l~umerische Mathematik 2, 245-- 262 (1960)
Approximation to the solution of partial differential-equations by the solutions of ordinary differential-equations* By RUDOLF F. ALBRECHT I. Introduction
This paper is concerned with general methods to approximate the solution of a given boundary-initial-value problem for a linear partial differential equation in two variables x, t b y the solution of a system of ordinary differential equations with respect to t. Such an approach is needed to obtain approximate solutions of partial differential equations b y means of an analog computer for such computers can only solve ordinary differential equations. More precisely, the problems we will consider are of the following type. Let a~ (x, t),
k = I, 2 . . . . . m,
b~(x, t),
l = 0, ~. . . . . n,
and /(x, t) be real and continuous functions of x and t on the rectangle R : O<_x<_a, O<:t<:b, with a,~(x,t) ~ l, b~(x,t) :~:O on/Y, and let
T = ~, ~k (x, t) 0t~ 0h
(I, t)
e~ x = ~, b~(x, t) 0x~"
(I, 2)
and I=0
Then we will consider the linear differential equation
(T + X) u = / ( x , t)
(I, 3)
and assume t h a t it has in R : O < x < a , O < t < b a uniquely determined solution u = u ( x , t) which satisfies for O < x < a the initial conditions
Ok u(x,O+)~-u~)(x) Olk
k = 0 , t, ,
m - - 1,**
(1,4)
...,
and for 0 < t < b the b o u n d a r y conditions
C~u(x,t):c~(t),
v = 1 , 2 . . . . . n.
(1,5)
* This work was supported by NASA Contract No. NAw-6557. 8o ** 0t o u(x, 0 + ) = u(x, 0+). Numer. Math. Bd. 2
18
246
RUDOLF F. ALBRECHT:
The functions ul~)(x) are real and continuous in O < x < a, the operators C~ are linearly independent and defined b y lv
[r 0J /~
for
v = 1 , 2 . . . . . s,
j=o
,,
j=0~
with l, G n - - l , O<_t<_b. If
(I, 6) --LV'~J
x=a--
and the functions G](t) and G(t) are real and continuous Oil
O=Xo
... < x , < . . . < X N _ I < X N = a
is a given equidistant partition of the interval [0, a~, we substitute for (I, 3) finite-difference expressions with respect to x at the points xi such t h a t the solution U~(t) of the resulting system of ordinary linear differential equations with respect to t is an approximation to the functions u(x~, t). I n the following section we will describe general methods for finding such finite-difference expressions. In section I I I we will use t h e m for the approxim a t i o n to the solution of the partial differential equation. For a restricted b u t i m p o r t a n t class of problems we will give an estimate of the error and a convergence test for our approximations. I n the last section we will consider some applications to illustrate the principle of our approach. There are already several papers written on particular cases of the methods described here (see, for example, references [1~, [21, E31, [4J) but, as far as the a u t h o r knows, no general t r e a t m e n t was given and no a t t e m p t has been made to prove the convergence. Finally it should be pointed out t h a t the methods described here m a y be extended to more independent variables and to systems of partial differential equations. A variant of our approach with a complete convergence and existence proof was given b y E. ROTHE for the heat equation with a nonlinear source t e r m [5~ 1. II. R e m a i n d e r Operators Let {~, r, },. . . . } be the set of elements x,/3, y . . . . represented b y real functions ~(t), /3(t), y(t) . . . . of t which are continuous on OGt<_b. This set is a c o m m u t a t i v e ring with respect to ordinary addition and multiplication of the functions. I n particular, the field of real numbers is contained in it. Using this field as a domain of multipliers and introducing the norm
I[ all = t max ~'l ~ (t)[} E[O,b]-' for a n y ~, we obtain a B a n a c h space ~ with the elements ~, r, y . . . . . Further, let {], g, h . . . . } be the set of elements ], g, h . . . . represented b y real functions ](x, t), g(x, t), h(x, t) . . . . of x and t which are continuous on ~'. This set is a c o m m u t a t i v e group with respect to ordinary addition of the functions. Using ~ as a domain of multipliers with respect to ordinary multiplication of 1 Substantially the same method is described (but not proved) in reierence [6~.
Approximation to the solution of partial differential-equations
247
the functions and introducing
I]111-- ma~_{I/(~,~)l} (x,t)ER
II/ll satisfies
for a n y element /, we see t h a t
all the norm axioms except t h a t
I1~/11 < I1~11 Iit11. Using 11/-~11 a s the distance between any two elements J, g we have a metric space ~3 with the elements [, g, h . . . . . The convergence in both ~ and ~3 of the functions is uniform and the spaces and ~3 are complete. B y ~ we denote the subspace of ~3 consisting of all functions which have continuous partial derivatives with respect to x through order ~ on-R. We define ~ o ~ . Finally, let h be a real variable, O
y(P~;I;h, tlC!R,
_p~ t = 1,o ~,(p~;/; g, t), P 3 " f o r a n y t , O~t<=b,
lim ~ (P~; ]; h, t) ~ O.
h--~O
For any z ~ R , for the same h and any two different remainder operators Pi~, Q~, defined on the same subset of ~, we set
(~ ~o1 / = o~(P~ 1),
(~ + Q~') 1 = P~ i + Qf 1.
Then ~ P~ and p o + Q~ are remainder operators also. An important class of remainder operators involving a given differential operator Y and defined on ~e, o=>n, can be constructed as follows. Let
Y and
2 g~(x, t) a~ J l=O
C,=~c~(t)
"
/,
oxJ x=~'
/=o
1; v = t , 2
. . . . . ~o; ~ o < n ; ~q[O,a]
=
be given operators with domain ~3,. We assume g~, c~i~;
l = O , l ..... n" g,(x,t) =~=O on R; v = t , 2 . . . . . ~o; i = O , l . . . . . l~; c~z~(t)=t=0 on [0, b].
For a n y fixed xi~ [0, aJ, a n y x i + a = x i + f f h , ff an integer, xi+v~ [0, a t, and a n y we denote
/(x,+,,,t)
by
0~
t~+,,
~x~/(xi+~,t ) b y gi(xi+~,,t ) b y
/i~(~+~, ~ = 0 , t, .... e,
gl,i+,,
~, ~, (x,+., t) [s/] [~xl ]~='i+~
l=o
~(0) _ /i+~--[~+,,
by
Y/+~[.
18"
248
RUDOLF F. ALBRECHT:
Further, let o~+~, fli+., and Yn be E ~t, and let al, 31, a2, 32 be non-negative integers such that x~_~l, xi+ ~t, x i _ ~ , xi+~2 are C E0, a~. Then, for any /E~30, we consider linear expressions of the form ~t
z~
~, h"o~i+~,Yi+,,/+
~, fli+=/i+=4-
or
0,
Shl,.y~C~/'
if x i ~ . = ~ .
(II, 1)
x=l
Using Taylor polynomials with respect to x at x = x i and remainder terms, expression (II, t) can be written
/~= --at
l=0
v--0
~ --~
[o, or @ -/-~O/X~hl• [x~---i J
v=0
(II, 2)
} ~
....
+Rf/.
We want to find quantities c~i+v, fli+~ and y~ such that the coefficients of ]i, ]~ . . . . . /!01 in (II, 2) are identically zero. In addition we assume that our problem is formulated for the minimum number of quantities ~i+,, fl~+=, V, and t h a t e i = t. In other words, we have to find a solution of the linear system of e + 1 equations which one writes in matrix form .......
h~go,i+, . . . . . . .
I ........
hZ~c~o
. . . .
h~go,~]
I
CXi+t, T
"
*
9
I (II, 3)
3~§ i =
.
9
-t
where in the ( ~ + 1) X ( ~ + t) matrix of (II, 3) elements of the first and the z + 1st line are written, 0=< 3--< 0, and where we use the convention g,,i+~0 and The rank of system
c~0
for
3>n,
for
3>l~.
(II, 3) has to be e+
+
+
From the determinant of the ( 0 + 1 ) • ( e + t) matrix in
(II, 3) we
can factor h ~
0(o+l)
out of the 3 + l s t line, 3 = 0 , 1, ..., ~. Omitting the factor h z we have a determinant which is a polynomial in h. It is different from zero for all sufficiently small values of h, if it has a nonzero constant term. By assumption all the g,,i+a and c~,t~ are # 0 . After factoring them out we have the following
Approximation to the solution of partial differential-equations
249
determinant 0 .........
0 1...1...t
0...0...0 o t Cr
0 . . . . . . . . .
I .....
t...
(II, 4)
0 (7 n - - l~
t
( n - 1,~) !
/* /AQ - n
(q-n)!
2Z0
Gq--l~r
q!
(e-l~)!
which is now independent of the coefficients gl (x, t) and c~i (t), If this determinant (II, 4) is different from zero, then uniquely determined quantities ei+~, fli+~, V~C~R will exist. F o r applying CRAMER'S rule we see t h a t substituting the right h a n d column of (II, 3) for any column of the determinant of the coefficients we can 0(0+1)
again factor out h - - 2 . As a consequence these solutions of system (II, 3) are uniformly bounded with respect to h and are continuous functions of t on [0, b], and t h e y are not all identically zero because g.,i 4 = 0. A necessary condition t h a t the determinant (II, 5) be different from zero is t h a t al+Tl~O+l--n, and t h a t all the l~'s are distinct. Using these solutions ~i+., fli+., and V~ of written Rf/=h o
~
~i+. ~,gli+~ h n-I e i + . l +
p=--o- x
g=l
/=0
'
' '
(II, 3), expression (II, 2) ~, f l i + = e i + ~ + 0 ,
.....
can be
or
(ii, 5)
/'=0
I n the sum inside the brackets the coefficient of any of the e's is uniformly b o u n d e d with respect to h and continuous on [0, b]. The remainder terms e~+a,l, ei+~, and e!~) are C ~R, and lim ei+. ~= lim ei+~ = lim e!~ = 0.
h-+0
'
h-->0
k--+0
J
Hence lira h -~ R~ / -- 0. h--+0
I n addition, (II, 5) is of form P 2 and is defined on ~30. Thus expression (II, 1) with these coefficients ~/+~, fli+~, 7~ defines a remainder operator R~. The m e t h o d just described has the disadvantage t h a t we have to solve a linear algebraic system for each particular operator Y. Under somewhat more restrictive assumptions there is another convenient procedure to find remainder operators of the desired type 9 This method is based on an idea of H. SASSENFELD
250
RUDOLF F. ALBRECHT:
[7] and originally used by him to approximate the solution of ordinary differential equations. We assume that the functions gz (x, t) of the operator Y are C ~3z, and that
Then let ] E ~,, be arbitrary and set
"
~f
~, Y(~z/)
l=0
(ii, 6)
Z=0
The functions ez(x , t), el E ~ , are certain sums of the coefficients gl(x, t) and their partial derivatives with respect to x. They can be found b y integrating
[ [ . . . ? Y[dx n by 0
parts until no derivatives of [ appeal" under integral signs.
0 0 nitimes
If we differentiate the result n-times with respect to x we can obtain the form
(II, 6). For an arbitrary [ E !30, we consider the following expressions with remainder operators (ki/P3 or (k)p3 respectively. (a) if the operators Ci+1 are involved:
(kj)p~ [ =
. . . . ~, hk (ki)~ ~(k) ~'i+~,/i+~+
[0 ~
if
~' (i)fli+Ji+~+ thJCi§
k~i, if
k>i,
(II, 7)
(b) otherwise : Ck)Pi~/= ~, hk ( k ) ~ i +t(k) . . i + ~_ ~
~ fli+~/i+~.
(11,8)
In both cases k = l , 2 ..... n. al and rl depend on k a n d j ' i n case (a) and on k only in case (b); a2 and T2 depend on j" in case (a). All coefficients ei+v, and fli+~ are constants. Using in case (a) the operators (ki/Ai, (~ (ki)C, defined by (ki)A~/= ~ ~=
hk(~)~i+~/~+~, for
k~l,
--G t
rs
(~i)C/={ and in case (b) the operators
0
if k=
--hiCi+~/ if (l*)Ai, (~ defined
k>]', by
Tx
(~)Ai/=
~, h~(~lai+v/i+v
for
k=>l.
/~= --at ra
~=
for an a r b i t r a r y / ~ ~3, equations
--a z
(II, 7) and (II, 8) become
(~ilp~o/ = (~'/A~/(~1 _ (O~lA~/ _ I~/c/,
Approximation to the solution of partial differential-equations
25t
The operators {~ i are assumed to be the same for all k's when j is fixed while the operators (~i)A~ and (kilC (k>= t) depend on k; also the operator (~ i is the same for a n y k. Of course, these are restrictions b u t they enable us to find v e r y easily remainder operators involving the operator Y. We have already mentioned t h a t remainder operators of the same degree, at the same point xr with the same value of h and the same domain of definition, form an ~R-module. Thus, for l = 0, 1, ..., n, we conclude from (~J)P~ / = (-i)A i/(-) _ COJ)A,/ - - (,~i)C / and
(~-l, jlpQ i = I.-l, jlAi/I.-O _ (OilAi / _ (.-l,J) C / by subtraction because of
('q) C = ('~-td)C
n> l+ ]
for
t h a t in case (a) (li)Q~, defined b y J
0
n>l+j,
if
(li)Q~ / = (,q)Ai ](,~) _ (,~-td)Ar ](,~-z) _ [ ('q)C /
if
n =< l + j,
is a remainder operator, and similarly t h a t in case (b) (OQ~], defined b y
I~(?f / = (-IA~//-~ _ I--Z/A~/I--~/, is a remainder operator. Let us assume t h a t the operators (~ilp~o, (h)pio are of the t y p e described in our preceding method. Then b y (II, 5) for a n y polynomial / in x over ~R of degree =< Q ikJ/Ro / = ik/p~ 1 = O. As a consequence, (~i)~ and ( 0 ~ defined b y
(lj)~r
(,q)Ai](l)
(,,-Zd)Ai/_ I
0
if
n>l+],
[("J)C/(-"+l/ if n<=l+],
(II, 9)
and
(II, t0)
Cz)Q~/ = (.)Ai/CO _ (.-l)Ai /
are remainder operators for a n y f C ~3~_.+l. Finally, we show how the remainder operators given b y (II, t) and (II, 9, 10) a p p l y to the partial differential equation (I, 3) with the b o u n d a r y operators (I, 6). For Y =~ X, for u (x, t) C ~3Q, and for $ = 0 or $ = a, respectively, we have
[ 0,
~
~"
or
| ~ hI,W,,c.,
or
,i, hl"y,c,, ~=s+l
with Using the operators of (II, 9, to) we have
-- ~, ('i)Q~(e,u) = ('q)Ai(T u -- ] ) + ~, O'-"i'A,(ezu) + l=o
l=0
~ ('q)C(e,u) `-'+t, . l=n--j
252
RUDOLF F. ALBRECHT :
or
-- ~. (~)Q~'(e, u) =
(")Ai(T u --1) + ~. ("-')A,(e,u).
l=0
l=0
If el ~ ~_o-,~+z, then b y (li)~ (el/)
and
~ (I)0~(e~/)
l=0
l=0
remainder operators for /C ~30 are given. If n = 2n' and if X has the self-adjoint form n"
0l
~l
Z (-- t ) ~ [ g ~ ( x, t)O#
l:O
with functions glE !3l, the formulas for the e~'s and the sum ~
(~i)C(ez u)(-~+~
l=n-i
simplify somewhat. III. Approximation to the solution of the partial differential equation We will consider the boundaly-initial-value problems (I, 3, 4, 5) under the additional assumption that the solution u (x, t)C !3~. This assumption is convenient for the error estimate we shall give. We use the notations
T~I~= ETI]~=x,, X~L =
EXl].,=~,
for any ] E!3~ and any point x i in the interval [0, a]. Then we assume A 1: There is a sequence {N,} of integers N~, r : t , 2 . . . . . 0
or or
h o~i,i+~, i+a/i+a -~-:~=-a~ n=l+s
exist for any /C ~Q, where k=> 0 and l ~ 0 are fixed integers independent of N,. The integers al, z~, a2, T2 may depend on i, the sums al+T1 and a2+~2 are bounded by a fixed integer, and boundary terms hr~Ti,~ C~/ occur only in operators P~ assigned to points x i in a certain neighborhood of the boundary points x----0 or x~-a. Moreover, the coefficients ~i,i+,, fli, i+~, 7i,~ are C 3, and all ~i, i (t) ~ 0 on [0, b]. We set : % i + ~ 0 for if>T1 and # ~ - - a 1 when T1 and a~ are the values corresponding to i, and consider the matrix
A=
OCk,l~
O~k,k+l
...
O~k+l,k
(Xk+l,k+l
......
O~Nr_l, k . . . . . . . . . . . . .
Then we require
O~k,Nr-- 1
O~Nr--l, Nr-- l
Approximation to the solution of partial differential-equations
253
A2:detA=~0 on [0, b]. For the exact solution u (x, t) the following system of equations holds Tl
T~
Z ~i,i+,,T~+."i+. - h - "
Y, / ~ i , i + . u i + . = 0,
~,
or
S" h t,,- n v .
~162
- - h e - n ri -~- ~=lZ'~
~ , ~ c~,
~, h l ' ~ - n v , n=s+l
or c~,
where we have used the relations X~+,~ ui+~, -
-
T/+~ u i + ~ + / i + ~ ,
and the abbreviation Pi ~ u = h ~ ri (t),
ri C ~ .
With the undetermined quantities vi, i - - k . . . . . N , - - l , and with v i = u ~ for i = 0 , t . . . . . k - - I and i = ~ - - l + t ..... N, Tz
~i
/ z = --G I
~ - - --q~
(III, I)
/o~ or
is a system of N ordinary linear differential equations of order m with a solution vi z u (x i, t) on ~0, b~ which satisfies the initial conditions dk dtk u ( ~ , 0 + ) = u ~ ) ( x ~ ) ,
k=0,
a . . . . . . ~ - - 1.
We assume t h a t for i + # < k and for i + # > N ~ - - l the functions u(xi+ ~, t) can be a p p r o x i m a t e d b y the known functions ~7i+. (t) which are m-times continuously differentiable on [0, b]. Then we consider the a p p r o x i m a t e system r~
T1
E o~.i+, T~+~ Cri+, -- h -~ E ~.~+, U~+,~=
v_A,_a o g , , + ~ / , x ~ +, + ~st,=|ZlhZ,~_nyi,~%,
or ~.=s+l ~ ht~-n Yi,~ c~,
with the undetermined quantities U~, i = k . . . . . N , - - l , and with U / = u i for i----0, t . . . . . k - - t and i ~ - - N , - - l + l . . . . . IV,,. In this system ( I I I , 2) all coefficients are known functions ~ ~. For d e t A 4-0 this system has a unique solution U~= U/(t) satisfying the initial conditions
~ - u ~ ( 0 + ) --- u<~)(x~) *
dtk
,
k = 0, ~,
9
m
-
!
9
(III, 3)
We investigate the differences zi = u (xi, t) -- U~(t) of the solutions of systems ( I I I , 1) and ( I I I , 2). T h e y are the solutions of another system of ordinary differential equations
~" og,i+,,Ti+t, zi+t,-- h - " ~
fli,~+,~zi+:,~--
-
-
h~
(TIT, 4)
254
RIIOOLF F. ALBRECHT:
z~=u(xi, t)--Yti(t) for i
with conditions for
(III, 4)
and for
i>N,--I
and these z~'s 6 ~. The initial
are dk
dtk Zi(O+)=O, k----O,1,...,m-- l. By analogy to (III, 1) being assigned to the given problem, to the homogeneous problem
(III, 4)
is assigned
(T+X) z=O, Ok etk z(x, 0 + ) = 0, C,z(x,t) = 0 . For zero initial values the system (III, 4) has a uniquely determined solution which is still dependent on the unknown function If we can estimate ]zi[ we have a measure of the accuracy of our approximation. For the following particular class we give such an estimate and sufficient conditions for the convergence of our method.
u(x, t).
We define the matrices
.
L --1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
.
~Nr--l,Nr--.J
.
~=~aOr lt+I* Tk+t* (r a=
.
-- Uk+t*)
],
0 rl
~l~.-l.N.-l+,
rN.-~+, (uNr-~§ -- ~N.-t+,)
-- --1
i
b=h -~ 0
J,
Tt
r--1
~
--l
where the number of nonzero components of a is n~ and of b is rib, n~ and n b independent of N,,. for :r and for ~ < - - a ~ . Then we assume A 3 : The operator T is independent of x. For any of the partitions of A 1 the matrix A in the corresponding system (III, 4) is independent of t. The
fl;i+,~=--O
matrix B is given by Z (t)B where Z (t)C ~, and where B is independent of t.
Approximation to the solution of partial differential-equations
255
A and B are symmetric and commutative and all eigenvalues of A 2 - - B distinct.
are
I n this case (III, 4) yields A(Tz)
--zBz
(III, 5)
~ -- a -- b - - h e - " r .
To solve the corresponding homogeneous system A(TZ)
= 0
--zBZ
(III, 6)
we set Z=g(t)
h=
h,
hk+l
khN,-,J where g(t) is a function of t, and h is independent of t. Separating the variables we have a system of algebraic equations for h (A2--
B) h = 0.
As a consequence of A 3 det (A ,~ -- B) = 0 has real and distinct roots ~ , v = 1, 2 , . . . , N.
Then for each v the equations
T g -- 2~ X g = 0
(III, 7)
and the adjoint equations T*s--
2~zs=O
have each m linearly independent solutions g~i(t) and s ~ - l ] ( t ) ; i = l , 2 . . . . . m; gvi, .~;[m--1] ~i C~. Moreover, the systems (A 2, -- B) h = 0 have N solutions
k hN,--l,v J
which we can assume to be orthonormal.
Therefore b y SCHWARZ'S inequality
N~--l
Y r=k
Ih,,~l
gV N.
(III,
8)
Then let the solutions g~i and s~7 -1] be such t h a t -g~l
%=
I
#
/
g.~l
gv2
k6vl
#
'''
(7i.... 11
gym
~i~-11
by2
"
"
"
ovt;r
is a fundamental matrix for the first order system equivalent to -Svl r
~
Sv 2
9 9 9
S~I]
.
-.
.
.
.
.
Sv m
_[1]
~:~
(III, 7),
and t h a t
256
RUDOLF F. ALBRECHT:
is a f u n d a m e n t a l m a t r i x for the adjoint system of this first order system with the p r o p e r t y -m--1 i=o ~/)v* r
=
m--1 ~vlSvl
" " "
.
m--1
/'
]=0
_j=0~
|
.
m=l
~s ,,(J) where s,i[~s _ ~,, o~,~(~ and % [8]. For equations (III,
Ovlbvm
"
1 ".
"
=
P s[/]~,(]) " " "
"'I
j/'~=o-Vm~
.
.
.
0
It is always possible to find f u n d a m e n t a l matrices %
6)
the vectors
h~g~i,
v=t,2,...,N,
i=1,2
..... ,m
are N m linearly independent solutions: for if there exist constants k,i (not all zero) such t h a t v, i
then v
with not all ~ = ~
k~i g,i=~O because not all k~ are zero and the g~i's for a fixed i
v are linearly independent. independent. Therefore
This is a contradiction; for the h,'s are linearly
Lh,
-1)
is a f u n d a m e n t a l m a t r i x for the first order system equivalent to ( I I I , 6) and similarly [hT,
Svi
{h
s [17
47 is a f u n d a m e n t a l m a t r i x for the corresponding adjoint system. We find m-1 - - [ X~ s [']
t0 1
h*h "(~)l
". 9
.
9
0
Thus the solution z of ( I I I , 5) with zero initial values can be written t
z = -- [h, g,~(t)] f [ s ; 7 - ~ ( z ) h* (a(~) + b('c) 4:- hO-'*r(z))] dT 0 01" t
m = -- ~ h . g . , ( t ) fs~m-~l(~) (h*~ a(~) + h* b(~) + hO-"h * r(z)) d~. V~$
0
Approximation to the solution of partial differential-equations
257
We define [ [ z l [ = ' = k, .m. . , a x - - ,{[[zill},
Ilr[] =
max
i=k,...,Nr--1
{][r,[l },
similarly for II all, II bit, and Ilh~ll =
max {[hi ~[}. i=k, ..., Nr--l v=l, 2,...,N
[]h~l] =<1 and (III, 8) ll~ll _--<][h~[[ (~o[laIl + ~11 bll + h~-" VNlir][) Z/{vi(t)s~--l]('~)
Then, because of
dT.
(II1, 9)
The functions gvi(t) and s~'~-~l(t) depend on 2., are continuous on [0, b1, and are therefore bounded for each particular 2~. B y P 3 lim [r i(t)] = 0, h-+0 further
VNh < V~. We assume A 4: For the partitions considered in A t limllh~ll[laI[=o, h-+0
,}~llh~IllI~lt=o,
limho-"-~llh~llllrll =o, h--~0
and 2 /
{.~(t).,. ~[.m--1] (~)d~ G I ,
where I is a uniform bound. Then on account of (III, 9)
II~ll < IIh, II (~11~II +%llbII +ho-=-~V~ll~tl) I
(III, 10)
and our procedure is uniformly convergent for the partitions of A t.
IV. Applications We now consider some simple but important examples to illustrate our methods and the principle of the convergence test. t. Given the equation ~u O~u + / (x, t), ~t = c z -b~-
0=
0=
c~ > 0 ,
c~constant,
/ff93,
with the initial condition u(x, o) = . * ( x ) ,
o < x < 1,
the boundary conditions
u (0, t) = u0 (t),
u(t,t)=5l(t ),
OGtGb,
which we assume has a unique solution u(x, t) C934. For partitions xi=ih, h = t / N ~ , N , > t , i = 0 , t . . . . . N,, a set of remainder operators p a like those re-
258
RUDOLF F. ALBRECHT:
quired in A t is given by P? / :
h2 t " -- (1i-1 - - 2/i + 1i+1), "~ ,~
i=l,2,...,N,
with
IP?/I :< ~ II/'ll. In this case we have k = l = 1, N = N,--1, a l = z l = 0, a ~ = T2= t, constant coefficients ~i,i+~, fli, i+~ and ~i,i~-t. D e t A = t and thus A 2 is satisfied. System (III, 2) is given b y dUi dt
c2 h2 ( U / _ 1 -
2 U~ + ~ + a ) :
]i
with U0=120, U N + I = U l , and initial conditions Ui(0)=u*(xi). becomes dz i -dl
c2
System (III, 4)
2Zi + Z'+I) = h c ~ r i '
h 2 (Zi-1--
with Zo----z~+l=0. The matrix B = B is symmetric, A and B are commutative, the operator T is independent of x, the eigenvalues of ( A 2 - - B ) are given b y 2v =
~Zv = ~ h v ,
2C2 (1 - - COS//v)
v = t, 2,
N,
and are all distinct. Thus A 3 is satisfied. We find gvl = gv = exp E-- 2~ t~, S v l = S v = exp ~2~t3,
h,,~ = |/2h ~in (~ i), II h=ll _-< Vzh, and
Further, ~
2c~(1-cos~) N
Thus A 4 is satisfied, from (III, t0) it follows that
II~ll < h~l/2 IluC')ll =
6(12--~)
'
and our approximate solutions converge uniformly as h 2 toward the exact solution. 2. We next consider the same problem as in example I but with boundary conditions u (0, t) = Uo (t), 0 u(t,t)=~(t), Ox
O<_t
Approximation to the solution of partial differential-equations
259
We use the remainder operators P~a and the relation , --
IN 2 1 - / N + I =
--
ha
h/N+I
--
-2-
,, /N+I
valid for a n y / E ~32, where ]N+I denotes the second derivative of ] with respect to x at a point x , X x < X < X N + ~. Thus we write tt
P2 u = h ~ ux
-t
h2
to m a t c h the b o u n d a r y condition at x = t. System
(III, 2)
c2 (U/_ 1 -- 2 U / + U/+I)
dUi dt
It
u,v) + h u l + -?- u x + l
-- (UN-~ --
is given b y
/i
=
h2
with Uo ~ - ~o , - - UN - ~ UN + I = h u-i,
and with initial conditions Ui(0)=u*(xi). dz i dt
c2 h2
System (III, 4) becomes
( z i - 1 - - 2 z, + z i + 1) = h c ~ r i ,
with Zo = O, - - zN + ZN+I = U ~ + I . The constant matrices A and B are again symmetric and commutative, and the eigenvalues of ( A 2 - - B ) are 2c 2
2~ = - ) y (1 - - c o s ~ ) ,
2v--I
/~ = ~-~T,
v = I, 2 . . . . . N ,
and t h e y are all distinct. Thus A 3 is satisfied. We find g.1 = gv = exp [-- 2~ t ] , s. 1 -- s. = exp [2. t ] , sin (/~ i) )"h sin ~u.(N+ 1)1 cos-a -~• with
=0
l i m []h.l]
h--+0
for all v values. F u r t h e r t
dz
N
h~
t
and
I[-]l=o,
c2
Ilbl[-<- 2-t1r
~b=4
Thus A 4 is satisfied. F r o m (III, 10) it follows t h a t
[/zlI < [IhvH (llu"ll +
~-hai1u<4)ll) 12-~1
and our procedure is uniformly convergent.
260
R U D O L F F. ALBRECHT:
3. We again consider the same problem as in example t but now we use higher order approximations. Let e = 5, u(x, t ) C ~ 6 , and let the remainder operators ~5 be given b y
p s l = . ~L2/'~H ~I,-1+ t 0 t.~" + 1 i ,t+ 1 ) - - 1 2 ( 1 i _ 1 - - 2 / ~ + / ~ + 1 ) ,
i=t
2..... N
with h~
(~)
I~/1-< 20 II/ II We have k = l = 1, 0"1='t'1=0'2='~2 = 1, and constant coefficients 0%i+~ , fli,i+,. Using the recursion formula det A N = 10 detAN_ 1 -- det A2v_ 2, it can be shown t h a t detA-->10. becomes --dtdU/-1 47 10 dUidt 47 dUi+ldt
Thus A 2 is satisfied.
System (III, 2) then
12C2h ~ (V/-1 -- 2 V~" 47 U/+a) = ]i-147 10/' +/*+1
with
Uo=go,
duo dt
a--
UN+I=gI,
dUN
dt u~
dt
and with initial conditions U~(0)=u*(x,). dzi_ 1 dz i dzi+~ dt 47 ! 0 ~ 47 dt
Z0:ZN+I--
12 h~
System
(III, 4)
e--
dt ul'
becomes
(Zi-1 @ 10Zi 47 zi+l) =
d z o __ d z N + t dt dt
h3 c2
~'i'
_0.
B o t h matrices A and B are symmetric and commutative, and the eigenvalues of (A 2 -- B) are t2c 2 l - - c o s h ~v 2 , - - h~ 5 + c o s h ~ v ' v=I,2,...,N, and t h e y all are distinct. Thus A 3 is satisfied. We find g. = exp [-- 2~ t~,
s. = exp [2. t],
hi, , = V z h s i n ( ~ h v i ) , and
~,
,=1
t
dr
:
h
/5+cosh
v=ll2C~(l--c~
<
Thus A 4 is satisfied and
I1~11 ---< 1o(12_n~) V 2 h ~ [lu<~>ll 4. Given the equation
~2u _ ~t ~
O~u Ox 2 '
O<=x<_a, --
O<=t
Approximation to the solution of partial differential-equations
26t
with t h e initial conditions
0 < x < ~,
o u (x, 0 + ) = u ~ ( ~ ) , -a~
u(x, o) = ~*(x), the b o u n d a r y conditions
. (o, t) = , 7 o (t), u(a,t)=g~(t),
O<_t<_b,
which we assume has a unique solution u (x, t) C f34. W e use the s a m e r e m a i n d e r o p e r a t o r s as in e x a m p l e 1. S y s t e m ( I I I , 2) is given b y
d2Ui dis
t (Ui_, - - 2 U/@ U/+I) = O, he
i=t,2,
...,
N,
s y s t e m ( I I I , 4) is given b y
cl~z~ _ 1 (zi_: - - 2z~ + zr dr= ha Z 0 = ZN+ 1 =
= ~ r~,
O,
a n d t h e eigenvalues of (el ~ - - B) are again 2 ;tV=-h2 (l -- cos ~z h v) ,
v=l,2
..... N.
W e find
g., - ,t;-~ ~os i g t,
g,2 = t;-I sin V2~ t,
S[1] v l --" - - a~ -~ Sill V a ; t ,
s ~ = a;~ cos l/a; t, h2
[I ~[I--< -,2 II"(')lI, and 2
f g,i(t) si~ ( r ) d T =< Zx ~ 1 It __ COS V ~ t] --< 2 ZN ).~-X< t 2 - -4- = ~ 9
i u 0
v=l
v=l
Thus
h~ V~
II*l[ < 5 Z i 2 - ~ : f 11ua>ll 9 Acknowledgement. The author wishes to thank Professor D. T. GREENWOOD and Professor 1R. M. H o w r for a number of helpful discussions in connection with the above development. References
[1] HowE, C, E., and R, M. HowE: Application of Difference Techniques to the Lateral Vibration of Beams Using the Electronic Differential Analyzer. Univ. of Mich. Eng. Res. Inst. Report 2115-1-T, Ann Arbor, Feb., t954. [2] HOWE, R, M. : Application of Difference Techniques to H e a t Flow Problems Using the Electronic Differential Analyzer. Univ. of Mich. Eng. Res. Inst. Report 2115-3-T, Ann Arbor, May, 1954. [3] FISHER, M. E.: Higher Order Differences in the Analogue Solution of Partial Differential Equation. Int. Analogy Computation Meeting, Sept., 1955. Numer. Math. Bd. 2 |9
262
ALBRECHT:Approximation to the solution of partial differential-equations
[4] FlSt~ER, M. E.: Higher Order Differences in the Analogue Solution of Partial Differential Equations. J. Assoc. Computing Machinery 3, No. 4, 325--347 (Oct., 1956). [5] ROTHE, E. : Zweidimensionale Parabolische Randwertaufgabe als Grenzfall eindimensionaler Randwertaufgaben. Math. Annalen 102, 650--670 (1930). [6] HARTREE, D. R., and J. R. WOMERSLEY: A Method for the Numerical or Mechanical Solution of Certain Types of Partial Differential Equations. Proc. Roy. Soc. London, Ser. A 161, 353--366 (1937). [7] SASSENFELD, H." Ein Summenverfahren flit Rand- und Eigenwertaufgaben linearer Differentialgleichungen. Z. angew. Math. u. Mech. 31, 240--241 (1951). [8] CODDINGTON,E. A., and N. LEVlNSON: Ordinary Differential Equations, p. 71. New York 1955. Mathematisches i n s t i t u t der Technischen Hochschule Mfinchen ArcisstraBe 21
(Received December 1, 1959)