EQUIVALENCE THEOREMS FOR
NONLINEAR FINITE-DIFFERENCE METHODS
by
M.N. Spijker
I.
Assume
U
equation and assume
INTRODUCTION
is a solution of a given nonlinear (integro-)differential
u
to be a corresponding solution of a finite-difference equa-
tion approximating the original infinitesimal problem. Let the finite-difference
equation and its solution
u
depend on the so-called mesh-width
h > 0 . In this
paper we deal with the following general problem: What conditions on the finitedifference equation are necessary and at the same time sufficient for the convergence of
u
to the true solution
U
as
h + 0 ?
For linear and semilinear differential equations there exists a series of
equivalence theorems in which such necessary and sufficient conditions for convergence are given (see [2], [6], [7], [12],
[13], [16]). For differential equations
which are not semilinear there seem to exist no such equivalence theorems with any
general applicability. Still for arbitrary differential equations which need not to
be semilinear Stetter [15] and Ansorge
[i] obtained a general condition that is suf-
ficient for convergence - see also Kinnebrock
[ii]. Their sufficient condition es-
sentially is that the linearization of the finite-difference scheme at the true solution
U
be stable. Their results only apply in full if the order of accuracy of
the finite-difference scheme is high enough. Furthermore, if the linearization of
the finite-difference scheme at
Wasow (cf.
U
is only stable in the sense of Forsythe and
[12, p. 95]) their results do not apply either.
In this paper an attempt is made to get rid of the limitations of the
theories mentioned above. We shall present equivalence theorems giving conditions
which are both necessary and sufficient for convergence. These theorems can be applied to differential equations that are not semilinear and furthermore they can be
234
used in a number of cases where the theories of [i], [ii],
[15] mentioned above do
not apply.
In this article we shall proceed in an abstract setting in order to make
it possible to apply our equivalence theorems to different kinds of (integro-)differential equations (e.g. of hyperbolic, parabolic or elliptic type). The framework
in which we proceed is similar to the one used by Stetter
[15].
Part of the proof of the equivalence theorems will essentially be based
on an idea used by Strang in [17] for proving convergence under conditions that are
not subject to the limitations mentioned above in the case of nonlinear hyperbolic
differential equations.
As a consequence of this method of proof we are naturally
led to a condition under which not only the global discretization error
tends to zero as
h + 0
but also
u - U
u - U
admits an expansion in powers of
h .
In this way we obtain the same result as derived in [14] but under weaker conditions
than those stated there.
In chapter 2 we have collected a few results from functional analysis
which are needed in the following chapters.
In chapter 3 we derive the equivalence
theorems as well as a number of closely related theorems.
the application of the abstract considerations
In chapter 4 we discuss
of chapter 3 to a nonlinear parabo-
lic differential equation and we display a numerical example in which repeated
(Richardson) extrapolation to
2.
h = 0
is performed.
SOME RESULTS FROM FUNCTIONAL ANALYSIS
In this chapter we review some simple results from f u n c t i o n a l a n a l y s i s .
The proofs of the following lemma's have been included for completeness and because
the Lipschitz condition being imposed here on the (n-th order) derivative of the
operator
~
is slightly weaker than required in most textbooks on functional ana-
lysis.
In the following lemma a version of Taylor's formula in normed vector
spaces is presented.
The proof will be given along the lines of Collatz
[4, p. 223].
235
LEMMA
open subset
i.
of
Let
R
and
R . Let
%
he a m a p p i n g
x° , xI
and the segment
the n-th
order
Fr6chet
S
be real n o r m e d
x ° + t . ( x l - x o)
derivative
of
~
from
E
exists
the Lipschitz
(2.1)
II ~(n)(x) ~(n)(xo) 11 !~'llxxoll
x
I lell -< - ~ ~
domain
S
S . Assume
E
be an
that the p o i n t ~
belong
to
E . Assume
in all points
of
E
that
and that
it
with
0 < t < 1 . Then
+ ~. ~(n)(xo)(Xl-xo)n
+ e
where
IlXl-Xoll n÷l
There
such that
exists
fILl]
f(J)
(j = 1,2,...,n;
(2.3)
exist
= L(e)
fined by
0 < t < i) . In view
of (2.1)
vanishes
at
line
- f(n)(0)l
of (2.2)
- f(0)
= f(t)
t = i . Since
t I 6 (0,i)
. Since
Continuing
in this way
Hence
g'(0)
in (2.2)
The following
function
: 0
theorem
the strong
follows
lemma
(see [10, p 708])
easily
consists
space
of this
with
We define
on
[0,i]
and the
we have
(0 < t < i)
e
.
we have
. Hence
the function
g
de-
tn.f(n)
tn+l,
- ... - n[
(0) llell
= 0
easily
- (n+l)[
L
= m ~(J)[xo+t(xl-Xo) ] (Xl-Xo)j
the vector
it follows
it follows
of function
relationship
g(0)
operator
[i0~ p. 142]).
i r(n).^,
- ... - ~-[..r
ku)
- t-f'(0)
also
f(n)(t n) - f(n)(0)
the inequality
defining
continuous
is continuous
! P't'llXl-Xoll n+l
- f'(0)
- f(0)
(see
0 < t < 1 . f
f(J)(t)
= f(1)
g(t)
for
linear
= lle]l
and satisfy
If(n)(t)
From the first
a real-valued
= i~ L(e)
= L ~ Ix O + t-(xl-Xo) ]
derivatives
Ilel[
and let
n : 1,2,3,...).
Proof.
f(t)
x = x ° + t . ( x l - x o)
*(x 1) = ~(x o) + ~ ' ( × o ) ( X l - X o) + . . .
(for
spaces
condition
of the form
(2.2)
into
(0 < t < i)
satisfies
for all
vector
we have
that
that
tn. Ilell
that
g'(t I) = 0
g"(t 2) : 0
g(n)(t n) = 0
= 0
for some
(see
formulation
[i0, p. 687],
lemma to the w e l l - k n o w n
t n 6 (0,i)
(2.3)
and the lemma has thus been
theory
t 2 ( (0, t I) .
for some
and by using
in a quantitative
for some
[15]).
with
t = tn
proved.
of the inverse
We also note
Newton-Kantorovich
.
theorem
236
LEMMA 2,
complete (i.e.
Let
R
E C R
into
%'(x)
exists on
and
S ' Let
xo e E
E
~'(Xo)-i
all
in this ball.
y e S
and
(e.6)
-
]Ix(y)
Proof.
tot
G
by
mapping
belongs to
with
llX-Xol I < i/~u
be a mapping from the open subset
S
onto
E
and
lly-yo]I <
exists and is bounded,
2
1/26 U
there is a unique
x(y)
for
x = x(y) 6 R
with
-~-282uliy-Yol[) ! 28"IlY-YolI.
Choose any
y E S
Ily-yol I < 1/9~
with
G
(2.7)
(for
Using (2.5) and lemma 1 with
u
llX-Xol I < i/6u)
n = i, x I = x
and define the operaG
are solutions of
exists,
and by (2.4), (2.5)
llG'(x)ll ! 6U'llX-Xol I
2
Then fixed points of
Cx : y . The Fr6chet derivative of
l]GX-Xoll
and
o
satisfies
G'(x) : ~'(Xo )-I [¢'(Xo) - ¢'(x)]
we have
x
I[}'(x) - }'(Xo)[I ~ z. IIX-Xol I
@x = y . Furthermore this
xoll
R
such that the open ball with center
Gx : x - ~'(x ° )-i [O(x)-y]
the equation
be
%x ° : Yo " Assume that the Fr6chet derivative
u > 0
i/p6
x
}
R
and that it satisfies the following two conditions
radius
Then for all
be real normed vector spaces and let
and let
There is a constant
(2.5)
S
is a Banach space). Let
The inverse
(2.4)
R
.
it follows that for
llX-Xo] I < i/6u
= I]x-x o - ~'(Xo )-I [~(x o) + ~'(x o) (x-x o) + e - y]
= ll
'(Xo)-i
y] II
Eye + e -
II@'(Xo)-III
]I
• {IIy-yo[ I + [leII}
2
6~
llyyoll +T llXXoll
Furthermore it is easily verified that a number
611y-yoll
r
+ y~p. r 2
1
o = ~.(i
r
if we choose any
ball with center
By (2.7)
whenever
-Vi-2~2~[ly-yoll),
Consequently,
closed
±
G
x°
r° < r < rI
1
rI = ~.(i
r
r
satisfies the inequality
where
+gl-282~lly-yoll)
with
r
and radius
r
is a contracting mapping from
o
< r < I/SU
--
then
D
G
into
.
and define
D
is a mapping from
D .
to be the
D
into
D .
237
Hence there is a unique
r ~ 1/6~
it
and radius
follows
i/~
x
that
with
this
x & D, Gx = x (see e.g.
x
is
while by taking
also
r = r
0
llx-XO[ [ ~ r O . Since the fixed points of
~x = y
unique
in the
[i0, p. 627])
open ball
it follows that this
G
. By letting
with
x
center
x
o
satisfies
are just the solutions of the equation
the lemma has thus been proved.
The following lemma, which is a partial converse of the preceding lemma
shows that a condition of type (2.4) is indispensable for the result of lemma 2 to
be valid•
LEMMA 3.
Let
R
and
mapping from the open subset
that for all
x~
E
R
-
provided
be real normed vector spaces and let
E C R
into
S • Let
the Fr~chet derivative
to-one mapping from
(2.8)
S
onto
,(Xo)ll
X-XoI ]
S
!
~'(x)
xo 6 E
and
exists, that
~
be a
~x ° = Yo " Assume
~'(x )
is a one-
0
and that
'11×-×oll
is small enough. Furthermore assume that there exist constants
such that
i.
for all
y e S
]IX-Xol I < ~
2.
This
x(y)
Then the inverse
and
[]y-yo[ ] < T
there is a unique
x = x(y) E R
llx(y) - Xol I ! 6.11y-yol I .
is bounded and
[l~'(Xo)-iIl _< 6 .
Choose an arbitrary vector
z e S
with
[[z[l = i
and define
by
~'(x o)
v
We shall show that
=
z
•
IIvlI ~ 6 , from which the lemma immediately follows.
Choose a constant
r > 0
such that the following conditions
(a)
-
are fulfilled:
(a)
the open ball with center
(b)
II
(o)
with
~x = y , and
satisfies
¢,(x o)-I
PrOOf.
veR
with
r
'(x)<
~
,
'(Xo)ll!
x
.llx-xoll
O
and radius
for
r
is contained in
IIX-Xoll < r ,
E ,
(d)
238
(d)
ll®(x)-Yoll
Let
t
<~
if
IIx-~oll
be a real parameter
¢'(x o) (x-x o) = t.z . Hence
<r-
and define
x
by the relation
tv : x-x ° . Now we take
ensure that
liX-Xol i < r . We define
y
and
we have by the assumption
of the lemma that
Cx = y
Hence by (b) and lemma i it follows
IIx-xoll
i
~'{t
: t6"{1
llX-Xoll
By letting
t
: t'llvll
$
0
* lie
+ ~'tlx-x °
*~.t. ll~
}
we have
that
•
vll ~ 6"{1 + t ~" I l v l l
I vll m ~
conditions
FINITE-DIFFERENCE
is convergent
theorems
theorems
scheme and in section
In section
the last of which implies
EQUATIONS
the notations
and
and sufficient
In
for
3.3 we shall state necessary
3.4 we shall finally prove
that a finite-difference
if and only if it is both stable and consistent
3.1
}
will be formulated.
which are necessary
for consistency.
2
and the lemma has thus been proved.
3.1 of this chapter we shall introduce
of a finite-difference
two equivalence
(see theorem
method
5).
BASIC DEFINITIONS
The finite-difference equation
Assume
A
and
ping with an open domain
satisfy the equation
(3.1)
}
2
by means of which the equivalence
and sufficient
3.1.1.
}
3.2 we shall derive conditions
stability
llx-xoll : ~'11y-yoll.
that
A THEORY FOR NONLINEAR
In section
section
[[y-%If <
(by (d))
2
it follows
3.
definitions
Cx : y . Since
~ 6.{ll®'<x o) <~-×o ) + ell}
: ~'{lltzlt
Since
by
so as to
0 < t < r/IIv]l
F(U)
:
0
.
B
are real normed vector spaces.
D C A
and with range in
Let
F
be a given map-
B . Let the vector
U 6 D
239
In the applications
(3.1) will stand for a given differential or integro-
differential equation with solution
U . Initial or boundary conditions
ting the differential equation are assumed incorporated
supplemen-
in the mapping
F . For an
example we refer to chapter 4.
Let
let
Ah
H
and
Bh
Ah
A
into
and
Lh
(0, ho]
with
inf H : 0
and
denote real normed vector spaces depending on the so-called mesh-
width parameter
ping
be a subset of the real interval
h e H . There are given families
Ah
and
B
into
B h , respectively.
are linear and bounded uniformly for
to be complete for all
of operators
Ah
and
Lh
map-
It is assumed that the operators
h • H . The space
Ah
is assumed
h e H .
We consider the approximation
of
U
by solving instead of (3.1) the
equation
(3.2)
where
Ch(U) = 0
Ch
is a mapping from an open set
is Fr6chet differentiable
Ah
maps the set
D
into
on
Dh
Dh C A h
for each
h e H . Furthermore
Ch
it is assumed that
(3.2) will stand for a finite-difference
with discrete initial or boundary conditions.
h
B h . We assume that
Dh .
In the applications
depends on
into
The solution
and will denote an approximation
to
u = uh
equation
of (3.2)
A h U . For an example see
chapter 4.
3.1.2.
A ~
D
Ah ~
Dh
F
>
B
>
Bh
Convergence
For the difference
uh - £h U
where
u : uh
and
U
satisfy
(3.2) and
(3.1), respectively the term global or accumulated discretizatlon error is used.
The finite-difference
method
(3.2)
(or briefly:
the operator
vergent if for
h
llm
h÷0
: 0 . In actual numerical applications
fluh- £hUll
sufficiently
small a solution
u = uh
Ch ) is called con-
to (3.2) exists with
it is not only desirable
that
240
~h
is convergent according to this deflnl~ion but it is also highly desirable that
it satisfies the following two requirements:
i.
Perturbations
w
in the finite-difference
convergence, at least if
llwil
scheme should not destroy the
O(h ~) - thi~ is
is of some order
desirable since perturbations like round-off errors cannot be avoided in
actual computations,
2.
The global discretization error should admit an expansion in powers of
h - this is desirable since such an expansion makes Richardson extrapolation possible.
In order to formulate
a
concept of convergence in which these two
requirements are included we give the following definition.
DEFINITION I.
for each
at
zh
h e H . Then
Let
6
and
q
under perturbations of order
llWhl I = O(h ~) (for
for all
h 6 H
w h & Bh
h+0)
with
> 0
and let
zh ~
is said to be convergent with an accuracy of order
~h
each family of vectors
be real numbers
~
if the following statement is true: for
depending on
h
, there exists a number
h ~ hI
q
there is a
vh ~
Dh
in such a way that
hI > 0
with the property that
with
and
(3.3)
¢h Vh = Wh '
(3.4)
llVh-ZhlI = O(h q)
(for
h e H, h ~ h I) .
We illustrate this definition by choosing for
q
an integer
~ i
and
by defining
(3.5)
where the
z h = A h {U +
U.
I
qi 1
i=l
h i Ui}
denote fixed elements of
accuracy of order
q
at this
zh
A . Suppose
under perturbations
follows from (3.4) that a solution
vh
tion in presence of a perturbation
w = wh
lim
h+0
IIVh- AhUII
#h
is convergent with an
of order
6 . It then
(see (3.3)) of the flnite-difference
= 0 . Hence the requirement
with
llWhl I = O ( h ~)
equa-
satisfies
i stated above is fulfilled.
Further,
241
by choosing
wh = 0
we have
uh - Ah U :
v h : uh
and consequently
q-I
[ h i A h U i + O(h q)
i=l
which is the content of requirement 2.
3.1.3.
Consistency
Throughout this chapter 3
Fo,FI, .... ,Fr_ I
tor
Ch
r
denotes a fixed integer
denote given operators from
applied to elements
Ah X
with
D
into
X ~ D
> I
and
B . We assume that the opera-
can be expanded in powers of
h
in the following way:
(3.6)
~h Ah X =
r-i
~ h j L h Fj(X) + O(h r)
j=0
DEFINITION 2.
Let
p
(for
h e H)
be an integer with
i <_ p <_ r . Then the operator
is said to be consistent of order p if
Ch
Fo(U) = FI(U) . . . . . .
For the element
Fp_l(U) = 0 .
Ch(AhU)
the term local discretization error is used.
From (3.6) and definition 2 it follows that the local discretization error is
O(h p)
if
~h
sistency
is consistent of order
p , the smaller is the difference between
(see (3.2)). Further it is clear f ~ m
> i
&f
p . Hence the higher the order of con-
~
--
(3.1) that
Cb
~h(U)
and
%h(hhU)
as
h-~O
is consistent of an order
= F .
0
3. I. 4.
Stability
In most current definitions of the concept of stability of finite-difference operators
9
~h
it is required that the difference between solutions
of the finite-difference
bations
(3.7)
where
w
and
and
and
equation obtained in presence of two different pertur-
W , respectively can be estimated by an inequality of the form
IFv~]l _<~
y > 0
v
h -~.
a ~ 0
ilw~ll
are independent of
stability in the sense of Lax and Richtmyer
h . The case
a = 0
[12, p. 45] , while
to stability in the sense of Forsythe and Wasow
~2,
corresponds to
a > 0
corresponds
p. 9 4 . For nonlinear problems
242
stability
w
and
is a local property
of
k e B h • The following
Ch
(3.7) does not hold for all
and in general
definition
is similar to the one used by Stetter
[15].
DEFINITION
(for
h • H) . Then
3.
Let
~h
is said to be
there exist constants
I.
if
v = v(w)
Dh
P
is
m-restricted
with
p > m,
By choosing
~ 0
and let
~
zh e
Dh
at
zh
if
such that:
llw- ChZhll
and
< ~h m
this definition
stable of order
a
then there is a unique
llV-Zhl I < yBh m-a,
IIV-Zhl I ~ yh-~.llCh
v - @h Zhll
by choosing
at this
and
zh = A h U
zh
(see (3.1)).
and is consistent
Assume
of an order
h
p > e .
w = 0
in definition
sufficiently
small.
Similarly
it can be shown that
at
under perturbations
Ah U
3 it follows
@h
z h.
th
at
¢h
stable of order
a
at each family
THEOREM
at
~
zh •
with an accuracy
a
at
zh
if it is
will be said to be stable of order
zh
with
zh
with
is convergent.
of order
p -
0-restricted
~
if it is
in the domain of definition
of
Ah
i
STABILITY THEOREMS
(Stability of linear operators) . Let
into
Bh
(for each
h e H ) . Then
Th
if and only if:
~h
Ch
Dh
h e H ) .
3.2
from
u = v(0) ~
p .
stable of order
a
llw-¢ h Zhl I = O(h p) < 6h ~
Consequently
is convergent
of order
stable of order
operator
an element
flu-& h UII ~ yh-a'O(h p) = o(hP-a).
We shall call
(for each
easily that
Hence there exists
and
i)
be real numbers
m-restricted stable of order
@h v = w
Ch u = 0
%h
~
satisfying
(3.8)
for
and
satisfies
We illustrate
¢h
and
~, y, h I > 0
h ~ hl, w e B h
v = v(w) E
2.
m
is a one-to-one
mapping
of
Ah
onto
Bh ,
~h
be a linear bounded
is stable of order
243
2)
the inverse
3)
IITh-l]I
Proof.
IIyll
~h v = w,
x 6 Ah
there
-i
~ yh -~
is bounded
~h
with
~h x = ty,
is a unique
Ilxll
~ e Ah
A h . Hence
~h
~
satisfying
ii~hi yll
= Ilxll
conditions
i), 2), S) of theorem
= t-l. IIxll
ll~h -I w - ~h-l(~h
a
it follows
" Since
t
that
may tend
in the whole of
B h . Furthermore
= ¥ h-~
• Thus the
i are fulfilled.
I), 2), 3) to be true.
Then
from which
it follows
easily that
~h
zh .
and let
open ball with center
of
that there is a unique
is also unique
(Stability of nonlinear operators)
2
be given constants
at
with
with
x = t-l.x
< YBh-a't-i
onto
hI
y • Bh
v e ~
~ t-l'¥ h ~ IItyll
Zh)ll ~ ¥h -e llw-~ h Zhl I
is stable of order
THEOREM
assume
Ah
some
is the constant
it follows
~h ~ = y
= t-l'IIV-Zhll
S
is a unique
IIxll
mapping of
h ~
z h . Let
where
x = v - zh
~h ~ = y'
is a one-to-one
Conversely,
there
at
< y6h -~ . By defining
with
to zero it is clear that
~
0 < t < B
h e H, h ~ h I
IlV-Zhl I < yBh -~ . Writing
with
y , provided
is stable of order
w = ~h Zh + t.y
3. Hence for
and
for some constant
Assume
: i . Put
definition
Th
z h e D h (for
and radius
zh
h • H)
p hc
. Let
. Assume
is contained
P > 0
and
that for each
in
Dh
c h 0
h ~ H
and that for
the
x
~n this ball
(s.9>
ll~A<x)-
where the constant
one-to-one
l
mapping
Proof.
follows
is independent
from
ted stable of order
stable of order
llx-zhll
~(Zh)IJ ! ~h-C
Ah
a
at
onto
zh
of
B h (for
h
and
x . Assume
h ~ H)
. Then
%b
that
is
if and only if the linear operator
%~(z h )
(c+2~)
is a
- restrie-
~h = %h(Zh)
~ .
Assume
~ ( z h)
is stable of order
a . Applying
theorem
i it
that
l l~(Zh)-1]l
i Y h-~
We shall apply lemma 2 with
Condition
(h i h I, h ~ H) .
R = 4'
(2.5) is fulfilled whenever
S = B h, E = Dh, ¢ = ¢h' Xo = Zh' 6 = yh
~
satisfies
~ > lh -c
and
is
244
h e y -i u -i = 6-1 p -i ! Ph c . Therefore we choose
(I, y-i p-i h~)
and lemma 2 may be applied.
iiw_%(~h)l I < (2~ = P )-l
iiV_Zhll
= (2Y = M) -~
< (6~)-I : (yM)-l.ha+C
satisfies
h=~+c
and
is
ma 3 may be applied with
~
there is a unique
at
~ ( Z h )-I
theorem i
v e Ah
with
with
(see (2.6)) this
- ~h(Zh)II
. Hence
(2a+c) - restricted stable of order
~h
v
is
~
at
z h . Lem-
R = Ah, S : Bh, E = Dh, } = ~h' Xo = Zh' ~ = lh-C'
is bounded and
%{(Zh)
w & Bh
zh .
= Bh m, o : YBh m-~, 8 = yh -~, m = 2~+c, h ~ h I
that
M = max
It follows that for all
= 2yh-elI~h(V)
(2e+c) - restricted stable of order
~h
with
#h(V) = w . Furthermore
llV-Zhl I ~ 28 IlW-~h(Zh)ll
Now assume
p = M h -c
(see definition 3) . It follows
II~{(Zh)-iII ! 6 = ¥h -a (for
is stable of order
h ~ h I) . By
~ . The theorem has thus been proved.
We note that theorem i is related to a result in [5, p. 106] and that
theorem 2 is an extension of a result in [15, p. 114].
3.3
3.3.1.
SOLVING THE EQUATION
th u = 0
WITH AN ERROR
=
O(h r)
Preliminaries
In this section 3.3 we assume that the operators
F.
3
appearing in (3.6)
satisfy the following three conditions.
Condition I.
and for
h E H
For each closed ball
X 6 D
a unique
The
and is c o n t i n u o u s
V° e D
of finite radius contained in
the relation (3.6) holds uniformly for
Condition II.
for
C
with
A
Condition III.
on
into
If
in
C .
(r-j)-th order Fr6chet derivative
D
Fo(V o) : 0
one-to-one mapping from
X
(for
j
and if
= 0,1,
r _> 2
....
,r-t)
F!r-J)(x)
]
. Furthermore
the derivative
F'(Voo )
p > i
r h 2
the operator
if and only if
--
We define the function
exists
there
is
is a
B .
F~(V o)
is a mapping onto
We note that from condition II and definition 2 it follows that
consistent of order
D
V
: U .
o
G
mapping
D X ~ho,ho]
into
B
by
~h
B .
is
245
G (X, h) :
(3.107
r-i
~ h J F.(X) . Now formula (3.6) can be written as
j:O
]
~h Ah X : L h G (X, h) +
O(h r) .
From the conditions II and III we have
is a one-to-one mapping from
A
onto
G (Vo, O) = 0 , and
B (if
~
G (Vo, O) = F'o (Vo)
r > 2) . In view of the implicit
function theorem (see [i0, p. 687] ) we expect the existence of a function
with
X(h)
G (X(h), h) : 0, X(0) = V
. Since the functions F. are smooth we also ex°r_ 1
]
pect a truncated Taylor series
[ h i V. of X(h) to satisfy
r-i
.
i=0
i
G ( Z hiVi , h) = O(h r) . Since A h maps D into D h and in view of (3.10) the
i=0
function
(3.11)
V(h) :
r-i
hi Vi
i=O
is thus expected to satisfy condition (3.12):
[or sufficiently small
(3.12)
atisfles
~h Ah V(h) :
h e H
the element
V(h)
satisfying (3.12) is that when
it is more appropriate to prove convergence of
Ch
by comparing
Ah U . The advantage is that for
the difference equation
¢h Uh : 0
D h and
O(h r) .
The reason for considering a
rather than with
A h V(h) belongs to
A h V(h)
uh
with
V° = U
A h V(h)
the perturbation to
can be made arbitrarily small by taking
r
suf-
ficiently large. The need for this arises when one wants to treat the case where
(3.8) is violated with
paring
uh
with
m = o + 2a
Ah V(h)
(see theorem 2). We note that the idea of com-
was used with succes by Strang [16],
[17].
In the following we shall prove (3.11), (3.12) rigorously and show that
%h
is consistent of order
3.3.2.
p
if and only if (3.15) holds.
Recurrence relations for
r-i
We put
V1, V2, .... ,Vr_ 1
V = i=0~ h i V.z where the elements
still to be determined. We choose
VI,V 2, .... ,Vr_ I
in
A
are
with 0 < h I ~ h ° so small that the closed
r-i i
ball with center Vo and radius ~ = ~ h I IIvilI is contained in D . For
i=l
r-i
r-I
0 < h ~ h I we have V e D and G (V, h) = ~ h j F (V ° + ~ h i Vi) =
j=0
J
i=l
r-i
j:0
hI
i F(I
hi
h j { Fj(V o) + ~., j )(Vo ) (r-I
[
V i) +.
+
i:l
"'" ~
i
F(r_I_J)(v
j
o
)(r-i
[ h i vi)r-l-J }
i:l
246
+ W . An application of lemma i with
R = A, S : B, E : D, Xo : Vo, x I : V,
n = r-l-j, ~ = sup I IFIr-]")(Z) II < ~
where the supremum of the continuous function
r-I
II jF(r-J)(z)II
is over all Z of the form Z : Vo + t • [ h i V.
1
i=l
(0 < t < 1, 0 < h < h 1) ( s e e c o n d i t i o n t i and [10, p . 6 6 0 ] ) , y i e l d s t h e r e s u l t t h a t
r-1
I IW]I : [ hJ'o(h r-j) : O(h r) . Since F (V o) : 0 (see condition II) we thus have
j:0
(3.13)
where the
W. ~ B
i
o
r-i
~ hi Wi +
i=l
G (V, h) :
O(h r)
are defined by
wi = ~ 1
~, F
the summation b e i n g
(jMI(Vo)
for
all
Vii Vi2 .... ViM
integers
M, j ,
i
with
m
0 _< M, 0 _< j, I _< im (i < m < M), j + i I + i 2 + .... + iM = i .
The largest index
obtained when
dependent
i
appearing in this summation is
m
i
m
= i
and this value is
j = O, M : i, i I : i . Further the only term in the sum that is in-
of the
V.1
m
is
F ~ ° ) ( V o ) = F i (V o )
. Hence
W i = F'o (Vo) Vi + Fi (Vo) + Yi
where
Y'l
when a l l
is an expression which only contains
Vk ( w i t h
1 ~ k < i)
equal
G (V(h), h) : O(h r) (0 < h _< h I )
if
zero.
V(h)
It
Vk
with
follows
k < i
from (3.13)
and which vanishes
that
is defined by (3.11) and the
V.l
satis-
fy
(3.14)
F' ( V )
0
0
V. , F. ( V )
1
1
0
+ Y. : 0 (i : 1,2,....,r-1)
l
.
From the conditions II, llI it follows that (3.14) is a series of recurrence relations defining the
V.
i
uniquely.
Using the relations
THEOREM 3. Let the conditions
there is an element
Vo
in
D
(3.14) we shall prove:
I, If, III stated above be fulfilled.
and a series of elements
VI,V2,....,Vr_ I
in
Then
A
with the following two properties:
i.
If
V(h)
2.
Let
p
p
($.15)
is defined by (3.11) then (3.12) holds,
be an integer with
i ~ p ~ r . Then
if and only if
Vo : U, V 1 : V 2 : .... : Vp_ 1 = 0 .
%h
is consistent of order
247
Proof.
We choose for
V. (i < i < r-l)
For
the element defined by condition II and for
o
the elements defined by (3 14).
1
i.
V
h ~ hl, h ~ H
where
o
we have
G (V(h), h) =
O(h r)
and
llV(h) - Vol I ~ o
is defined above. In view of condition I the relation
holds uniformly for
X
in the closed ball with center
V
(3.10)
and radius
o
o
Hence (3.12) is fulfilled.
2.
By examining the relations
V I = .... = Vp_ I = 0
(3.14) it follows in view of condition II that
if and only if
F.] (U) = 0 (0 _< j _< p-l)
F I (V o) = .... = Fp_I(V o) = 0 . Hence
if and only if
U = Vo, V I =....= Vp_ I = 0 .
The theorem has thus been proved.
Remark.
exist elements
In the above condition III has only been used to prove that there
VI,V2, .... ,Vr_ I
satisfying (3.14). Therefore theorem 3 remains
valid if condition IIl is replaced by the following weaker requirement
Condition III.
satisfying
If
r ~ 2
there exist elements
III :
VI,V2, .... ,Vr_ I
in
A
(3.14).
Furthermore the uniformity of (3.6) on each closed ball in
been used to prove that (3.10) holds with
X
replaced by
V(h)
D
has only
. Consequently the
result of theorem 3 still holds if the conditions I and llI are replaced
simultaneously by I * and III ~ where condition I * is as follows:
r-I
~ h i V.
i:0
i
the conditions II and III ~) satisfies
Condition I ~.
Ch Ah V(h)
=
V(h) :
r-i
[
j=O
h ]-L h Fj
3.4
(where
V.
i
are the elements in
A
from
(V(h)) + O ( h r) (for h+O)
EQUIVALENCE THEOREMS
Throughout this section 3.4 we assume the following condition to be fulfilled:
Condition IV.
in
A
Vo
with the properties
is an element in
D
and
VI,V2,. ...,Vr_ I
i and 2 stated in theorem 3. Furthermore
are elements
H C (0, h ~
248
V
where
h
is so small that the closed ball with center
0
r-i
:
[ hio l lVill
is contained in D .
and radius
0
i=l
Since
Ah
maps
Ah { q i 1 h i Vi }
i=O
element
Throughout
D
into
belongs
Dh
to
there
Dh
follows
whenever
from condition
h e H
and
q
IV that the
is an integer
~ r
this section we write
V(h)
=
r-i
[ h i V.
i=O
l
and we assume
Condition V.
open ball with center
for
x
There are real numbers
A h V(h)
and radius
p > O, I ~ O, c ~ 0
oh c
is contained
in
such that the
Dh
and such that
in this ball
l[~h (x) - ¢{(A h V(h))ll
Condition VI.
mapping
Bh
onto
THEOREM
%
! lh-Cll x - A h V(h)ll
The linear operator
(for each
, for
}~ (A h V(h))
h ( H •
has a bounded
inverse
h e H) .
(Equi~lence of convergence and stability) . Let the conditions
4
IV, V and VI be fulfilled.
Assume
q ~ 1
is an integer and
a ~ 0
a real number
with
(3.16)
Let
c + 2a < q + e < r .
zh 6 D h
(3.17)
Then
~
~h
bations
order
be defined by
q-i
: A h { [ h i Vi}
i=0
is convergent
of order
q + ~
.
with an accuracy
of order
q
at this
if and only if the linear operator
~
under pertur-
~i'(~)
n n
is stable of
a •
Proof.l. Assume
¢{(Zh)
a bounded
inverse
(3.1~)
II~(%)-llt ~ yb-~
is stable
of order
a . By theorem
1
¢{(~)
and
From the definition
of
V(h)
we have
IIz h - A h V(h)I] :
O(h q) . In view of
has
249
(3.16) we thus have
lJz h - AhV(h)Jl
< oh c
for
h
sufficiently small and by
condition V:
(3.19)
As
J l}~(Zh) - ~{(AhV(h))lJ <_ kh-c.O(h q) .
- c + q - ~ > 0
l
it follows from (3.18), (3.19) that
h) -
for
h
• ll
(Zh) ll[ = O(hC÷q
sufficiently small. By Banach's lemma
<
¢~(AhV(h))
thus also has a bounded
inverse and
I I~h(Ah v(h))-lll _< Yo ° h
for some fixed
Yo > 0
and
h
-~
sufficiently small in
Applying theorem 2 with
restricted stable of order
8, YI' hl > 0
a
such that for
Jw - ~hAhV(h)lJ
< 8h c+2a
at
z h = AhV(h)
AhV(h)
Now let
l lVh - AhV(h)ll
v E Dh
< 8h c+2~
and there is a unique
order
w ~ Bh
with
with
~h v = w,
h
v h ~ Dh
sufficiently small
with
~hV
= Wh,
I lVh-Zhll < flyh - AhV(h)II
¢h
zh
under perturbations of order
2.Assume
~h
is convergent with an accuracy of order
q + a . We shall prove that
q + e .
~ ( z h)
q
y(h) 6 Bh
with
J Jy(h)Jl = 1
h & H . We choose an element
1
lJFhY(h)II ~ ' J l F h J
and
w h = ~hAhV(h) + hq+ey(h)
for
I
and we define
.
By virtue of (3.12) (which holds in view of condition IV) we have
at
zh
is stable of
e .
Fh = ~ ( A h V ( h ) ) - i
+
is convergent with an
at
We define
•
l]Vh - AhV(h)II <__Ylh-~ll¢hV h - %hAhV(h)J] <
I IAhV(h) - Zhl I = O(h q) + O(h q) . It follows that
under perturbations of order
(c + 2a)
= O(h q+a) + O(h r) . In view of (3.16)
Ylh-~'{O(h q+~) + O(hr)} : O(h q) . Consequently
q
is
h÷0) . Then
= o(h c+2e) . Hence for
< Yl. Bh c+~ . Moreover
accuracy of order
~h
] Iv - AhV(h)J I <--Y1 h-~'[ JCh v - ~hAh V(h)II
w h ~ Bh, l[whJ j = O(h q+a) (for
][wh - %hAhV(h)II
I]Wh - ~hAhV(h)ll
and all
there is a unique
Jwh - ~hAhV(h)II < J lWhJl + ]J#hAhV(h)J[
we thus have
there follows that
. Consequently there are constants
h < hl, h 6 H
Jv - AhV(h)J I < YlShC+a; moreover
H .
250
IIWhl I = O(h r) + O(h q+a) : O(h q+a) (for
h÷O)
sufficiently
with
Yo
small there exist
is independent
where
YI
h . Hence
is some constant
we have for
~(AhV(h))
of
h
v h 6 Dh
(v h - AhV(h))
!
2
! Yo hq
~ Yo'h q + Ilz h - AhV(h)ll
h
where
~ Yl'h q
h . In view of condition V and lemma i
h q+e y(h) = }hVh - %hAhV(h)
=
Ahv<h)[12
Fh(hq+~ y(h)) : (v h - AhV(h))
hq+~.llrhll
]IVh-Zhll
1 for
with
i
IIeII~7~hC'IIVh
Since
of
small that
+ e
%hVh : w h,
IIv h - AhV(h)II
independent
sufficiently
. According to definition
< Ilrh (h q+~ y(h))ll
+ Fh(e)
we get
< ]lVh _ AhV(h)ll
--
+
--
Ilrhll.llel
I
°
Consequent ly
hq+~ ]lrhl I L 2Yl'hq
Dividing both members
IIFhl I
÷
~h-C.(ylhq)2.11rhl I
of this inequality by
and combining the terms in which
enters we obtain
(i- ~y12.hq-e-~) IIFh] I ±2y 1
Since
h q+~
q - e - @ > 0
h -~
(see (3.16)) we have
llFhl I ~ 4 71 h -e
provided
h e H
is sufficiently
small. By using Banach's
in part i of the proof it follows that
~ ( z h)
lemma in a similar way as
also has a bounded inverse and
II ~{(Zh) ~II m Y2 h-~
for some constant
of order
Y2
and
h
sufficiently
small. By theorem 1
%~(z h)
is stable
~ . The theorem has thus been proved.
We now are in a position to derive our main result which is formulated
the following equivalence
theorem 5. In addition to the conditions
in
IV, V, VI we
assume here:
Condition VII.
Condition VIII.
If
If
X E A
v
and
and
9
lim
h+O
llAhXll
satisfy
: 0
then
th v : ~h 9
X = 0
for some
0
h { H
then
251
V
=
V
•
THEOREM
the conditions
i < p < q
(3.16)
Then
(Equivalence of convergence to stability and consistency)
5
IV - Vlll be fulfilled.
and
a > 0
Assume
p
and
q
are integers
. Let
with
is a real number with
c + 2a < q + e < r .
@h
q + a
is convergent
at some
zh
with an accuracy
of order
q
under perturbations
of order
of the form
q-l.
(3.20)
with
X
z h : A h {U +
hlu'}
l
i:p
(h 6 H)
Up, Up+l, .... , Uq_ I @ A , if and only if the linear
is stable of order
Proof.
consistent
a
and simultaneously
i.
Let
of order
¢~(AhV(h))
p . By virtue
Hence this
zh
I1~ h - f l h V ( h ) l l : O(h q )
h
sufficiently
small)
be stable of order
is equal to the
der
at
defined by (3.20)
2.
perturbations
Let
of order
We define
v h e Dh
condition
and
VIII that
q + a
(h÷0)
¢~z h)
at some
zh
Ch
be
V
= U,
o
and we define
of Banach's
y
and
lemma as in
a . From
with an accuracy
of order
zh
Since
some c o n s t a n t
with an accuracy
of or-
q + a .
of order
q
under
of the form (3.20).
. By condition
. Consequently
h ~ some hl, h ( H . Since
AhV(h)
and let
p .
is stable of order
is convergent
under perturbations
be convergent
w h = %hAhV(h)
[lWhl I = O(h r) = O(h q+e)
some
@h
(for
application
4 that
¢h
a
p _< i _< q-l)
~ yh - a
it follows by a similar
4 we thus obtain the result that
of order
defined by (3.17).
II~(AhV(h))-lll
and
theorem
zh
zh
¢~(AhV(h))
IV (see (3.15)) we have
U.I = V.I (for
the first part of the proof of theorem
q
is consistent
of condition
V I = V 2 = .... = Vp_ I = 0 . We define
by (3.20).
Ch
operator
IV (see (3.12)) we have
@hVh = Wh,
v h = z h + O(h q)
@hVh = @hAhV(h)
= v h = z h + O(h q) (h ~ hl, h e H)
for
we have from
. Hence
r-i
q-l.
A h {V ° + [ h i V i} : A h {U + ~ hiUi } + O(h q) .
i=l
i=p
Combining
the terms in which the same powers
get (cf. condition
VII):
of
h
appear
and by letting
h÷0
we
252
A h (Vo-U) ÷ 0, A h V i ÷ 0 (i < i < p-i),
A h (Vi-U i) ÷ 0 (p ! i ! q-l)
(for
V ° : U, V i : 0 (i < i < p-i),
Since
our
zh
of the form (3.20)
theorem 4 it thus follows
Applying
Banach's
IV that
equals
the
#h
zh
that the linear operator
the proof of theorem
In the applications
usually
c, a
and
p
then yields
r ~ q + a
IIUh-A h UII
on
p
(which ensures
of the linear operator
(among other things)
finite-difference
equation
= O(h p)
(like e.g.
4.
a .
. We emphasize
is stable
of order
~
q
(3.16) to hold)
(A h V(h))
determined
In order to prove convergence
and
r
so large that
and to check for stability
. An application
the result that for
~h(U) = 0
has a solution
h
of theorem
sufficiently
u = uh
5
small the
satisfying
the fact that this holds without
any restriction
(3.8)).
AN APPLICATION
4.1
In this chapter
on the infinite
is stable of order
are given numbers
method one only has to choose
e)
Applying
5.
of the finite-difference
(of order
p •
of order
defined by (3.17).
#{ (A h V(h))
scheme under consideration.
and
is consistent
#~'(%)n
by the finite-difference
q > c + a
.
lemma in exactly the same way as in the first part of the proof
of theorem 4 it follows that the linear operator
This completes
and
V.I : U.I (p -< i _< q-l)
(3.15) thus holds we have by condition
Furthermore
h÷0)
TO PARABOLIC
DIFFERENTIAL
EQUATIONS
THE DIFFERENTIAL EQUATION
it is assumed that
U
is a real function which
strip
G : {(s,t)
] - ~ < s < ~, 0 < t < T}
and which solves the initial value problem
(4.l.a)
U (s,O) - f(s) = O, - ~ < s < ~ ,
(4.1.b)
g [Ut (s,t), Uss (s,t), U s (s,t), U (s,t), s, t] : O, (s,t) ~ G
is defined
253
the subscrips
in (4.l.b)
denoting partial
In the following
differentiation.
we shall use the notation
note the first order partial
derivative
z i (i=1,2,3,4)
we shall denote by
functions
. Furthermore
defined
on
E
lowing smoothness
(A)
for
on the set
this chapter
All partial
derivatives
Izil ~ e (i=1,2,3,4),
such that for
r
f, g and U
and for any value of
~
zi, s, t
(s,t) ~ G . Furthermore
satisfy the fol-
g [Zl,Z2,Z3,Z4,s,t ]
the function
g
exist
and each of
vary in such a way that
there are constants
ul > 0, ~2 > 0
i
The function
(C)
U
belongs
. For
~ - ~i'
f
belongs
to class
X & A
max
sup
0 < i < r+l
G
I
an arbitrary
by
B = {Y I Y = (Yo,YI)
in
B
32 g [Zl'Z2'Z3'Z4's't]
to class
C ~ (G)
C = (- ~, ~)
h U2
'
,
.
of chapter
3 we define the vector
space
A
we define the norm
. X(s,t)[ +
~t I
max
sup
0 < i < 2r+2
G
but fixed integer
where
I
'i
3s
x(s,t)[
> i . The vector space
Yo e C ~ (-~,~),
YI E C ~ (G)}
. For
B
is defined
Y = (Yo,YI)
we define the norm
[IYII
the supremums
where
deriva-
- ~ < z. < ~, (s,t) ~ G :
A = C ~ (G)
fine
that each of their partial
of the function
remains bounded when
(B)
denoting
C~(E) the class of all bounded real
it is assumed that
In order to use the concepts
I IXII :
to the variable
E .
derivatives
81 g [Zl'Z2'Z3'Z4'S't]
by
with respect
to de-
conditions:
- ~ < z i < ~, (s,t) ~ G;
its partial
g
which have the property
tives exists and is bounded
Throughout
of
3 i g [Zl,Z2,Z3,z4,s,t ]
D = A
Yo(S)
: sup
IYo(S)I
being for
- ~ < s < ~
and the operator
: X(s,0)
With these definitions
+ sup
- f(s),
F
IYI (s,t) I
and
mapping
Yl(S,t)
(s,t) ( G , respectively.
D
into
B
is defined by
Finally we deF(X) = (Yo,YI)
: g [Xt(s,t) , Xss(S,t) , Xs(s,t) , X(s,t),
(3.1) is equivalent
to (4.1).
s,t]
254
4.2
THE FINITE-DIFFERENCE EQUATION
In order to construct a finite-difference
lution
U
to (4.1) can be approximated we choose increments
the variables
t
and
s , respectively
a constant independent of
constant
scheme by means of which the so-
< T . For
in such a way that the quotient
h . We define the interval
h 6 H
h = At > 0, As > 0
H = (0, ho]
of
h/(As) 2
where
h°
is
is a
we define
G(h) = {(s,t)l(s,t) ~ G and s : m&s, t = nh, m : 0,+1,+2 ..... , n = 0,1,2, .... } ,
Go(h)= { s I s = m&s,
m = 0,+i,+2,....]
,
Gl(h): {(s~t)l(s,t) ~ G(h) and t+h ! T} .
Using the shifting operator
finite-difference
u(s,0) - f(s) = 0,
(4.2.b)
g [h-l'{u (s,t+h)
E i u(s,t) = u(s+iAs,t)
O
-
u
(s,t)}
(As)-2" [ ~i Ei u(s,t)
i
(s,t) E Gl(h) •
and B i
,
are constants independent of
h
[ai
i
,
(As)-l.[ 8i E i
i
u
(s,t),
satisfying the conditions
(4.3) and vanishing ~or all but a finite number of values
(4.3)
the
s e G (h) ,
u(s,t), s, t] : 0,
~i
defined by
equation we shall deal with can be written in the following way:
(4.2.a)
In (4.2.b)
E
i = 0,+1,+2, ....
: ~ i'8i = - Bi '
i:°'Zh i2:2'ZBii:l
i
We note that since
i
~i g ~ - ~ i
mined by (4.2.b) once
< 0
u(s,t)(s
(see condition (A))
u(s,t+h) is uniquely deter-
~ G (h)) is given. Consequently
u(s,t)(t:0,h,2h, .... ~ T)
O
can be computed uniquely from (4.2) by an application of (4.2.a) and by applying
(4.2.b) successively with
t : 0,h,2h, .... ~ T-h .
In order to use the concepts of chapter 3 we define
space consisting of all real bounded functions defined on
l lxll : sup
Ix(s,t)l
for
x e A h , the supremum being for
the vector space consisting of all elements
real bounded functions defined on
IlYll : II(Yo,Yl)II
s 6 Go(h),
: sup
(s,t) ~ Gl(h)
G (h)
O
lYo(S)l + sup
, respectively.
and
y = (yo,Yl)
Gl(h)
lYl(S,t)]
For
X E A
G(h)
Ah
to be the vector
with the norm
(s,t) e G(h)
where
Yo
. Bh
and
Yl
is
are
, respectively with norm
the supremums being for
we define
AhX : x ( ~
by
255
setting
x(s,t) = X(s,t) (for all
Lh (Yo,YI) : (yo,Yl) E B h
with
(s,t) ~ G(h)) and for
(Yo,YI) ~ B
we put
Yo(S) = Yo(S), Yl(S,t) : Yl(S,t) (for all
s ( G o ( h ) , (s,t) E Gl(h) ) .
To simplify the notations we introduce operators
I
(4.4)
C. = C.(h)
]
]
defined by
lX(S,t) : h -I {x(s,t+h) - x(s,t)} ,
C2x(s,t) : (As) -2 [aiElx(s,t)
i
.
C3x(s,t) = (As) -I [~iElx(s,t)
,
,
i
C4x(s,t) : x(s,t) .
We define
Dh = Ah
~h(X) : (yo,Yl)
and the operator
where
~h
mapping
Yo(S) : x(s,O) - f(s)
Dh
into
Bh
is defined by
and
Yl(S,t) : g [ClX(S,t), C2x(s,t), C3x(s,t), C4x(s,t) , s, t] . With these definitions
(3.2) is equivalent to our finite-difference scheme (4.2).
Let
z E Dh . Then
%~(z)
exists and is given by
{}~(z))[x] = (yo,Yl)
where
Yo(S)
=
x(s,O),
(4.S)
4
Yl (s't) =
~ ~i g [ClZ(S't)' C2z(s't)' C3z(s't)' C4z(s't)' s, t].Cix(s,t)
i=l
It follows that all general conditions of section 3.1.1 are satisfied here.
4.3
THE
CONDITIONS
I - VIII
The purpose of this section is to show that with the definitions of the
sections 4.1, 4.2 the conditions I - VIII of chapter 3 are fulfilled.
Condition
I.
Let
~IXII ~ B
X
where
belong to a given closed ball
B
C
contained in
is some constant depending only on
A . Then
C . We have
~hAh X = (yo,Yl) with
Yo(S)
: X(s,0)
- f(s)
,
Yl(S,t) = g [CiX(s,t) , C2X(s,t) , C3X(s,t) , C4X(s,t) , s, t]
Using (4.3) and Taylor's formula it follows that
256
lYl(S,t) - g [Xt(s,t) , Xss(S,t) , Xs(s,t) , X(s,t),
(4.6)
r-i
f
hJ {fj[X](s,t)}l
i
K"
s, t]
-
hr
j=l
for all (s,t) e Gl(h)
. The
by the partial derivatives
fj
of
are operators
mapping
g . The constant
K
A
into itself determined
can be estimated
in terms of
bounds for the derivatives
(~i/~tl)X
(2 < i < r+l),
(~i/~si)X
(3 < i < 2r+2)
.
Hence there exists a constant
K
IIXII
of the norm in A). Defining
! 8
(4.7.a)
and
(see the definition
F
O
= F
for all
X
with
,
F.(X)] = (0, fj(X))
X6C
such that (4.6) holds uniformly
(i < j < r-l)
it follows that (3.6) holds uniformly for
.
Conditions II, III.
Using the smoothness
the definition
of
lity requirements
f.
]
condition
(A), the definition
it follows that the operators
of condition
F.
]
of the norm in
A
and
satisfy the differentiabi-
II.
From (4.7.~) it follows that
(4.7.b)
V
satisfies
O
= U
Fo(V o) = 0 . The assumption that
shown to lead to a contradiction
value problem for
V° - 9°
Fo(9 o) = 0
by deriving a linear homogeneous
by subtracting
Fo(V o)
from
value problem obtained in this way has a unique solution
shows that
V ° - 9 ° = 0 . ?onsequently
We shall now show that
onto
B . The equality
F'(U)X = Y
for
X(s,0) = Yo(S
(-~ < s < ~),
(4.8.b)
gl(s,t)Xt(s,t
+ g2(s,t)
Yl(S,t)
6 G)
where
there is a unique
Fg(Vo)
(4.8.a)
((s,t
for some
= F'(U)
9o ~ Vo
can be
parabolic
initial
Fo(V o) . For the initial
(see [9, p. 171]), which
Vo
with
is a one-to-one
X E A , Y = (Yo,YI) E B
Xss(S,t ) + g3(s,t) Xs(s,t)
Fo(V o) = 0 .
mapping from
A
is equivalent
to
+ g4(s,t) X(s,t) =
257
(4.9)
gi (s't) : ~i g [Ut(s't)' Uss(S't)' Us(s't)' U(s,t], s, t]
(i=1,2,3,4) . Using the result in ~ , p. 1 7 ~ ,
a straightforward extension of
[9, p. 175] and the smoothness assumptions (A), (C) it can be proved that the initial
value problem (4.8) has a unique solution
element in
X E A
whenever
Y = (Yo,YI)
is a given
B . This completes the proof of the conditions II, III.
Condition IV.
Since the conditions I, II, III are satisfied there follows from theorem 3
that condition IV can be fulfilled provided
H = (O,ho]
where
h°
is small enough.
Condition V.
r-i
Let
z : AhV(h) : Ah
in condition IV. For
x ~ Ah
i
~
hlV i
we have
where the supremum is for all
where
V.
i
are the elements in
A
occuring
l l~(x) - @~(z)][ : sup l l{~(x) - ~ ( z ) } v If
v ~ ~
with
I Ivll = i . From (4.5), (4.4) we thus
have
(4.10)
4
[ 13i~ - ~ig 1
i=l
[l}{(x) -}{(z)l [ <_ ~h-l.sup
where the argument of
3i~
equals
[ClX(S,t), C2x(s,t), C3x(s,t), C4(x,t), s, t~
and the argument of
Dig
is the same with
supremum is over all (s,t) E Gl(h),
the factor
h
-i
n
x
replaced by
z . In (4.10) the
is a constant depending only on
~i,Bi
and
in (4.10) stems from the inverse powers of the mesh-width occuring
in (4.4). By the mean value theorem we obtain from (4.10)
(4.11)
ll}{(x) - ~{(z)l I ~ (~h-l)2.sup
the first 4 arguments of
~i~jg
4
4
[
[ l~i~jgl
i=l j:l
being values between
Ck x(s,t)
and
Ck z(s,t)
(k:i,2,3,4) .
Let
Vi E A
and
A h,
llx-zll
< h,
the first 4 arguments of
(s,t)
vary through
H
and
H . Sinoe
3i3.g
]
z
is composed of smooth functions
in (4.11) will then remain bounded if
Gl(h) , respectively. In view of condition (A) it
thus follows that the supremum appearing in (4.11) is finite, uniformly for
llxzll < h
Consequently
h
iI% (x)
£(z)ll m
-2 11xzll
h 6 H,
for some constant
I .
258
Hence condition V is fulfilled with
(4.12)
p : i/h
and
o
c = 2 .
Conditions
YI, VII, VIII.
From (4.4) and condition (A) there follows that for any fixed
z e Ah
and for arbitrary
y = (yo,Yl) e B h
fying (4.5). Moreover
llxll ~ Y'IIyII,
z
y . Consequently VI is fulfilled.
but independent of
y
there exists a unique
h E H,
x e ~
being a parameter depending on
satish
and
It is easily verified that
the conditions VII, VIII are also satisfied here.
4.4
A SUFFICIENT CONDITION FOR CONVERGENCE
In this section we prove the following theorem 6 by combining theorem 5
with a well known sufficient condition for stability of linear difference schemes
due
to
John
[9].
THEOREM 6. Let the conditions
solution of the finite-difference
Let
(A), (B), (C) hold. Let
equation (4.2) where
ai,6 i
Uh(S,t)
be the
are subject to (4.3).
K(s,t) = {-hg2(s,t)}/{(&s)2gl(s,t)}
(see (4.9)) and assume the function
~(e)
:
i
+ K(s,t).{%
+ 2 Z
satisfies the inequality
I*(e)l~
(whenever
-~
exp
< s < ~,
~.
]
j>O
cos(je)}
(-Me 2)
0 < t
< T,
lel ~ w ),
M
denoting some constant
Then there exists an infinite series of functions
which is in
(4.13)
Ca(G)
such that for all positive integers
Uh(S,t) - U(s,t) :
(for h+O) uniformly for
Proof. I.
> 0 .
UI,U2,U3, .... each of
q
q-i
.
[ hzUi(s,t) + O(h q)
i=l
(s,t) e G(h)
.
With the definitions of the sections 4.1, 4.2 theorem 5 may be
applied since the conditions IV - VIII have been shown to be fulfilled here. From
(4.7.a) and definition 2 (section 3.1) it follows that
Ch
is consistent of order i.
259
By choosing
%h
a = O, r : q > 2
in (3.16) we thus obtain from (4.12) the result that
is convergent with an accuracy of order
some
zh
of the form
defined by
q
• ]
z h : A h {U + qi 1 hiUi
i=l
under perturbations
of order
q
provided the linear operator
at
~h
~h : %'h (AhV(h))
is stable of order
on
9(~)
0 . It will be shown that this is the case under the condition
stated in theorem 6. Hence (4.13) holds for each integer
definition
i, section 3.1). Consequently
2.
y = (yo,Yl) 6 B h
Solving the last equality for
x(s,t+h)
x(s,t+h)
: Yl(s,t),
O(hllxlE) + 0(hllyTl)
t : O, h, 2h, .... )
with
g2
replaced by
where
initial condition
an inequality
h
and
x(s,O)
= Yo(S)
of the form
~h
for
(s,t) • Gl(h)
0 . It
x C Ah~
llxll ~ Y'IlYll
is stable of order
~i = e-l"
( 7
+
K(s,t)
(see (4.3)) an application
x(s,t)
determined by the
relation
(4.14) satisfies
being some constant
exists with the property
0
of (4.4)
is defined by the same formula as
and by the recurrence
M > 0
.
+ (As)L(s,t)-[ 6iEZx(s,t)
i
[9, p. 166] now shows that
y ) provided a constant
Consequently
L(s,t)
g3 " Using the fact that
of John's stability theorem
~h x : y
we obtain by an application
: x(s,t) + K(s,t)-[ ~iEZx(s,t)
i
(for
is stable of order
q h I .
to
x(s~O) : Yo(S), s e Go(h)
4
{gi(s,t) + O(h)).Cix(s,t)
i=l
(4.14)
~h
(4.9) that the equality
is equivalent
(see
(4.13) also holds for each integer
It remains to be shown that
follows from (4.5), (4.7.b),
q > 2
independent
of
stated in theorem 6.
(see theorem i) and the theorem has thus
been proved.
As an illustration
(4.2] with
ei,6i
of theorem 6 we consider the finite-difference
from table i. The expressions
in (4.2.b) which the
ei
method
and
8i
260
i
< -3
-2
-1
0
1
2
> 3
12 ~.
0
-i
16
-30
16
-i
0
12 B i
0
i
-8
0
8
-i
0
1
Table i.
are involved
A choice for the parameters in (4.2.b)
in, now represent
order partial
derivatives
O((As) 4)
with respect
shows that the function
~(e)
~(0) = i - {K(s,t)/3).{7
- 8(cos
i and i - 16 K(s,t)/3
h
- (As)2
(4.15)
we h a v e
inf
that for
filled.
M > 0
satisfies
[3]). An easy calculation
6 now takes the form
e) + (cos e) 2)
the values
the constant
> -
sufficiently
ratio
1 . Since
6 we thus arrive
When
ei,6i
of which vary between
h/(As) 2
Uh(S,t)
4.5
in such a way that
on
~(e)
2 + O(e 4 )
it
follows
of theorem 6 is ful-
at the following
i and (4.15) holds
of the finite-difference
scheme
(4.2)
q ~ i .
NUMERICAL ILLUSTRATION
We consider the following
= 0,
= 1-K(s,t)'e
are taken from table
for each integer
- f(s)
~(e)
small the condition
then the solution
(4.13)
(see e.g.
g2 ( s ' t )
By virtue of theorem
(see (4.9))
s
to the first and second
- gl (s't)
16 K ( s , t ) / 3 }
Conclusion.
to
of theorem
. By choosing
< ~-inf
8 G
{1 -
approximations
initial value problem of type (4.1):
(4.16.a)
U(s,0)
-~ < s < ~,
(4.16.b)
-3.{1 + [Us(s,t)]2}-Ut(s,t)
+ 2Uss(S,t)
+ cos
[Uss(S,t) ] - R(s,t)
= 0,
-~ < s < ~, 0 < t < T = i
where
f(s) = exp
R(s,t)
= {-7
[-s2],
+ 8s 2 -
It is easily verified
12s2"exp
that
[2t
U(s,t)
-
2s 2 ] ) ' e x p
= exp
~
~
- s 2]
- s 2]
+ cos
satisfies
{(2
- 4s2)'e×p
(4.16).
[t
- s2]}
261
The initial value problem (4.16) has been solved numerically by the method
(4.2) with
ai, Bi
from table 1 and with
h/(As) 2 : 0.37.
It can be verified (without any knowledge about the true solution
U ) that (4.15)
is satisfied here. Hence the conclusion at the end of section 4.4 applies to the
numerical approximations
Uh(S,t)
obtained and (4.13) thus holds.
1/16
2.86062
1/32
2.78461
2.70861
1/64
2.74953
2.71446
2.71640
1/128
2.73335
2.71716
2.71807
Table 2.
Approximations
to
2.71830
U(0,1) = 2.71828
In table 2 we have listed the approximations
(s,t) : (0,i)
for various values of
Ux
(x = 21h; i = 0,i, ..... j)
obtained at the point
h . The approximations
table 2 have been obtained by fitting a polynomial
data
uh
and setting
P.(x)
]
~
(j = 1,2,3)
of degree
u~ = Pj(O)
j
in
with the
(by performing Neville
extrapolation [8, p. 208]). The expansion (4.13) (with (s,t) : (0,I)) can be shown
to imply that
~
- U(O,I) = O(h j+l)
(h÷O)
(see e.g. [8, p. 240]). The results in
table 2 (and further results not listed here) are in accordance with these considerations and confirm the theory described above.
Remarks I.
By applying (4.13) with
under the conditions of theorem 6 not only
also each finite-difference quotient of
partial derivative of
2.
U(s,t)
(for
q
high enough it follows easily that
Uh(S,t)
uh(s,t)
tends towards
U(s,t) , but
tends towards the corresponding
h÷O) .
In order to relax the smoothness requirements (A), (B), (C)
and still to get a result similar to (4.13) it might be appropriate not to deal with
condition III, but instead with I I l ,
or to prove directly that the results i, 2 of
262
theorem
3 hold.
boundary
A similar remark
value problems
3.
m = a+2~:2
for nonlinear
In the example
(see definition
consequently
applies
a combination
to the more difficult
parabolic
treated
3 and theorem
of theorem
Acknowledgements.
2). Hence
offering me the opportunity
Dalhousie
University
of
following
Uh(S,t)
to Professor
to work a few weeks
at the Department
the computations
IBM 360 - 65 of Leiden University.
reported
here and
definition
towards
H. Brunner
where part of th~s paper was written.
Mr. C. den Heyer who performed
p = i, c : 2, a = 0,
(3.8) is violated
2 with the argument
I am indebted
of initial-
equations.
above we have
would not even yield a proof of mere convergence
treatment
U(s,t)
3
(h÷0)
from Halifax
of Mathematics
I also wish to thank
in section
4.5 on the
for
of
263
REFERENCES
[i]
ANSORGE, R.:
Konvergenz yon Differenzenverfahren f~r quasilineare Anfangs-
wertaufgaben. Numer. Math. 13, 217 - 225 (1969).
[2]
ANSORGE, R., HASS, R.:
Konvergenz yon Differenzenverfahren f{ir lineare und
nichtlineare Anfangswertaufgaben. Lecture Notes in Mathematics 159.
Berlin: Springer-Verlag 1970.
[3]
COLLATZ, L.:
The numerical treatment of differential equations. Berlin:
Springer-Verlag 1960.
[~
COLLATZ, L.:
Funktionalanalysis und numerische Mathematik. Berlin: Springer-
Verlag 1964.
[5]
GODUNOV, S.K., RYABENKI, V.S.: Theory of difference schemes. Amsterdam: NorthHolland publishing company 1964.
[6]
HASS, R.:
Stabilit[t und Konvergenz von Differenzverfahren f{ir halblineare
Probleme. Thesis, Hamburg University 1971.
[7]
HENRICI, P.:
Discrete variable methods in ordinary differential equations.
New York: J. Wiley & Sons 1962.
[8]
HENRICI, P.:
[9]
JOHN, F.:
Elements of numerical analysis. New York: J. Wiley & Sons 1964.
On integration of parabolic equations by difference methods. Comm.
Pure Appl. Math. 5, 155 - 211 (1952).
[i0] KANTOROVICH, L.V., Akilov, G.P.:
Functional analysis in normed spaces. Oxford:
Pergamon Press 1964.
[Ii] KINNEBROCK, W.:
Stabilit[t und Konvergenz nichtlinearer Differenzenoperatoren
bei Anfangswertaufgaben. Report, Gesellschaft f~r Kernforshung mbH Karlsruhe 1971.
[12] RICHTMYER, R.D., MORTON, K.W.:
Difference methods for initial-value problems.
New York: John Wiley & Sons 1967.
[13] SPIJKER, M.N.:
Stability and convergence of finite-difference methods. Thesis,
Leiden University 1968.
[14] STETTER, H.J.:
Asymptotic expansions for the error of discretization algorithms
for nonlinear functional equations. Numer. Math. 7, 18 - 31 (1965).
264
[15] STETTER, H.J.:
Stability of nonlinear discretization algorithms. In: Numerical
solution of partial differential equations, ed. J.H. Bramble. New York:
Academic Press 1966.
[16] STRANG, W.G.:
Difference methods for mixed boundary-value problems. Duke Math.
J. 27, 221 - 231 (1960).
[17] STRANG, G.:
Accurate partial difference methods II, Non-linear problems.
Numer. Math. 6, 37 - 46 (1964).