one such difference scheme, using the method of fractional steps proposed in [3]-[5]. To justify the validity of the method the authors of [3]-[5] replaced the difference scheme by another which they considered equivalent to it and established the convergence of the latter as a consequence of stability and approximation. But in the class of boundary conditions considered these schemes are not equivalent. Further, even in the case of the heat conduction equation an example can be found in which the method does not approximate the differential equation for any small t and h, if the function is not linear at the boundary of the domain. If, however, zero boundary conditions are chosen, the analysis of this method given in [3] and [5] can be used, although not for such a wide class with constant coefficients: The result of [4], however cannot be obtained even for zero boundary conditions. Nevertheless, the range of application of the fractional step method can be extended, and it will be shown here that under zero boundary conditions a somewhat modified method is absolutely stable and has a convergent difference scheme: (a) for equations where L is a self-conjugate, elliptical negative definite operator with separable variables of the order 2m, (b) for equations with a right hand side. For m > 1 convergence of the method means convergence to the generalized solution. The possibility of increasing the time steps is also mentioned. * Zh. vych. mat. 2: No.. 1, 57-79, 1962. 55
Ye.G.
56 4
1. FRACTIONAL We
equation
STEP METHOD
D'YAKONOV
FOR THE HEAT CONDUCTION
EQUATION
shall examine the principle of the method by taking the heat conduction [3], g+g+g
(1.1)
in the cylinder sz{(x,y):
Qr=~?xtO
o
1, o
l}
with the initial condition r&o = dX,Y) and the boundary
(1.2)
condition r+ = py(%Y, ‘),
@,Y)ER
(1.3)
S is the lateral surface of QT. Let h = l/N be the step for x and y; r the time step; x1 = ih, i = 0, 1, . . . . . . N; yJ =jh,j = 0, 1, . . . ,N; t, = nt,n = 0, 1, . . . . . . , T/t;Q,,{(i,j) : i = 1,2, . . . ,N-1; j= 1,2 ,... . . . 9N-- 11; aw) :i=O,Norj=O,N}; 52,=8,US,. z@ will denote D(x*, yJ, tn). Conditions (1.2) and (1.3) are approximated
in the usual way:
r,l$’ = vij, $’ = y)$‘, The method of fractional
(1.2’)
(i,i) E z. @+l) z &-+l), (i,i> E&l.
(1.3’)
steps consists in forming an equation
system with
intermediate functions of z$+~’ for finding u$+‘) when (i, j) E Sz, from the known values of r@) :
(1.4)
where dB,,V, = 4~;Vij =
Wtl,j-2Qj+Wi-l.j
,
h2 j-1
%,jt*-2Qj+Q, hZ
O
When (i, j) E S,, the values of TIE+*)are determined from the condition (n+h)=
Vij
@0_
Vij -
@I
lyij*
We rewrite (1.4) in the form (E-o(TZ4i;) z$;+*) = (E+(l
-_a)z4&$‘, (i&E
gh,
Difference schemcss for boundary problems
is the identity operator. The followjng notations will be used:
where E
j=2,3,,..,N-22),
D’{Ii,j):i=2,3,..,,i\l-2;
S,{i,j):i
= 2,3, ..,) N-2;
S,(i,j>:i=
I,.#--1;
S,,(i,j):i=l,N-1;
j=
j=2,3,
1,&-l), _.,N---3),
j=l,N-+
The authors of [3]-[S] replace the scheme (1.5) by the following scheme without the intermediate
values of T$+*):
(E-~d~;)(E-~d~~~~~~l~
= (E+(I-~~til~~~f.E+fI-~~~~~~~~~~-
(1.6)
This scheme, as shown in [3], is &sofutely stable and approximates equation (1.1) in the metric I+ ,with the order of approximation O(z)+O(V). Therefore its convergence in L, with the order O(t)+O(/P) follows as a result of some well-known theorems (see [6])_
and balls is the value of vri at the point of the boundary closest to (ij), Let us see how o:*+) is eliminated. If fi,j) E I)‘, each of the two relationS (1.5) wilt be satisfied at the points (i&-l), &j)_ f&J_tQ1 fr‘-I,& i+l,j). Hence applying the operator (E+(f --~)r~If;l, to the first equation and (E-SLY&;) to the second we obtain respectively
YE. G. D’YAKONOV
58
But if (ij) E S, (for the sake of definiteness we take, for example, j = N-l), the second of the relations (1.5) is satisfied at the points (i- 1,j), (i,j) (i + 1, j) and applying to it the operator (E--or&) we again obtain (1.8’). The first of the relationships (1.5) holds only at the points (i, j- l), (i, j), while at the point (i, j+ 1) we have instead the equation (E-oatd:;)v$?) --0)rd~-)2+-zL3~;yiN @). Hence, instead of (1.8) we have XX 1 @+(I
= (E+ (l-
--a)z~~~)(E--azd~;)v~+*’ = (E_t(l-~)rd~;)(Et(l-~)td~~v~~‘-(l-~)~d,~W~~~~s,.
Adding this equation to (1.8’) we find that (1.7) is true for points (i, j) E S,This elimination process is carried out similarly for the points S’,,and S,,. If y(x, y, t) = 0 or even Py/ax2 = Py/,lav” = 0, R$) becomes zero for all (ij) E J2,, and (1.5) and (1.6) are equivalent. But if R$) # 0, it is easy to construct an example showing that scheme (1.7) (and hence also scheme (1.4)) does not approximate equation (1.1). Let us suppose that azu atz'
a a54 -at ax29
a a=u --at ay ’
adu 32,
adu ax2ay2 9
ab ay’
(1.9)
exist and are finite in QT. Under these conditions scheme (1.6) approximates (1.1). Hence substituting u(x, y, t), the exact solution of (1. l), in (1.7) we obtain (E-aazd~;)(E-o(~~d~~)uUI~+~’ = (E+(1-~)zd~;)(E+(1-o)z~y2;)u~~~+~{O(~)+O(h2)+~~~~}.
(1.10)
Let us take u = sin xxe-%“. Then obviously all the conditions for the derivatives are satisfied and R$) = (c- 1) $ sin2 $
sin nXieWnZt,
if (i,j) E S,.
As usual we introduce the norm of the function on the layer t = t, = nz llzI(“)ll =
(h22
(lj$q.
i, j=l
Then llR”II > c/h, where c > 0 does not depend on r and h. Therefore (1.10) it follows that (1.7) does not approximate (1.1). Thus the following theorem is established.
from
THEOREM 1. The method oj’jractional steps is an absolutely stable and convergent scheme with a mean order oj’convergence equal to 0(T)+O(h2) jor eqn. (1.1) with the initial condition (1.2) and the boundary condition us = 0. In the case oj’the non-homogeneous boundary condition (1.3) the method, generally speaking, may not approximate eqn. (1.1).
Difference schemes for boundary problems
59
It can be easily verified that both the example and the theorem are perfectly applicable to the case of p space variables, only the requirement that -___ 8% a~:,ax~,...ax~p ’ should be continuous, where si # s,, if i the conditions (1.9). We may point out further that for the established for y # 0 also. The scheme in can be easily extended to the case of any only for p = 2. $ 2. METHOD
OF FRACTIONAL VARIABLES
# j, si = 1,2, . . . , p, must be added to schemes in [I], [2] convergence has been [2], proposed only for p = 2 and p = 3, p, while the scheme in [I] is applicable
STEPS FOR EQUATIONS WITH AND RIGHT HAND SIDE
SEPARABLE
1. We shall now describe a modification of the method of fractional variables for equations with separable variables and right hand side.
Suppose in the cylinder Q,=~x [O
at
satisfying the boundary
s=l,...,p)
’
cS=l Lu+f,
(2.1)
conditions
au (u,--_, av
am-h
a**,
___ avm_,
1
s = (O,O, . . ..O)
(2.2)
and the initial condition U]t=l?=Q)(%.%?, .5x,).
(2.3)
Here
Lsu= -p a=0
(-lY-$+.(xs,~), s
s
us&,) > 0, a,,(~,) 3 0, S is the lateral surface of QT, Y the normal to S. Let us construct a difference approximation of this problem. Let t be the time step, h = l/N the space variable step. The following notations are introduced: (%)is=kh7
i,=O,l,...,
N;
s=1,2
,..., p;
v(i,h, i2h, . . . . iPh , nz) = v~~..~,= ~2); SZ,{A:i,=O,l,...,N; Q,{A:i,
= m,m+l,
s, = .ni\rn, C,{A:i,=O,l,...,
N;
A=(iI,i,,...,ip);
s=1,2 . . . . N-m;
&=m,m+l,...,
t,=n~;
s=
)...) p}, 1,2, . ..) p},
*
N-m
for s’fs}.
YE. G. D’YAKONOV
60
Where no confusion is possible we shall simply write v instead of vA and omit the index. The relation between the net functions and the net also will not be indicated. ~2) is determined from condition (2.3): vy = p)d.
(2.3’)
If A E S,, we write v$)=O
(k=O,l,...,
T/z),
(2.2’)
To find vg+r) at the points Qh from the known ~6”)we shall consider p systems of equations with p- 1 intermediate functions v$‘+“~): o(n+s/p)_~t”+~S-l~lPl
=
z
~(L:v(n+s/P)+L:v[“+(“-l)IPl)
.(n+l)_&l+(S-UP/l
(s =
+Lh pv [n+(~--lV~l)
$
1,2,
. . . ,p__l),
Thy-W,
(2.4)
t
where m LpJ =
c
(_
1,a-1+
Jsv,
(a,,(j,q~~~v))
=
“il...(is+l;;.fP-“il...‘~,
a=0
d,,v,=
vij...ip-W,...(is-l)...ip
____.
,
Thy’“’
=
(fj
(&
+
L;))-'l'"';
h
E is the identity operator, jp) denotes J&h, izh, . . . . . . , i,h, (n+&)z. In this approximation (-Lh,) is a self-conjugate and positive definite operator and L’&Li, = Lt,L’& (see, for example, [7], [8]). The computing scheme is as follows : v(“+~‘~) is found from the known (N+ l)P-dimensional vector 7P)(v(“) is determined from the initial condition) as the solution of the first system (2.4) with the conditions (2.2’). The vector v(“+~‘~),satisfying the second system of (2.4) and the conditions (2.2’) is determined from the vector v(“+~@)obtained till v(“+(P-~)‘P)is found. Each of these systems has matrices containing unknowns, in which the nonzero coefficients are arranged in 2mf 1 diagonals, so that x l/hP arithmetical operations are required for finding the solution of the system. The determination of
is also carried out in steps: p_
~L~)w~+s/(p-l)l
=
wy+s/(P-l)l =
WY’ =
1,2, . . . . p-l,
s =
A E-f&,
w~+(s-lMP-l)l,
0
for d ES,,,
fY’
for d E&,
0
for A E&.
Difference. schemes for boundary problems
61
It can be easily verified that the determination of w(“+l) requires ,~l/hparithmetical operations. pi(“+‘)is found from the values found of ~r”+(p-~)‘~land w(‘+l) of the last system (2.4) and the conditions (2.2’). The transition from v@) to zPil) is thus completed in x l/P arithmetical operations, i.e. the number of these is of the same order as the number of points on the layer t = t,. 2. To examine the problem of stability of scheme (2.4) we rewrite the latter in the form (E-$L:)V’“+~~~)
= (E+;L+“+‘“-‘,?
(&
= (&
;+P+”
$= 1,2,...,p__l;
~Lhp)D[“+(p-l)~pl+TThf(~)
(2.5)
and eliminate the intermediate values of v f”+@), taking into account the co~utability of L,,and Ls,and conditions (2.27, the importance of which has already been mentioned in 8. 1. Because of conditions (2.2’) it can be considered that (2.5) is true not only at the points Q,,, but also at all points C. Therefore the first p- 1 of the systems (2.5) are equivalent:
Multiplying this equation by (E+(z/2)@
and the last of the equations
(2.5)by
P-l n
s=l
(~-t@w:~,
and then adding, we obtain
since P-1
g-h@’ = f d f*) f “Lh 2 *)
E-
ni s=1
Af8.
Under the conditions (2.2’) ti is a self-conjugate and negative definite operator (see (81). Consequently there is a system of functions yl$)(x,), k = 1, 2, .....,N-2m+1, satisfying (2.2’), such that and
(wiy’,wg> = 41k* = 3
0 for kl-;f;k,. 1 for kl==kgr
where
Then the operators A = fi s-1
(E-$LJh)and
.=fi(E++L:) .S=1
62
D’YAKONOV
YE. G.
have the eigenfunctions @k,...k&$,
and correspondingly
Therefore,
... ,
the eigenvalues
as can be easily verified,
and ]IA-‘BII < 1, where [ICI1= maxI&] ({A,} are eigenvalues of the matrix C). Thus the stability not only of scheme (2.6) but also of each of the intermediate steps is proved. The intermediate steps in the schemes of [l] and [2] can be unstable. In the proof of the stability the condition a S(l>, 0 was used only for negative definiteness of Lt. It can therefore be relaxed and it may be required simply that Lt be negative definite. The stability of the right hand side can be easily derived from the stability just established and the initial data (see [6]). 3. We shall now show that the scheme (2.6) approximates equation (2.1). Later we shall often need the class of functions D;(c) introduced in [9], viz. we shall say that the function F(x1,‘x2, . . . . . xp) defined in the p-dimensional hypercube abelongs to D;(c), if all the partial derivatives F(x,, x2, .., . , xp) containing not more than 4 differentiations with respect to each coordinate are bounded in modulus by the constant c. (2.6) is rewritten in the form
=- 1
&+1)-w(“)
t
2
+
c p
L;(v(“+l)
$v(“)) j-f’“‘+
s=1
f(I)2
L;, L;,(v("))
- (n+l))-+... + v
s1,s*=1 s1>sa
. ..
+(+)'-'L~
. ..L~(o(“)-(--~)P~(“+~)).
(2.7)
VS’ = 0 Let a,,, j; p be such that A E S,; 1) the following exist and are continuous in QT a=+la,,
a) yeax:+13
b) $i;
(2.8)
l, ***v2m;
ki=O,
arm+su
k=0,1,...,2m;
~axy+2
9
s = 1,2,
.. ..p.
(2.8’)
Difference schemes for boundary problems
63
2) for each t(0 < t ,< T) (2.8”)
2.8 E dlgm(c)*
Then, substituting the u-solution of (2.1) in (2.7) we obtain
..Lhp(zP)-(-- l)W+1)), i.e. the scheme (2.5) approximates equation (2.1) with a mean order of 0(r2)+O(h). As can be easily verified, the order of approximation for equations with constant coeliicients will be O(9)+ O(h2). The boundary conditions (2.2) are approximated in the general case with an error O(h). It is known [6] that, if a difference scheme approximates to a differential equation and the initial and boundary conditions, and is stable with respect to the initial and boundary conditions as well as the right hand side, the scheme is convergent. But the theorem of [6] can be applied to the problem (2.1)$2.2)-(2.3) only when m =I, since the boundary conditions are approximated accurately only when m = 1, and stability to boundary conditions cannot be obtained in the general case. The fact that all eigenvalues of the operator
are greater than unity under the conditions (2.2’) is also insufficient for convergence, because this establishes the boundedness of A-l qnly for the conditions (2.2’), which do not approximate (2.2) accurately. Therefore, for m > I we confine ourselves for the moment to the statement that the method of fractional steps is an absolutely stable and approximating scheme. The question of its convergence will be discussed in 0 4. For m = 1, consequently, the following theorem is true. THEOREM 2. The method oj jkaktional steps jor eqn. (2.1) with the conditions (2.2) and (2.3), when the conditions (2.8), (2.83, (2.8”) are j~~~lied, is an absolutely stable and convergent scheme with a mean order oj’convergence 0(x2)+0(h). Ij‘the equation coeficients are constant, the order oj’convergence will be 0(z2)+O(h2). Some difference schemes with variable time steps were studied for the heat conduction equation in [2] and [IO]. In the next section we shall consider the possibility of a similar modification of the method of fractional steps. $ 3. FRACTIONAL
STEP METHOD
WITH
VARIABLE TIME STEPS
1. Let us analyse the fractional step method with increasing time steps for a somewhat more general equation than (2.1).
YE. G.
64
D’YAK~N~V
We look for the solution of the following equation in the region QT (see 0 2)
(3.1) us(xs14 I=- 0;
with the boundary
b&9 0 2 0,
condition u/s=0
and the initial condition (2.3). The difference approximation form
(3.21
of this problem with variable z will be of the
(#=; 1,2, . . ..p).
(!
Ls,BIu(n)=~4,~a, x,,
t,+2-
if de&.
t.+‘-)a(“,, =a) dxs’u+,i,x,,
T,= (f+-%L,..)J-';
t,= qj+zlf...+z,_l.
S-l
The remaining notations are the same as in Q2. As we see, here L,, n(s= 1, . . . , p) depends on t,, but its self-conjugateness, negative definiteness and the commutability of L,,, n and L,, ,, are established as in 171. Eliminating the intermediate stages we obtain the analogue of (2.6):
To establish the stability of this scheme we have to consider how l/w&I changes on transition to the next layer, if at the points of fl, wCn)satisfies the relation P
E-~L,‘~)w'"tl,=~(~+qL~,.)~(", c( "=I
(3.4)
and the condition ~6”’ = 0, d E S,. The process of inves~gat~on is the same as in 0 2, with only this difference that the family of eigenfunctions of the operators
changes from t,, to tn+r during the transition. Using the orthogonality of the different eigenfunctions, we conclude from the fact that the coefficient of each eigenfunction in the expansion of w@+l) has a modulus less than that of ~(“f that IIw(“+1)/1 < ]Iw(“)//.
(3-3
Difference schemes for boundary problems
65
2. We shall prove the following theorem.
THEOREM3. Ij the jollowing conditions are jufirlfilled: 1) ,
Pas y-g@ s
= 1, . ...P).
exist and are jinite; 2) a,, b,, v and j’are such that a) all the derivatives -$@=O,l,..., s
4),
$
(a=o,1,2,3),
- frI
(
3
s Sl 1
(a,@=O,1,2)
in Qr have a modulus less than &feed*; b) jk any t(0 < t < T) u E Dg(h4e-d’). then the vector v@‘)(0 < n < T/t) obtained from the scheme (3.1’) with z,, = t,edtn converges to u(“) unijormly with respect to t(0 ,< t < T) in the metric L, with the order 0(+)+0(h). Proof. Let us substitute u (x1, . . . . . . , x,, t) in (3.3). Then because of the conditions of the theorem we obtain P
I-If E-
~Ls,,)d’+‘)
= fi
S==l
(E+~L,,.)u(“)+z.f’“)+
s=1
+z,e-d’O(z~)+z,e-dt~O(h).
(3.6)
Here t, < 1. For e(“) = u(“)--v~“) we shall then have E-
$f-L,.)e@‘+lJ = fr (I?+ TLS,“) s=1
+z,e-~nO(t~)+t,e-dt~O(h),
(3.7)
eg) = 0, if A E S,,, and naturally
e(O)= 0. Considering the expansions of e(“+l), e(“) and all the terms in (3.7) in terms of the eigenfunctions of the operators A=
fi(E-$LS,“)
and B=fi(E+sL_), s11
s=1
we obtain Ile(“+l)/l< I/e(“);/+e-‘**z,(O ($)+O
(h)),
or, considering the form of z, Ile(“+l)l/ S /le(“)ll+zo(O(t~)+O(h)). Hence, for any t(0 < t < T) we easily obtain I/u(~)-v(~)~[= 0($)+0(h),
(3.8)
where k is, such that tO+tI+ . . . +tk-l = t. For equations with constant coefficients, as can be easily verified, the right hand side of (3.8) will contain 0 (t3+O(h2).
YE. G. D’UKONOV
66
3. We shall now demonstrate the possibility of using a variant of the fractional step method for eqn. (3.1) without any restrictions regarding the sign of b,. We introduce the notations: &,.v
= &S!%(&, t,)Q%
B,v=
-
b&x, 9tn)7.J * W-1
Let us consider the following variant of the fractional step method: (E-_zL, *~)vt~~“p) = ~~~~B~)~(n)+f(~), (E--L,,) vfn+slPf= &+(s-WPl* Eliminating
v("+~'~) we
(3.9)
obtain
P rI( S==l
E--L,,.f~f”+~f
= (E+zB,,)o(“)+zf’“‘.
(3.10)
The operators L,,, and B. may not have common eigenfunctions, but it can be easily verified that P
1lI-I S=l
(E-tL,,.)e(“+l)/l
> (l+c,z)[lef”+x)II
and ]l(E+r4)ec”)[j
< (f-l-e,-Qljet”)~/.
From this it is easy to obtain the sufficient condition for stability from the initial data /je(“+l)li < (l+cr)jle(“)l(. The expression #&*f-0 11= O(r)+O@) is derived in the usual manner. If the ._. . coefficients of the equation are constant, I~.P~--v(~)jj = O(r)+0 (h”). § 4. CONVERGENCE
TO THE GENERALIZED
SOLUTION
We consider eqn. (2.1) with the initial condition (2.3) and the boundary condition (2.2). The coeflkients a,, > 0; a&a < m) can have any sign but are such that Lt is a negative definite operator. For simplicity we assume everywhere that a&~~) E C(O) @,)(a < m). Let x = (x1, x2, . . . , x,,). We shall call the function u(x, t) the generalized solution of eqn. (2.1) with the conditions (2.2) and (2.3), if 1) u(x, t> E Wj” (QT> 2) t/(x, t) E We-) as u functjo~ of x for every t E [O, T] 1.
3) u,s ,... s =(O,O ,......, 1 ( 4) jtrnod (24(x, dt)-i(x))2dQ = 0
O)inthesenseofthemetricL,(S),
5) th; following equation will be true for any sanction G(x, t) sathfying conditions 1), 2) and 3)
s
$;iTdQ=
QT
-
(4.1)
Difference schemes for boundary problems
67
Here !&, denotes the set of points 0 < x8 < 1 (S = 1, . . . . . p), t = to. F$J~(~)(D)is a space of functions having generalized derivatives up to the order m inclusively, summable with the square on f) (see [1 I]). It can be easily verified that every classical solution of a mixed problem is generalized and the generalized solution is unique. It will be shown below that a generalized solution exists and can be obtained as the limit of the approximations obtained by the method of fractional steps. Here the chief tool is the “energy inequalities” (see [I l], [12]). The derivation of such inequalities for parabolic equations is discussed in [13], [14], [15], [16J, [17], the most general results being given in [13], [14]. 2. As mentioned in 0 2, the fractional step method leads to the relation (2.6). Let S,ZLk denote a sum of the type
e Then (2.6) for p = 2r can be rewritten as follows:
(4.2) v&‘=O(n=O,l,...,~ft), But if p = 2r+l,
if de&.
(4.3)
(2.6) becomes
(4.2’) The following notations will be used: a;,,, is the set of all lattice points lying on the layer t = to and belonging to the region Q’, G Q; t/t F=o
Q6, kr
will be denoted by QL,,, and the sums over all the points QLSt; Qk, t QA,t; Q,, t wilf be denoted respectively by
68
YE.G. D’YAKONOV (il . . . . . . i,>
rf A =
lies outside L?,,, we write
V~~=0(n=0,1,...,
T/z),
Q&s, h) = 0
(4.4)
and extend (4.2) and (4.2’) to all these points. Of course in that case the right hand sides of (4.2) and (4.2’) at these points will not involve$ But if (4.2) at all points is multiplied by thP(v(“+1)+~9/2 and the relations obtained are summed Q' h.t over Q;,,, then because of (4.3) and (4.4) we obtain zhp
c
0’ x
(f)(“+l)
_&P
_ &)
c
@+1)+&Q 2
p ic
. ..+(~)p-2~~s_.L~,...L~p_~~~~=rhp~~(”’
Lff... Oc”+l:+w’“, .
(4.5)
Q
Since under the conditions (4.3) Li is a negative definite self-conjugate operator, and Lhs1Lh% = Lh s2LhSl’ I* (
-fL)
= _Zhp c
‘(“+~
Q
{T
L;+
s-1
To obtain the difference energy inequalities we transform the remaining terms on the left hand side using the formulae for “summation by parts” (see [12]) and taking into account the conditions (4.3). We shall illustrate these transformations with the example of L’&Lia, writing 2 brevity :
for A”_, and 0: for At:, xs
for the sake of
m
=
cc
aslDllas,dD:: D:: @)J2 - c 2 aslor,as,,,(D~:Dz cp)“, n’ a,a2=0 Q’ cr,.aa=O
where k = t/z.
(4.6)
Difference
schemes for boundary
problems
69
As we see, use is essentially made of the fact that each operator L3 depends only on x,. Thus instead of (4.5) we shall have
On the left hand side of (4.7) we leave only terms having the second index equal to m in all the coefficients, terms with the index tl, < m being transferred to the right hand side. The coefficients a,, in terms on the left hand side are replaced by their minimum in Qr, the terms on the right hand side are increased by replacing a,, by the maximum of a,,] in QT, and the sum of the squares of the differences with CI,< m by their upper bound expressed as the sum of the squares of similar differences containing only the indices u, = m (this can be done because of the conditions (4.3)). The expression zhp
c
f’“’
w(n+l)+wW 2
Q
is estimated in the form frhp
2
(f (n))2+ -+ zhp c Q
Introducing
(v(“))~. Q
the notation
J, (dk’) = hp c {(v(~))~-/-T~&,,( D”’ SlD”sau(k))2+ . . . + zP(Of”. . . Dp”v(~))~}, I?’ we obtain from (4.7)
1,
&+U+&
(f (n92 + Jo(v) + +
) -!- J&J(~)) < Co(T) {dzp c
2
Q
Y
n=o
J,,,@(“)))
9 J
(4.8)
where C,(T) is a constant independent of z and h. This relation can also be obtained for p = 2r+l if a similar transformation is carried out on (4.2’), introducing the notation 1,
w(“+l)+v(“) 2
&+l)+yW
. = &’
)
F
2
YE. G. D’YAKONOV
70
(4.9)
Let fo C(O)(Qr) and v E D;(C).
Then, using the method described in [12], it is easy to obtain the following expressions J,(U@)) < C,(T) = const ., 1, ( *‘:e
1
Q C,(T) = const.
(4.10)
Here it is naturally also assumed that p: satisfies the compatibility conditions, i.e. q~and all its derivatives up to the (m-1)th order become zero on the boundary of fi. @+$+...+bp~ (aI+az+...aP
(4.11)
LEMMA 1. Let the conditions (4.9) and (4.9’) be satisjied. Then the following inequalities hold for ~6”’
;chP &i‘
< C,(T) = con&,
x F,(D!~zJ(~))~< C,(T)
hP
a’
Proof. We multiply
= const.
(4.12)
(4.12’)
s=l
(4.2) at all points Qi,r by ~hp~~~~~)=I
&P
elfn+rt+w(")
and sum up the equations obtained over Q6,z: ~hp~Y....(l;iH4SI:+...+~iiJ.lI:i:jX
(4.13)
Since the operators Li, , Lh,, . , . L\, are positive,
Difference schemes for boundary problems
71
The remaining terms are transformed: thp
WI &,&n)~‘@(“+l) + ?I(“))= ZhP u&l: D,d”)) D;(v(“+l) + d”)) c P ' a=0 0' =
th’ )y F u~.D,(D:v(“))~ = h" )j R a,.[(o:u(k))2-(D:41)2], 0’ a=0 Ll’ a=0
where k = t/z. Similarly L;, L~,L~,(o("+')+o("))
th”
.D,d")
c
Hence from (4.13) it follows that zhp
c
( D,v("))~+ hP
us,( @ zJ(ky + . . .
Q
m
p-2
as,,,..4
sIzp--I
c ='...=p_l=O
p_l’“p_lm~“’
x(Dz... D:;z;d2)+zhp c
Q'
(4.14) f(")Dt(v)(").
Only terms with all differences of the lmth order are left on the left hand side of (4.14), the others being taken over to the right hand side. The coefficients on the left are replaced by their minimum in Qr, the coefficients on the right by the maximur_, of their modulus, and the square of the differences in terms of the squares of the largest differences. Bearing in mind that under the conditions (4.9) and (4.9’) the expression within curly brackets on the right hand side of (4.14) is finite and that
cf.
Dtv(“)< + x[(f)2+(Dtv)2],
Q'
we
Q
obtain
. . . t~P-2slZp_,(D~...
D;_lu(k))2] < C,(T) +
YE.G.
72
D’YAKONOV
we SIB11now show Bow tie terms oft the rig& hand side of (4x) can be estimated. That is, we shall prove the inequality hP
(-D~-““a@))~ < &hP (D~et~~))~2+C,hP (zI(@)Z, cP c c w&e 8 > 0 can be as sm& as desired. 3ecause of conditions (4.3) and {4,4), (4.16) can be rewritten as
(4.16)
if we consider the expansion of 0 in terms of sines in the cube L&: -1 g x, < 2@ = 1,2, *.*p) containing the cube 9, 7 h&+~l agf sm 3
QC”> =
f
5: then, taking account of (4.4) we find that (4.16’)is equivalent to the inequality (4.16”) Since for any smaB E > 0 and any A it is possi@e to Snd a C, ~~de~~de~~ of 2) so large that ASrnN2 < E,?~~-+- C,, therefore (4”16”) is true. Expression (4.16) is therefore proved, Evidently similar inequalities are true for the other terms:
ffence, choosing c: so small that the- coe&%& of the s.sm of &e squares of the differences of the m(2g+-I)th order is Ie;ss than 87’4 and taking into account the finiteness of Jt(~(k))obtained earlier, we obtain for p = 2r
73
Difference schemes for boundary problems and hence, for p = 2r+l th”C(D~~‘.)~“+hPC(~~~~(k))~+~2~~=~~~D~~~D~~(k~~~+*.* P‘ s=1 e ...+rP-l(D~D;... D;T.J(~))S)Q C(T) = const.
(4.17’)
This ends the proof of the lemma. 4. Inequalities stronger than (4.17) and (4.17’) are needed to apply the inclusion difference theorems of [ll] and [12]. We therefore prove the following theorem. THEOREM 4. Ij’ the conditions of’ Lemma 1 are satisfied jbr the problem (2.1)(2.2)-(2.3), the jbllowing inequalities are true jbr the sequence ojjicnctions v$‘) determined on the lattice points by the method ojjiactional steps (2.4), (2.27, (2.3), (4.4): n
zhp c
((D,&y+
hrx
{ Q’
Prooji Inequality
(D:e)(B))2+(v(e))2}
2
‘~D;~...D;Pv”?,]
c
g
<
y =
y =
COnSt.
COnSt.
at+...+ap
(4.18) (4.18’)
(4.18) is a trivial corollary of Lemma 1, since r & (D$@))2
is easily estimated in terms of F (Ev(“))~ as a result of conditions
(2.3’), (4.4).
The proof of (4.18’) is carried out as in the case of (4.16), i.e. it is sufficient to obtain D;~v(~f)~
hPxD;r... M
<
y1 =
const.
aI+u2+...+ap=m.
It is easy to verify that under conditions (4.4) we shall have hP
c
(@I..
.D;P v(k))8 =
(-
(w(~),&D;‘,
1)“hP
, .@PpD;pdk))
c
a
P
(D?v(~))* = hP
h”
i2a” S3
whereat,...I~ are the expansion coe~ci~nts of 99’) in terms of sines in the cube fz,; (--AZ) is the eigenvalue of the operator index lI . . . , . . 1,. But since
from the boundedness
tiSD: for products of sines with the
of hp x p: (D~v’~‘)~ I?’ a=1
YE. G. D’Y.~K~NOV
74
follows the boundedness
of C#
+.J
=/T...QW~.
I
P
and consequently (4.18’). We shall now formulate a theorem which is a corollary of the theorems in [12] (see Chapter 1, sec. 6). THEOREM5. Let the inequalities (4.18) and (4.18’) be tnre for Qgiven sequence of functions VAdetermined on the lattice points Q’. If (VA)’ denotes the multilinear function with respect toxI, x2, . . . , xp , t coinciding with VAon the lattice points, there exists a sequence v48 for which 1) the junctions (DtvAS)’ and (D;l Dt.. . DIVAS)‘, u,+ aZ+ . . . + up = m, converge weakly in QT and 52, respectively to the functions u, and u~,=~..,~~; 2) the functions L&A)
(Di’Dia . . . D~PvJ’,
a,+a,+
. . . +ap = m,
- 1, converge
in
to u~*olg...op;
3) the limiting functions
have the properties:
u = uo.o...o E W)(QT), u E
IlullW!‘)(Q,) II4lwp)(n,)
am)G9~
G
< c
c(Y),
(7)
and
au at
aal+a’s+...+apu
= ‘,
a,a$....=p~ al+ar+
-_=u
axfiax;a . . . a+
5. To prove convergence to the generalized further require the following lemma.
. . . +ap 6 m.
solution of (2.1)-(2.2)-(2.3)
we
LEMMA 2. Let 1) q E Db(c) and for every t (0 < t < T)f E Ok(c) 2) the functions y and f together with all their derivatives up to the (2m-1)th order with respect to x, (s = 1,2, . . . . . . p) become zero on the boundary of G’ and Qr respectively. Then for any n(n < T/Z) we shall have h’x(L:,L;B *t
. . . LfI v(“))~ < C,(T) = const., Si #
Sj
for i # j;
i= 1,2,...p;
si = 1,2, . ..p.
Proof As in 6 2, we consider the expansions of v(“+l), v(“) and f(“) in eigenfunctions of the operators Lt (s = 1, 2, . . . . . . p): @+1)
=
c
at?P+kl...kp;
kl...kp v(n)
-
= c kl...kp
f(“) =
c k,...kp.
The
notations are the same as in $ 2.
at!..kpvk,...kpi
bt!..kpl))k,...kp.
75
Difference schemes for boundary problems
We then have, because of (2.6), P
n(
-T(p)2
l-
a?+.‘,, = ’
P
2
s
S=l
4%1
P
1+
+
1
N S=l The
P
+(A&~
coefficient of jGk,,,.kpin the expansion of Li, LtZ . . . L”,,#+I)
will
be
ug+$
= (- l)‘(Apl))2 . . . (J.J”1)2a(“+,1) kl kp’ Sl
81
From the conditions
c
kl...kP
of the lemma we have
) < c6, (@‘)” . . . (@z)4(a$:‘.,k S P' Sl
c
k,...kP
)” < c,. (lp’)4 . . . (A,#4(b$“f,,k $1 q ’ P
Hence c
k,...kP
(@+:))2 < C,(T) = const. 1
P
and the lemma is proved. We now have all that is necessary to prove the convergence of the fractional step method. 6. THEOREM 6. Let the conditions of Lemma 2 be satisfied. Then jbr z + 0 and h --f 0 the sequences (VA)‘, (D, vJ, (Of’, . . . . . . , D~PvJ tend, in the sense mentioned in Theorem 5, to the generalized solution and the corresponding derived generalized solution. Proof. The inequalities (4.18) and (4.18’) are true when the conditions of Lemma 2 are satisfied. Therefore Theorem 5 is true. We shall prove that the function u, obtained from this theorem as the limit of the subsequence (v,,)’ when t + 0 and h --) 0, is the generalized solution of the problem (2.1)-(2.2)-(2.3). It remains to verify that u satisfies the conditions 3), 4) and 5) from the definition of the generalized solution. The functions (Di’Dx2 . . . . . . D;r,vJ for aI+xq+ . . . ap < m- 1 are elements of the space I@)(!&) which become zero near the boundary. Therefore, from the theorem on the absolute continuity of the insertion operator (see [l 11) the limiting functions a~l+-+apujaxp . . . . . . ax;P will become zero on S, so that condition 3) is satisfied. Condition 4) is satisfied because the functions (v,)’ are elements of the space W.jl)(QT) and are therefore continuous to the same degree as a whole in L,(J2,) [ll]. To verify condition 5) it is sufficient to obtain (4.1) for any function Q, having the.usual derivatives up to the mth order with respect to x1, x2, . . . xp, of the first
order with respect to t, and becoming zero near S, since any function3 of the class of allowed functions can be obtained as the limit of such smooth functions Qi. Let us take a function 5, multiply (4.2) and (4.2’) by zhp(@(n+l)+@(n))/2 sum the equations obtained over Q,,, T. We obtain
and
76
YE.G,
DVAKONOV
As can be easily verified, because af Lemma 2 the expression in square brackets will be bounded in LZ(QT). Bearing in mind that @ satisfies (2.2), eqn. (4.19) can be written as
The term tf 1 obviously tends to zero when z+ 0. Consequently we obtain (4.1) from (4-20) at the limit, whkb isI&& was required,. Since the generalized sol&ion of the problem (~.~~~~.~~~2.3) is ~~~q~~, it is easy to verify that convergence to it occurs Ghen t -+ 0, k -P 0 not only for (v,)l’ but for the entire sequence 94. We have thus proved the convergence of the disc&c: functions obtained by the fraction&l step method to the generalized solution of the problem ~2.~~(~.~)-(2_3)~ without any restrictions of the w zj@= < e3 as in the case of an explicit scheme, Convergence to the generalized sohztion without such restrictions can also be obtained for an implicit scheme 1131,but the amount of computing work with such a scheme will be incomparably greater than for tha method of fractional steps.
Difference
schemes for boundary
problems
77
NOTE 1. If a somewhat more general scheme is considered instead of (2.4), o(n+s/P)_erCn+(s-l)lPl
=
-bS)@p+wIPl
b,ep+s/P)+(l
(S = 1,2. . . . ,p-1).
t o(n+l)_&+(P-lvP1
t
.bp~;v(~+q..(~
_,,)L:,.[“+(P-~)/PI+T~~(“),
all the results of 5 2, 3 and 4 will be applicable to the case 4 < os < 1 the calculations becoming rather difficult only in 0 4, and in Theorem 2 instead of O(t2) there will be O(t), if a, #&. NOTE 2. All the results can be easily transferred
(&)i, = 0,
to the case of the net
(s = 1,2, .. . ,P),
when 0 is a p-dimensional parallelepiped. In conclusion the author expresses his gratitude to A. A. Samarskii for valuable advice on writing this paper. Translated by PRASENJITBA.W REFERENCES. 1. PEACEMAN, D. and RACHFORD, H., The numerical solution of parabolic and elliptic differential equations. J. Sot. Industr. and Appl. Math. 3: No. 1,28-41, 19.55. 2. DOUGLAS, J. and RACHFORD, H., On the numerical solution of heat conduction problems in two and three space variables. Trans. Amer. Math. Sot. 82: No. 2,421-39, 1956. 3. YANENKO, N.N., Dokl. Akad. Nauk SSSR 125: No. 6, 1207-10, 1959. 4. YANENKO, N. N., SUCHKOV V. A. and POGODIN, Yu. Ya., Dokl. Akad. Nauk SSSR 128: No. 5, 903-5, 1959. 5. YANENKO, N. N., Dokl. Akad. Nauk SSSR 134: No. 5, 10346, 1960. 6. RYABENKII, V. S. and FILIPPOV A. F., Ob ustoichivosti raznostnykh uravnenii. (The stability of difference equations.) Gostekhizdat, Moscow-Leningrad, 1956. 7. DYAKONOV, Ye. G., Dokl. Akad. Nauk SSSR 138: No. 2, 271-274, 1961. 8. DYAKONOV, Ye. G., Dokl. Akad. Nauk SSSR 138: No. 3, 522-525, 1961. 9. GEL’FAND, I. M., FROLOV, A. S. and CHENTSOV, N. N., Izv. vyssh. uchebn. zavedenii. Ser. matem. No. 5, 32-45, 1958. 10. DOUGLAS, J. and GALLIE, T., Variable time steps in the solution of the heat flow equation by difference equation. Proc. Amer. Math. Sot. 6: No. 5,787-93, 1955. 11. SOBOLEV, S. L., Nekotorye primeneniya funktsional’nogo analiza v matematicheskoi fizike. (Some applications of functional analysis in mathematical physics.) 12. LADYZHENSKAYA, 0. A., Smeshannaya zadacha dlya giperbolicheskogo uravnenii. (Mixed problem for hyperbolic equations.) Gostekhizdat, Moscow-Leningrad, 1953. 13. LADYZHENSKAYA, O.A., Matem. Sb. 39: No. 4, 491-524, 1956. 14. LADYZHENSKAYA, 0. A., Matem. Sb. 45: No. 2, 123-58, 1958. 15. LEES, M., Energy inequalities for the solution of differential equations. Trans. Amer. Math. Sot. 94: 58-73, 1960. 16. LEES, M., Approximate solutions of parabolic equations. J. Sot. Industr. and Appl. Math. 7: 167-83, 1959. 17. SAMARSKII, A. A., Zh. vych. mat. 1: No. 3,441-460, 1961.
BOUNDARY
PROBLEMS*
YE. G. D’YAKONOV Moscow (Received 27 September 1961)
VARIOUSdifference schemes are available for finding approximately the solution of a boundary problem of the equation au/at = Lu, where L is an elliptical operator with constant coefficients. The principal requirements for such schemes are stability and convergence. In explicit difference schemes the time step has to be chosen considerably smaller than the step for space variables, while in implicit schemes (which are free from this restriction) systems of linear algebraic equations have to be solved. If the number of space variables p 3 2, then both explicit and implicit schemes require much computing work. A series of investigations (see [I]-[5]) was therefore undertaken to study problems of constructing absolutely stable difference schemes in which the solution of equation systems with a large number of variables is not necessary. The present paper is devoted to the analysis of one such difference scheme, using the method of fractional steps proposed in [3]-[5]. To justify the validity of the method the authors of [3]-[5] replaced the difference scheme by another which they considered equivalent to it and established the convergence of the latter as a consequence of stability and approximation. But in the class of boundary conditions considered these schemes are not equivalent. Further, even in the case of the heat conduction equation an example can be found in which the method does not approximate the differential equation for any small t and h, if the function is not linear at the boundary of the domain. If, however, zero boundary conditions are chosen, the analysis of this method given in [3] and [5] can be used, although not for such a wide class with constant coefficients: The result of [4], however cannot be obtained even for zero boundary conditions. Nevertheless, the range of application of the fractional step method can be extended, and it will be shown here that under zero boundary conditions a somewhat modified method is absolutely stable and has a convergent difference scheme: (a) for equations where L is a self-conjugate, elliptical negative definite operator with separable variables of the order 2m, (b) for equations with a right hand side. For m > 1 convergence of the method means convergence to the generalized solution. The possibility of increasing the time steps is also mentioned. * Zh. vych. mat. 2: No.. 1, 57-79, 1962. 55
Ye.G.
56 4
1. FRACTIONAL We
equation
STEP METHOD
D'YAKONOV
FOR THE HEAT CONDUCTION
EQUATION
shall examine the principle of the method by taking the heat conduction [3], g+g+g
(1.1)
in the cylinder sz{(x,y):
Qr=~?xtO
o
1, o
l}
with the initial condition r&o = dX,Y) and the boundary
(1.2)
condition r+ = py(%Y, ‘),
@,Y)ER
(1.3)
S is the lateral surface of QT. Let h = l/N be the step for x and y; r the time step; x1 = ih, i = 0, 1, . . . . . . N; yJ =jh,j = 0, 1, . . . ,N; t, = nt,n = 0, 1, . . . . . . , T/t;Q,,{(i,j) : i = 1,2, . . . ,N-1; j= 1,2 ,... . . . 9N-- 11; aw) :i=O,Norj=O,N}; 52,=8,US,. z@ will denote D(x*, yJ, tn). Conditions (1.2) and (1.3) are approximated
in the usual way:
r,l$’ = vij, $’ = y)$‘, The method of fractional
(1.2’)
(i,i) E z. @+l) z &-+l), (i,i> E&l.
(1.3’)
steps consists in forming an equation
system with
intermediate functions of z$+~’ for finding u$+‘) when (i, j) E Sz, from the known values of r@) :
(1.4)
where dB,,V, = 4~;Vij =
Wtl,j-2Qj+Wi-l.j
,
h2 j-1
%,jt*-2Qj+Q, hZ
O
When (i, j) E S,, the values of TIE+*)are determined from the condition (n+h)=
Vij
@0_
Vij -
@I
lyij*
We rewrite (1.4) in the form (E-o(TZ4i;) z$;+*) = (E+(l
-_a)z4&$‘, (i&E
gh,
Difference schemcss for boundary problems
is the identity operator. The followjng notations will be used:
where E
j=2,3,,..,N-22),
D’{Ii,j):i=2,3,..,,i\l-2;
S,{i,j):i
= 2,3, ..,) N-2;
S,(i,j>:i=
I,.#--1;
S,,(i,j):i=l,N-1;
j=
j=2,3,
1,&-l), _.,N---3),
j=l,N-+
The authors of [3]-[S] replace the scheme (1.5) by the following scheme without the intermediate
values of T$+*):
(E-~d~;)(E-~d~~~~~~l~
= (E+(I-~~til~~~f.E+fI-~~~~~~~~~~-
(1.6)
This scheme, as shown in [3], is &sofutely stable and approximates equation (1.1) in the metric I+ ,with the order of approximation O(z)+O(V). Therefore its convergence in L, with the order O(t)+O(/P) follows as a result of some well-known theorems (see [6])_
and balls is the value of vri at the point of the boundary closest to (ij), Let us see how o:*+) is eliminated. If fi,j) E I)‘, each of the two relationS (1.5) wilt be satisfied at the points (i&-l), &j)_ f&J_tQ1 fr‘-I,& i+l,j). Hence applying the operator (E+(f --~)r~If;l, to the first equation and (E-SLY&;) to the second we obtain respectively
YE. G. D’YAKONOV
58
But if (ij) E S, (for the sake of definiteness we take, for example, j = N-l), the second of the relations (1.5) is satisfied at the points (i- 1,j), (i,j) (i + 1, j) and applying to it the operator (E--or&) we again obtain (1.8’). The first of the relationships (1.5) holds only at the points (i, j- l), (i, j), while at the point (i, j+ 1) we have instead the equation (E-oatd:;)v$?) --0)rd~-)2+-zL3~;yiN @). Hence, instead of (1.8) we have XX 1 @+(I
= (E+ (l-
--a)z~~~)(E--azd~;)v~+*’ = (E_t(l-~)rd~;)(Et(l-~)td~~v~~‘-(l-~)~d,~W~~~~s,.
Adding this equation to (1.8’) we find that (1.7) is true for points (i, j) E S,This elimination process is carried out similarly for the points S’,,and S,,. If y(x, y, t) = 0 or even Py/ax2 = Py/,lav” = 0, R$) becomes zero for all (ij) E J2,, and (1.5) and (1.6) are equivalent. But if R$) # 0, it is easy to construct an example showing that scheme (1.7) (and hence also scheme (1.4)) does not approximate equation (1.1). Let us suppose that azu atz'
a a54 -at ax29
a a=u --at ay ’
adu 32,
adu ax2ay2 9
ab ay’
(1.9)
exist and are finite in QT. Under these conditions scheme (1.6) approximates (1.1). Hence substituting u(x, y, t), the exact solution of (1. l), in (1.7) we obtain (E-aazd~;)(E-o(~~d~~)uUI~+~’ = (E+(1-~)zd~;)(E+(1-o)z~y2;)u~~~+~{O(~)+O(h2)+~~~~}.
(1.10)
Let us take u = sin xxe-%“. Then obviously all the conditions for the derivatives are satisfied and R$) = (c- 1) $ sin2 $
sin nXieWnZt,
if (i,j) E S,.
As usual we introduce the norm of the function on the layer t = t, = nz llzI(“)ll =
(h22
(lj$q.
i, j=l
Then llR”II > c/h, where c > 0 does not depend on r and h. Therefore (1.10) it follows that (1.7) does not approximate (1.1). Thus the following theorem is established.
from
THEOREM 1. The method oj’jractional steps is an absolutely stable and convergent scheme with a mean order oj’convergence equal to 0(T)+O(h2) jor eqn. (1.1) with the initial condition (1.2) and the boundary condition us = 0. In the case oj’the non-homogeneous boundary condition (1.3) the method, generally speaking, may not approximate eqn. (1.1).
Difference schemes for boundary problems
59
It can be easily verified that both the example and the theorem are perfectly applicable to the case of p space variables, only the requirement that -___ 8% a~:,ax~,...ax~p ’ should be continuous, where si # s,, if i the conditions (1.9). We may point out further that for the established for y # 0 also. The scheme in can be easily extended to the case of any only for p = 2. $ 2. METHOD
OF FRACTIONAL VARIABLES
# j, si = 1,2, . . . , p, must be added to schemes in [I], [2] convergence has been [2], proposed only for p = 2 and p = 3, p, while the scheme in [I] is applicable
STEPS FOR EQUATIONS WITH AND RIGHT HAND SIDE
SEPARABLE
1. We shall now describe a modification of the method of fractional variables for equations with separable variables and right hand side.
Suppose in the cylinder Q,=~x [O
at
satisfying the boundary
s=l,...,p)
’
cS=l Lu+f,
(2.1)
conditions
au (u,--_, av
am-h
a**,
___ avm_,
1
s = (O,O, . . ..O)
(2.2)
and the initial condition U]t=l?=Q)(%.%?, .5x,).
(2.3)
Here
Lsu= -p a=0
(-lY-$+.(xs,~), s
s
us&,) > 0, a,,(~,) 3 0, S is the lateral surface of QT, Y the normal to S. Let us construct a difference approximation of this problem. Let t be the time step, h = l/N the space variable step. The following notations are introduced: (%)is=kh7
i,=O,l,...,
N;
s=1,2
,..., p;
v(i,h, i2h, . . . . iPh , nz) = v~~..~,= ~2); SZ,{A:i,=O,l,...,N; Q,{A:i,
= m,m+l,
s, = .ni\rn, C,{A:i,=O,l,...,
N;
A=(iI,i,,...,ip);
s=1,2 . . . . N-m;
&=m,m+l,...,
t,=n~;
s=
)...) p}, 1,2, . ..) p},
*
N-m
for s’fs}.
YE. G. D’YAKONOV
60
Where no confusion is possible we shall simply write v instead of vA and omit the index. The relation between the net functions and the net also will not be indicated. ~2) is determined from condition (2.3): vy = p)d.
(2.3’)
If A E S,, we write v$)=O
(k=O,l,...,
T/z),
(2.2’)
To find vg+r) at the points Qh from the known ~6”)we shall consider p systems of equations with p- 1 intermediate functions v$‘+“~): o(n+s/p)_~t”+~S-l~lPl
=
z
~(L:v(n+s/P)+L:v[“+(“-l)IPl)
.(n+l)_&l+(S-UP/l
(s =
+Lh pv [n+(~--lV~l)
$
1,2,
. . . ,p__l),
Thy-W,
(2.4)
t
where m LpJ =
c
(_
1,a-1+
Jsv,
(a,,(j,q~~~v))
=
“il...(is+l;;.fP-“il...‘~,
a=0
d,,v,=
vij...ip-W,...(is-l)...ip
____.
,
Thy’“’
=
(fj
(&
+
L;))-'l'"';
h
E is the identity operator, jp) denotes J&h, izh, . . . . . . , i,h, (n+&)z. In this approximation (-Lh,) is a self-conjugate and positive definite operator and L’&Li, = Lt,L’& (see, for example, [7], [8]). The computing scheme is as follows : v(“+~‘~) is found from the known (N+ l)P-dimensional vector 7P)(v(“) is determined from the initial condition) as the solution of the first system (2.4) with the conditions (2.2’). The vector v(“+~‘~),satisfying the second system of (2.4) and the conditions (2.2’) is determined from the vector v(“+~@)obtained till v(“+(P-~)‘P)is found. Each of these systems has matrices containing unknowns, in which the nonzero coefficients are arranged in 2mf 1 diagonals, so that x l/hP arithmetical operations are required for finding the solution of the system. The determination of
is also carried out in steps: p_
~L~)w~+s/(p-l)l
=
wy+s/(P-l)l =
WY’ =
1,2, . . . . p-l,
s =
A E-f&,
w~+(s-lMP-l)l,
0
for d ES,,,
fY’
for d E&,
0
for A E&.
Difference. schemes for boundary problems
61
It can be easily verified that the determination of w(“+l) requires ,~l/hparithmetical operations. pi(“+‘)is found from the values found of ~r”+(p-~)‘~land w(‘+l) of the last system (2.4) and the conditions (2.2’). The transition from v@) to zPil) is thus completed in x l/P arithmetical operations, i.e. the number of these is of the same order as the number of points on the layer t = t,. 2. To examine the problem of stability of scheme (2.4) we rewrite the latter in the form (E-$L:)V’“+~~~)
= (E+;L+“+‘“-‘,?
(&
= (&
;+P+”
$= 1,2,...,p__l;
~Lhp)D[“+(p-l)~pl+TThf(~)
(2.5)
and eliminate the intermediate values of v f”+@), taking into account the co~utability of L,,and Ls,and conditions (2.27, the importance of which has already been mentioned in 8. 1. Because of conditions (2.2’) it can be considered that (2.5) is true not only at the points Q,,, but also at all points C. Therefore the first p- 1 of the systems (2.5) are equivalent:
Multiplying this equation by (E+(z/2)@
and the last of the equations
(2.5)by
P-l n
s=l
(~-t@w:~,
and then adding, we obtain
since P-1
g-h@’ = f d f*) f “Lh 2 *)
E-
ni s=1
Af8.
Under the conditions (2.2’) ti is a self-conjugate and negative definite operator (see (81). Consequently there is a system of functions yl$)(x,), k = 1, 2, .....,N-2m+1, satisfying (2.2’), such that and
(wiy’,wg> = 41k* = 3
0 for kl-;f;k,. 1 for kl==kgr
where
Then the operators A = fi s-1
(E-$LJh)and
.=fi(E++L:) .S=1
62
D’YAKONOV
YE. G.
have the eigenfunctions @k,...k&$,
and correspondingly
Therefore,
... ,
the eigenvalues
as can be easily verified,
and ]IA-‘BII < 1, where [ICI1= maxI&] ({A,} are eigenvalues of the matrix C). Thus the stability not only of scheme (2.6) but also of each of the intermediate steps is proved. The intermediate steps in the schemes of [l] and [2] can be unstable. In the proof of the stability the condition a S(l>, 0 was used only for negative definiteness of Lt. It can therefore be relaxed and it may be required simply that Lt be negative definite. The stability of the right hand side can be easily derived from the stability just established and the initial data (see [6]). 3. We shall now show that the scheme (2.6) approximates equation (2.1). Later we shall often need the class of functions D;(c) introduced in [9], viz. we shall say that the function F(x1,‘x2, . . . . . xp) defined in the p-dimensional hypercube abelongs to D;(c), if all the partial derivatives F(x,, x2, .., . , xp) containing not more than 4 differentiations with respect to each coordinate are bounded in modulus by the constant c. (2.6) is rewritten in the form
=- 1
&+1)-w(“)
t
2
+
c p
L;(v(“+l)
$v(“)) j-f’“‘+
s=1
f(I)2
L;, L;,(v("))
- (n+l))-+... + v
s1,s*=1 s1>sa
. ..
+(+)'-'L~
. ..L~(o(“)-(--~)P~(“+~)).
(2.7)
VS’ = 0 Let a,,, j; p be such that A E S,; 1) the following exist and are continuous in QT a=+la,,
a) yeax:+13
b) $i;
(2.8)
l, ***v2m;
ki=O,
arm+su
k=0,1,...,2m;
~axy+2
9
s = 1,2,
.. ..p.
(2.8’)
Difference schemes for boundary problems
63
2) for each t(0 < t ,< T) (2.8”)
2.8 E dlgm(c)*
Then, substituting the u-solution of (2.1) in (2.7) we obtain
..Lhp(zP)-(-- l)W+1)), i.e. the scheme (2.5) approximates equation (2.1) with a mean order of 0(r2)+O(h). As can be easily verified, the order of approximation for equations with constant coeliicients will be O(9)+ O(h2). The boundary conditions (2.2) are approximated in the general case with an error O(h). It is known [6] that, if a difference scheme approximates to a differential equation and the initial and boundary conditions, and is stable with respect to the initial and boundary conditions as well as the right hand side, the scheme is convergent. But the theorem of [6] can be applied to the problem (2.1)$2.2)-(2.3) only when m =I, since the boundary conditions are approximated accurately only when m = 1, and stability to boundary conditions cannot be obtained in the general case. The fact that all eigenvalues of the operator
are greater than unity under the conditions (2.2’) is also insufficient for convergence, because this establishes the boundedness of A-l qnly for the conditions (2.2’), which do not approximate (2.2) accurately. Therefore, for m > I we confine ourselves for the moment to the statement that the method of fractional steps is an absolutely stable and approximating scheme. The question of its convergence will be discussed in 0 4. For m = 1, consequently, the following theorem is true. THEOREM 2. The method oj jkaktional steps jor eqn. (2.1) with the conditions (2.2) and (2.3), when the conditions (2.8), (2.83, (2.8”) are j~~~lied, is an absolutely stable and convergent scheme with a mean order oj’convergence 0(x2)+0(h). Ij‘the equation coeficients are constant, the order oj’convergence will be 0(z2)+O(h2). Some difference schemes with variable time steps were studied for the heat conduction equation in [2] and [IO]. In the next section we shall consider the possibility of a similar modification of the method of fractional steps. $ 3. FRACTIONAL
STEP METHOD
WITH
VARIABLE TIME STEPS
1. Let us analyse the fractional step method with increasing time steps for a somewhat more general equation than (2.1).
YE. G.
64
D’YAK~N~V
We look for the solution of the following equation in the region QT (see 0 2)
(3.1) us(xs14 I=- 0;
with the boundary
b&9 0 2 0,
condition u/s=0
and the initial condition (2.3). The difference approximation form
(3.21
of this problem with variable z will be of the
(#=; 1,2, . . ..p).
(!
Ls,BIu(n)=~4,~a, x,,
t,+2-
if de&.
t.+‘-)a(“,, =a) dxs’u+,i,x,,
T,= (f+-%L,..)J-';
t,= qj+zlf...+z,_l.
S-l
The remaining notations are the same as in Q2. As we see, here L,, n(s= 1, . . . , p) depends on t,, but its self-conjugateness, negative definiteness and the commutability of L,,, n and L,, ,, are established as in 171. Eliminating the intermediate stages we obtain the analogue of (2.6):
To establish the stability of this scheme we have to consider how l/w&I changes on transition to the next layer, if at the points of fl, wCn)satisfies the relation P
E-~L,‘~)w'"tl,=~(~+qL~,.)~(", c( "=I
(3.4)
and the condition ~6”’ = 0, d E S,. The process of inves~gat~on is the same as in 0 2, with only this difference that the family of eigenfunctions of the operators
changes from t,, to tn+r during the transition. Using the orthogonality of the different eigenfunctions, we conclude from the fact that the coefficient of each eigenfunction in the expansion of w@+l) has a modulus less than that of ~(“f that IIw(“+1)/1 < ]Iw(“)//.
(3-3
Difference schemes for boundary problems
65
2. We shall prove the following theorem.
THEOREM3. Ij the jollowing conditions are jufirlfilled: 1) ,
Pas y-g@ s
= 1, . ...P).
exist and are jinite; 2) a,, b,, v and j’are such that a) all the derivatives -$@=O,l,..., s
4),
$
(a=o,1,2,3),
- frI
(
3
s Sl 1
(a,@=O,1,2)
in Qr have a modulus less than &feed*; b) jk any t(0 < t < T) u E Dg(h4e-d’). then the vector v@‘)(0 < n < T/t) obtained from the scheme (3.1’) with z,, = t,edtn converges to u(“) unijormly with respect to t(0 ,< t < T) in the metric L, with the order 0(+)+0(h). Proof. Let us substitute u (x1, . . . . . . , x,, t) in (3.3). Then because of the conditions of the theorem we obtain P
I-If E-
~Ls,,)d’+‘)
= fi
S==l
(E+~L,,.)u(“)+z.f’“)+
s=1
+z,e-d’O(z~)+z,e-dt~O(h).
(3.6)
Here t, < 1. For e(“) = u(“)--v~“) we shall then have E-
$f-L,.)e@‘+lJ = fr (I?+ TLS,“) s=1
+z,e-~nO(t~)+t,e-dt~O(h),
(3.7)
eg) = 0, if A E S,,, and naturally
e(O)= 0. Considering the expansions of e(“+l), e(“) and all the terms in (3.7) in terms of the eigenfunctions of the operators A=
fi(E-$LS,“)
and B=fi(E+sL_), s11
s=1
we obtain Ile(“+l)/l< I/e(“);/+e-‘**z,(O ($)+O
(h)),
or, considering the form of z, Ile(“+l)l/ S /le(“)ll+zo(O(t~)+O(h)). Hence, for any t(0 < t < T) we easily obtain I/u(~)-v(~)~[= 0($)+0(h),
(3.8)
where k is, such that tO+tI+ . . . +tk-l = t. For equations with constant coefficients, as can be easily verified, the right hand side of (3.8) will contain 0 (t3+O(h2).
YE. G. D’UKONOV
66
3. We shall now demonstrate the possibility of using a variant of the fractional step method for eqn. (3.1) without any restrictions regarding the sign of b,. We introduce the notations: &,.v
= &S!%(&, t,)Q%
B,v=
-
b&x, 9tn)7.J * W-1
Let us consider the following variant of the fractional step method: (E-_zL, *~)vt~~“p) = ~~~~B~)~(n)+f(~), (E--L,,) vfn+slPf= &+(s-WPl* Eliminating
v("+~'~) we
(3.9)
obtain
P rI( S==l
E--L,,.f~f”+~f
= (E+zB,,)o(“)+zf’“‘.
(3.10)
The operators L,,, and B. may not have common eigenfunctions, but it can be easily verified that P
1lI-I S=l
(E-tL,,.)e(“+l)/l
> (l+c,z)[lef”+x)II
and ]l(E+r4)ec”)[j
< (f-l-e,-Qljet”)~/.
From this it is easy to obtain the sufficient condition for stability from the initial data /je(“+l)li < (l+cr)jle(“)l(. The expression #&*f-0 11= O(r)+O@) is derived in the usual manner. If the ._. . coefficients of the equation are constant, I~.P~--v(~)jj = O(r)+0 (h”). § 4. CONVERGENCE
TO THE GENERALIZED
SOLUTION
We consider eqn. (2.1) with the initial condition (2.3) and the boundary condition (2.2). The coeflkients a,, > 0; a&a < m) can have any sign but are such that Lt is a negative definite operator. For simplicity we assume everywhere that a&~~) E C(O) @,)(a < m). Let x = (x1, x2, . . . , x,,). We shall call the function u(x, t) the generalized solution of eqn. (2.1) with the conditions (2.2) and (2.3), if 1) u(x, t> E Wj” (QT> 2) t/(x, t) E We-) as u functjo~ of x for every t E [O, T] 1.
3) u,s ,... s =(O,O ,......, 1 ( 4) jtrnod (24(x, dt)-i(x))2dQ = 0
O)inthesenseofthemetricL,(S),
5) th; following equation will be true for any sanction G(x, t) sathfying conditions 1), 2) and 3)
s
$;iTdQ=
QT
-
(4.1)
Difference schemes for boundary problems
67
Here !&, denotes the set of points 0 < x8 < 1 (S = 1, . . . . . p), t = to. F$J~(~)(D)is a space of functions having generalized derivatives up to the order m inclusively, summable with the square on f) (see [1 I]). It can be easily verified that every classical solution of a mixed problem is generalized and the generalized solution is unique. It will be shown below that a generalized solution exists and can be obtained as the limit of the approximations obtained by the method of fractional steps. Here the chief tool is the “energy inequalities” (see [I l], [12]). The derivation of such inequalities for parabolic equations is discussed in [13], [14], [15], [16J, [17], the most general results being given in [13], [14]. 2. As mentioned in 0 2, the fractional step method leads to the relation (2.6). Let S,ZLk denote a sum of the type
e Then (2.6) for p = 2r can be rewritten as follows:
(4.2) v&‘=O(n=O,l,...,~ft), But if p = 2r+l,
if de&.
(4.3)
(2.6) becomes
(4.2’) The following notations will be used: a;,,, is the set of all lattice points lying on the layer t = to and belonging to the region Q’, G Q; t/t F=o
Q6, kr
will be denoted by QL,,, and the sums over all the points QLSt; Qk, t QA,t; Q,, t wilf be denoted respectively by
68
YE.G. D’YAKONOV (il . . . . . . i,>
rf A =
lies outside L?,,, we write
V~~=0(n=0,1,...,
T/z),
Q&s, h) = 0
(4.4)
and extend (4.2) and (4.2’) to all these points. Of course in that case the right hand sides of (4.2) and (4.2’) at these points will not involve$ But if (4.2) at all points is multiplied by thP(v(“+1)+~9/2 and the relations obtained are summed Q' h.t over Q;,,, then because of (4.3) and (4.4) we obtain zhp
c
0’ x
(f)(“+l)
_&P
_ &)
c
@+1)+&Q 2
p ic
. ..+(~)p-2~~s_.L~,...L~p_~~~~=rhp~~(”’
Lff... Oc”+l:+w’“, .
(4.5)
Q
Since under the conditions (4.3) Li is a negative definite self-conjugate operator, and Lhs1Lh% = Lh s2LhSl’ I* (
-fL)
= _Zhp c
‘(“+~
Q
{T
L;+
s-1
To obtain the difference energy inequalities we transform the remaining terms on the left hand side using the formulae for “summation by parts” (see [12]) and taking into account the conditions (4.3). We shall illustrate these transformations with the example of L’&Lia, writing 2 brevity :
for A”_, and 0: for At:, xs
for the sake of
m
=
cc
aslDllas,dD:: D:: @)J2 - c 2 aslor,as,,,(D~:Dz cp)“, n’ a,a2=0 Q’ cr,.aa=O
where k = t/z.
(4.6)
Difference
schemes for boundary
problems
69
As we see, use is essentially made of the fact that each operator L3 depends only on x,. Thus instead of (4.5) we shall have
On the left hand side of (4.7) we leave only terms having the second index equal to m in all the coefficients, terms with the index tl, < m being transferred to the right hand side. The coefficients a,, in terms on the left hand side are replaced by their minimum in Qr, the terms on the right hand side are increased by replacing a,, by the maximum of a,,] in QT, and the sum of the squares of the differences with CI,< m by their upper bound expressed as the sum of the squares of similar differences containing only the indices u, = m (this can be done because of the conditions (4.3)). The expression zhp
c
f’“’
w(n+l)+wW 2
Q
is estimated in the form frhp
2
(f (n))2+ -+ zhp c Q
Introducing
(v(“))~. Q
the notation
J, (dk’) = hp c {(v(~))~-/-T~&,,( D”’ SlD”sau(k))2+ . . . + zP(Of”. . . Dp”v(~))~}, I?’ we obtain from (4.7)
1,
&+U+&
(f (n92 + Jo(v) + +
) -!- J&J(~)) < Co(T) {dzp c
2
Q
Y
n=o
J,,,@(“)))
9 J
(4.8)
where C,(T) is a constant independent of z and h. This relation can also be obtained for p = 2r+l if a similar transformation is carried out on (4.2’), introducing the notation 1,
w(“+l)+v(“) 2
&+l)+yW
. = &’
)
F
2
YE. G. D’YAKONOV
70
(4.9)
Let fo C(O)(Qr) and v E D;(C).
Then, using the method described in [12], it is easy to obtain the following expressions J,(U@)) < C,(T) = const ., 1, ( *‘:e
1
Q C,(T) = const.
(4.10)
Here it is naturally also assumed that p: satisfies the compatibility conditions, i.e. q~and all its derivatives up to the (m-1)th order become zero on the boundary of fi. @+$+...+bp~ (aI+az+...aP
(4.11)
LEMMA 1. Let the conditions (4.9) and (4.9’) be satisjied. Then the following inequalities hold for ~6”’
;chP &i‘
< C,(T) = con&,
x F,(D!~zJ(~))~< C,(T)
hP
a’
Proof. We multiply
= const.
(4.12)
(4.12’)
s=l
(4.2) at all points Qi,r by ~hp~~~~~)=I
&P
elfn+rt+w(")
and sum up the equations obtained over Q6,z: ~hp~Y....(l;iH4SI:+...+~iiJ.lI:i:jX
(4.13)
Since the operators Li, , Lh,, . , . L\, are positive,
Difference schemes for boundary problems
71
The remaining terms are transformed: thp
WI &,&n)~‘@(“+l) + ?I(“))= ZhP u&l: D,d”)) D;(v(“+l) + d”)) c P ' a=0 0' =
th’ )y F u~.D,(D:v(“))~ = h" )j R a,.[(o:u(k))2-(D:41)2], 0’ a=0 Ll’ a=0
where k = t/z. Similarly L;, L~,L~,(o("+')+o("))
th”
.D,d")
c
Hence from (4.13) it follows that zhp
c
( D,v("))~+ hP
us,( @ zJ(ky + . . .
Q
m
p-2
as,,,..4
sIzp--I
c ='...=p_l=O
p_l’“p_lm~“’
x(Dz... D:;z;d2)+zhp c
Q'
(4.14) f(")Dt(v)(").
Only terms with all differences of the lmth order are left on the left hand side of (4.14), the others being taken over to the right hand side. The coefficients on the left are replaced by their minimum in Qr, the coefficients on the right by the maximur_, of their modulus, and the square of the differences in terms of the squares of the largest differences. Bearing in mind that under the conditions (4.9) and (4.9’) the expression within curly brackets on the right hand side of (4.14) is finite and that
cf.
Dtv(“)< + x[(f)2+(Dtv)2],
Q'
we
Q
obtain
. . . t~P-2slZp_,(D~...
D;_lu(k))2] < C,(T) +
YE.G.
72
D’YAKONOV
we SIB11now show Bow tie terms oft the rig& hand side of (4x) can be estimated. That is, we shall prove the inequality hP
(-D~-““a@))~ < &hP (D~et~~))~2+C,hP (zI(@)Z, cP c c w&e 8 > 0 can be as sm& as desired. 3ecause of conditions (4.3) and {4,4), (4.16) can be rewritten as
(4.16)
if we consider the expansion of 0 in terms of sines in the cube L&: -1 g x, < 2@ = 1,2, *.*p) containing the cube 9, 7 h&+~l agf sm 3
QC”> =
f
5: then, taking account of (4.4) we find that (4.16’)is equivalent to the inequality (4.16”) Since for any smaB E > 0 and any A it is possi@e to Snd a C, ~~de~~de~~ of 2) so large that ASrnN2 < E,?~~-+- C,, therefore (4”16”) is true. Expression (4.16) is therefore proved, Evidently similar inequalities are true for the other terms:
ffence, choosing c: so small that the- coe&%& of the s.sm of &e squares of the differences of the m(2g+-I)th order is Ie;ss than 87’4 and taking into account the finiteness of Jt(~(k))obtained earlier, we obtain for p = 2r
73
Difference schemes for boundary problems and hence, for p = 2r+l th”C(D~~‘.)~“+hPC(~~~~(k))~+~2~~=~~~D~~~D~~(k~~~+*.* P‘ s=1 e ...+rP-l(D~D;... D;T.J(~))S)Q C(T) = const.
(4.17’)
This ends the proof of the lemma. 4. Inequalities stronger than (4.17) and (4.17’) are needed to apply the inclusion difference theorems of [ll] and [12]. We therefore prove the following theorem. THEOREM 4. Ij’ the conditions of’ Lemma 1 are satisfied jbr the problem (2.1)(2.2)-(2.3), the jbllowing inequalities are true jbr the sequence ojjicnctions v$‘) determined on the lattice points by the method ojjiactional steps (2.4), (2.27, (2.3), (4.4): n
zhp c
((D,&y+
hrx
{ Q’
Prooji Inequality
(D:e)(B))2+(v(e))2}
2
‘~D;~...D;Pv”?,]
c
g
<
y =
y =
COnSt.
COnSt.
at+...+ap
(4.18) (4.18’)
(4.18) is a trivial corollary of Lemma 1, since r & (D$@))2
is easily estimated in terms of F (Ev(“))~ as a result of conditions
(2.3’), (4.4).
The proof of (4.18’) is carried out as in the case of (4.16), i.e. it is sufficient to obtain D;~v(~f)~
hPxD;r... M
<
y1 =
const.
aI+u2+...+ap=m.
It is easy to verify that under conditions (4.4) we shall have hP
c
(@I..
.D;P v(k))8 =
(-
(w(~),&D;‘,
1)“hP
, .@PpD;pdk))
c
a
P
(D?v(~))* = hP
h”
i2a” S3
whereat,...I~ are the expansion coe~ci~nts of 99’) in terms of sines in the cube fz,; (--AZ) is the eigenvalue of the operator index lI . . . , . . 1,. But since
from the boundedness
tiSD: for products of sines with the
of hp x p: (D~v’~‘)~ I?’ a=1
YE. G. D’Y.~K~NOV
74
follows the boundedness
of C#
+.J
=/T...QW~.
I
P
and consequently (4.18’). We shall now formulate a theorem which is a corollary of the theorems in [12] (see Chapter 1, sec. 6). THEOREM5. Let the inequalities (4.18) and (4.18’) be tnre for Qgiven sequence of functions VAdetermined on the lattice points Q’. If (VA)’ denotes the multilinear function with respect toxI, x2, . . . , xp , t coinciding with VAon the lattice points, there exists a sequence v48 for which 1) the junctions (DtvAS)’ and (D;l Dt.. . DIVAS)‘, u,+ aZ+ . . . + up = m, converge weakly in QT and 52, respectively to the functions u, and u~,=~..,~~; 2) the functions L&A)
(Di’Dia . . . D~PvJ’,
a,+a,+
. . . +ap = m,
- 1, converge
in
to u~*olg...op;
3) the limiting functions
have the properties:
u = uo.o...o E W)(QT), u E
IlullW!‘)(Q,) II4lwp)(n,)
am)G9~
G
< c
c(Y),
(7)
and
au at
aal+a’s+...+apu
= ‘,
a,a$....=p~ al+ar+
-_=u
axfiax;a . . . a+
5. To prove convergence to the generalized further require the following lemma.
. . . +ap 6 m.
solution of (2.1)-(2.2)-(2.3)
we
LEMMA 2. Let 1) q E Db(c) and for every t (0 < t < T)f E Ok(c) 2) the functions y and f together with all their derivatives up to the (2m-1)th order with respect to x, (s = 1,2, . . . . . . p) become zero on the boundary of G’ and Qr respectively. Then for any n(n < T/Z) we shall have h’x(L:,L;B *t
. . . LfI v(“))~ < C,(T) = const., Si #
Sj
for i # j;
i= 1,2,...p;
si = 1,2, . ..p.
Proof As in 6 2, we consider the expansions of v(“+l), v(“) and f(“) in eigenfunctions of the operators Lt (s = 1, 2, . . . . . . p): @+1)
=
c
at?P+kl...kp;
kl...kp v(n)
-
= c kl...kp
f(“) =
c k,...kp.
The
notations are the same as in $ 2.
at!..kpvk,...kpi
bt!..kpl))k,...kp.
75
Difference schemes for boundary problems
We then have, because of (2.6), P
n(
-T(p)2
l-
a?+.‘,, = ’
P
2
s
S=l
4%1
P
1+
+
1
N S=l The
P
+(A&~
coefficient of jGk,,,.kpin the expansion of Li, LtZ . . . L”,,#+I)
will
be
ug+$
= (- l)‘(Apl))2 . . . (J.J”1)2a(“+,1) kl kp’ Sl
81
From the conditions
c
kl...kP
of the lemma we have
) < c6, (@‘)” . . . (@z)4(a$:‘.,k S P' Sl
c
k,...kP
)” < c,. (lp’)4 . . . (A,#4(b$“f,,k $1 q ’ P
Hence c
k,...kP
(@+:))2 < C,(T) = const. 1
P
and the lemma is proved. We now have all that is necessary to prove the convergence of the fractional step method. 6. THEOREM 6. Let the conditions of Lemma 2 be satisfied. Then jbr z + 0 and h --f 0 the sequences (VA)‘, (D, vJ, (Of’, . . . . . . , D~PvJ tend, in the sense mentioned in Theorem 5, to the generalized solution and the corresponding derived generalized solution. Proof. The inequalities (4.18) and (4.18’) are true when the conditions of Lemma 2 are satisfied. Therefore Theorem 5 is true. We shall prove that the function u, obtained from this theorem as the limit of the subsequence (v,,)’ when t + 0 and h --) 0, is the generalized solution of the problem (2.1)-(2.2)-(2.3). It remains to verify that u satisfies the conditions 3), 4) and 5) from the definition of the generalized solution. The functions (Di’Dx2 . . . . . . D;r,vJ for aI+xq+ . . . ap < m- 1 are elements of the space I@)(!&) which become zero near the boundary. Therefore, from the theorem on the absolute continuity of the insertion operator (see [l 11) the limiting functions a~l+-+apujaxp . . . . . . ax;P will become zero on S, so that condition 3) is satisfied. Condition 4) is satisfied because the functions (v,)’ are elements of the space W.jl)(QT) and are therefore continuous to the same degree as a whole in L,(J2,) [ll]. To verify condition 5) it is sufficient to obtain (4.1) for any function Q, having the.usual derivatives up to the mth order with respect to x1, x2, . . . xp, of the first
order with respect to t, and becoming zero near S, since any function3 of the class of allowed functions can be obtained as the limit of such smooth functions Qi. Let us take a function 5, multiply (4.2) and (4.2’) by zhp(@(n+l)+@(n))/2 sum the equations obtained over Q,,, T. We obtain
and
76
YE.G,
DVAKONOV
As can be easily verified, because af Lemma 2 the expression in square brackets will be bounded in LZ(QT). Bearing in mind that @ satisfies (2.2), eqn. (4.19) can be written as
The term tf 1 obviously tends to zero when z+ 0. Consequently we obtain (4.1) from (4-20) at the limit, whkb isI&& was required,. Since the generalized sol&ion of the problem (~.~~~~.~~~2.3) is ~~~q~~, it is easy to verify that convergence to it occurs Ghen t -+ 0, k -P 0 not only for (v,)l’ but for the entire sequence 94. We have thus proved the convergence of the disc&c: functions obtained by the fraction&l step method to the generalized solution of the problem ~2.~~(~.~)-(2_3)~ without any restrictions of the w zj@= < e3 as in the case of an explicit scheme, Convergence to the generalized sohztion without such restrictions can also be obtained for an implicit scheme 1131,but the amount of computing work with such a scheme will be incomparably greater than for tha method of fractional steps.
Difference
schemes for boundary
problems
77
NOTE 1. If a somewhat more general scheme is considered instead of (2.4), o(n+s/P)_erCn+(s-l)lPl
=
-bS)@p+wIPl
b,ep+s/P)+(l
(S = 1,2. . . . ,p-1).
t o(n+l)_&+(P-lvP1
t
.bp~;v(~+q..(~
_,,)L:,.[“+(P-~)/PI+T~~(“),
all the results of 5 2, 3 and 4 will be applicable to the case 4 < os < 1 the calculations becoming rather difficult only in 0 4, and in Theorem 2 instead of O(t2) there will be O(t), if a, #&. NOTE 2. All the results can be easily transferred
(&)i, = 0,
to the case of the net
(s = 1,2, .. . ,P),
when 0 is a p-dimensional parallelepiped. In conclusion the author expresses his gratitude to A. A. Samarskii for valuable advice on writing this paper. Translated by PRASENJITBA.W REFERENCES. 1. PEACEMAN, D. and RACHFORD, H., The numerical solution of parabolic and elliptic differential equations. J. Sot. Industr. and Appl. Math. 3: No. 1,28-41, 19.55. 2. DOUGLAS, J. and RACHFORD, H., On the numerical solution of heat conduction problems in two and three space variables. Trans. Amer. Math. Sot. 82: No. 2,421-39, 1956. 3. YANENKO, N.N., Dokl. Akad. Nauk SSSR 125: No. 6, 1207-10, 1959. 4. YANENKO, N. N., SUCHKOV V. A. and POGODIN, Yu. Ya., Dokl. Akad. Nauk SSSR 128: No. 5, 903-5, 1959. 5. YANENKO, N. N., Dokl. Akad. Nauk SSSR 134: No. 5, 10346, 1960. 6. RYABENKII, V. S. and FILIPPOV A. F., Ob ustoichivosti raznostnykh uravnenii. (The stability of difference equations.) Gostekhizdat, Moscow-Leningrad, 1956. 7. DYAKONOV, Ye. G., Dokl. Akad. Nauk SSSR 138: No. 2, 271-274, 1961. 8. DYAKONOV, Ye. G., Dokl. Akad. Nauk SSSR 138: No. 3, 522-525, 1961. 9. GEL’FAND, I. M., FROLOV, A. S. and CHENTSOV, N. N., Izv. vyssh. uchebn. zavedenii. Ser. matem. No. 5, 32-45, 1958. 10. DOUGLAS, J. and GALLIE, T., Variable time steps in the solution of the heat flow equation by difference equation. Proc. Amer. Math. Sot. 6: No. 5,787-93, 1955. 11. SOBOLEV, S. L., Nekotorye primeneniya funktsional’nogo analiza v matematicheskoi fizike. (Some applications of functional analysis in mathematical physics.) 12. LADYZHENSKAYA, 0. A., Smeshannaya zadacha dlya giperbolicheskogo uravnenii. (Mixed problem for hyperbolic equations.) Gostekhizdat, Moscow-Leningrad, 1953. 13. LADYZHENSKAYA, O.A., Matem. Sb. 39: No. 4, 491-524, 1956. 14. LADYZHENSKAYA, 0. A., Matem. Sb. 45: No. 2, 123-58, 1958. 15. LEES, M., Energy inequalities for the solution of differential equations. Trans. Amer. Math. Sot. 94: 58-73, 1960. 16. LEES, M., Approximate solutions of parabolic equations. J. Sot. Industr. and Appl. Math. 7: 167-83, 1959. 17. SAMARSKII, A. A., Zh. vych. mat. 1: No. 3,441-460, 1961.