Atkinson K. An Introduction to Numerical Analysis
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BOUNDARY
VALUE
PROBLEMS 441
Finite difference methods
We
consider the two-point
BVP
y"
=
f(x,
y,
y')
a<x<b
y(a)=yl
y(b)=y2
(6.11.25)
with the true solution denoted by
Y(x).
Let n >
1,
h =
(b-
a)jn,
xj
=a+
jh
for j =
0,
1,
...
, n. At each interior node point
X;,
0 < i < n, approximate
Y"(x;)
and Y'(x;):
{6.11.26)
for some
X;_
1
.s;
E;,
1J;
.s;
xi+l• i =
1,
...
, n - 1. Dropping the
final
error terms
and using these approximations in the differential equation,
we
have the ap-
proximating nonlinear system:
i = 1,
...
,
n-
1 (6.11.27)
This
is
a system
of
n-
1 nonlinear equations in
then
- 1 unknowns y
1
,
•••
, Yn-l·
The values y
0
= y
1
and
Yn
= y
2
are known from the boundary conditions.
The analysis of the error in {
Y;}
compared to {
Y(
x;)}
is
too complicated to be
given here,
as
it requires the methods of analyzing the solvability of systems
of
nonlinear equations. In essence, if
Y(x)
is
four times differentiable, if the
problem (6.11.25) is uniquely solvable for some region about the graph on
[a, b]
of
Y(x
), and if
f(x,
u,
v)
is
sufficiently differentiable, then there
is
a solution to
(6.11.27) and it satisfies
(6.11.28)
For an analysis, see Keller
(1976,
Sec.
3.2)
or Keller (1968,
Sec.
3.2). Moreover,
with additional assumptions on
f and the smoothness of
Y,
it can be shown that
{6.11.29)
with
T(x)
independent of h. This can be used to justify Richardson extrapola-
tion, to obtain results that converge more rapidly.
[Other methods to improve
convergence are based on correcting for the error in the central difference
approximations of (6.11.27).]
The system (6.11.27) can be solved in a variety of ways, some of which are
simple modifications of the methods described in Sections
2.10 and 2.11. In
!
\
.......
.!
442 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS
matrix form,
we
have
-2
1
0
1
-2
0
1
(
Y2-
Yo)
I
X1,
Y1•
2h
(
Y3
- Y1)
f
X2,
Y2•
2
h
1
0
(
Yn-
Yn-2)
I xn-1• Yn-1• 2h
which
we
denote by
-2
1
1 A
h2Ay = f(y) + g
0
Y1
Y2
1
-2
Yn-1
'Y1
h2
0
(6.11.30)
The matrix A
is
nonsingular [see Theorem
8.2,
Chapter
8];
and linear systems
Az
= b are easily solvable,
as
described preceding Theorem
8.2
in Chapter
8.
Newton's method (see Section 2.11) for solving (6.11.30) is given by
(6.11.31)
1
:::;;
i,
j:::;; n - 1
The
Jacobian matrix simplifies considerably because of the special form of f(y):
at(
. .
Yi+1
- Y1-1)
x,
y,, 2h
[F(y)]
ij
= a
Yj
This
is
zero unless j =
i-
1,
i, or i +
1:
[ (
)]
(
Yi+1-Yi-1)
F y
ii
=
/2
X;,
Y;,
2h
-1
(
Y1+1...,.
Y;-1)
[F(y
)L,i-1 = 2h
/3
X;,
Y;,
2h
· 1 (
Yi+1-Yi-1)
[F(y
)L./+1 = 2h/3
X;,
Y;,
2h
-·.
·---····-·-·-
-·-
----
------
. --··--·-- - --- -----·····--- ---····-
--:
BOUNDARY VALUE PROBLEMS
443
with f
2
(x,
u, u) and f
3
(x,
u,
v) denoting partial derivatives with respect to u and
v, respectively. Thus the matrix being inverted in (6.11.31)
is
of the special form
we call
tridiagonal. Letting
(6.11.32)
we can rewrite (6.11.31) as
y(m+l)
=
y(m)
_
8(m)
(6.11.33)
This linear system is easily and rapidly solvable, as shown in Section 8.2
of
Chapter
8.
The
number
of
multiplications and divisions can be shown to about
5n, a relatively small number of operations for solving a linear system of n - 1
equations. Additional savings can be made by
not
varying
B,
or by only
changing
it
after several iterations
of
(6.11.33).
For
an extensive survey and
discussion
of
the solution of nonlinear systems that arise in connection with
solving
BVPs, see Deuflhard (1979).
Example
We applied the preceding finite difference procedure (6.11.27) to the
solution
of
the BVP (6.11.22), used earlier to illustrate the shooting method. The
results are given in Table
6.30 for successive doublings of n =
2jh.
The nonlin-
ear
system in (6.11.27) was solved using Newton's method,
as
described in
(6.11.33).
The
initial guess was
y~O>(x;)
=
(e
+
e-1)-1
i =
0,
1,
...
, n
based
on
connecting the boundary values
by
a straight line. The quantity
dh
=
M~
IY/m+l)-
y/m)l
o:s;,:s;n
was computed for each iterate, and when the condition
Table
6.30 The finite difference method
for solving (6.11.22)
2
n=-
Eh
Ratio
h
4
.2.63E-
2
8
·s.87E-
3
4.48
16
1.43E-
3 4.11
32
3.55E-
4
4.03
64
8.86E-
5
4.01
444 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS
was satisfied, the iteration
was
terminated. In all cases, the number of iterates
computed was 5 or
6.
For the error, let
with
Yh
the solution of (6.11.27) obtained with Newton's method. According to
(6.11.28) and (6.11.29),
we
should expect the values
Eh
to decrease by about a
factor of 4 when
h
is
halved, and that
is
what
we
observe in the table.
Higher order methods can be obtained in several ways. (1)
Use higher order
approximations to the derivatives, improving (6.11.26); (2) use Richardson ex-
trapolation, based on (6.11.29); as with Romberg integration,
it
can be repeated
to obtain methods of arbitrarily high order; (3) the truncation errors in (6.11.26)
can be approximated with higher order differences using the calculated values of
Yh·
Using these values as corrections to (6.11.27),
we
can obtain a new more
accurate approximation to the differential equation
in
(6.11.25), leading to a more
accurate solution.
All
of these techniques have been used, and some have been
implemented as quite sophisticated computer codes. For a further discussion and
for examples of computer codes, see Fox
(1980, p. 191), Jain (1984, Chap.
4),
and
Pereyra (1979).
Other methods and problems There are a number of other methods used for
solving boundary value problems. The most important of these
is
probably the
collocation method. For discussions referring to collocation methods, see Reddien
(1979), Deuflhard
(1979),
and Ascher and Russell (1985). For an important
collocation computer code, see Ascher et al. (1981a) and (1981b).
Another approach to solving a boundary value problem
is
to solve an
equivalent reformulation as an integral equation. There
is
much less development
of such numerical methods, although they can be very effective in some situa-
tions. For an introduction to this approach, see Keller (1968, Chap. 4).
There are also many other types of boundary value problems,
some containing
some type of these singular behavior, that
we
have not discussed here. For
all
of
these, see the papers in the proceedings of Ascher and Russell (1985), Aziz
(1975), Childs et
al.
(1979), and Gladwell- and Sayers (1980); also see Keller
(1976, Chap. 4) for singular problems. For discussions of software, see Childs
et
al.
(1979), Gladwell and Sayers (1980), and Enright (1985).
Discussion
of
the Literature
Ordinary and partial differential equations are the principal form of mathemati-
cal model occurring in the sciences and engineering,
and
consequently, the
·numerical solution of differential equations
is
a very large area of study. Two
classical
books that reflect the state of knowledge before the widespread use of
digital computers are Collatz (1966) and Milne (1953). Some important and
\
·-··-·- -------- ---------
----
___________
..
___________________________________
,
________
-
-\
DISCUSSION
OF
THE
LITERATURE
445
general books, since 1960,
in
the numerical solution of ordinary differential
equations are Henrici (1962), Gear
(1971),
Lapidus and Seinfeld (1971), Lambert
(1973),
Stetter (1973), Hall and Watt (1976), Shampine and Gordon (1975),
Vander
Houwen (1977), Ortega and Poole (1981), and Butcher (1987). A useful
survey is given in Gupta et
al.
(1985).
The modern theory of convergence and stability of multistep methods, intro-
duced
in
Section 6.8, dates from Dahlquist (1956). An historical account
is
given
in Dahlquist (1985). The text by Henrici (1962) has become a classic account of
that theory, including extensions and applications of it. Gear (1971) is a more
modern account of all methods, especially variable order methods.
Stetter (1973)
gives a very general and complete abstract analysis of the numerical theory for
. solving initial value problems. A complete account up to
1970 of
Runge-Kutta
methods, their development and error analysis,
is
given in Lapidus and Seinfeld
(1971). Hall and Watt (1976)
gives
a survey of all aspects of the solution of
ordinary differential equations, including the many special topics that have
become of greater interest in the past ten years.
The first significant use of the concept of a variable order method is due to
Gear (1971) and Krogh (1969).
Such methods are superior to a fixed-order
multistep method in efficiency, and they do not require any additional method
for starting the integration or for changing the stepsize. A very good account of
the variable-order Adams method
is
given
in
Shampine and Gordon (1975)
and the excellent code
DE/STEP
is
included. Other important early codes based
on the Adams family of formulas
were
those in Krogh (1969), DIFSUB from
Gear (1971), and GEAR from Hindmarsh (1974). The latter program
GEAR
has
been further developed into a large multifunction package, called ODEPACK,
and it
is
described
in
Hindmarsh (1983). Variants of these codes and other
differential equation solvers are available in the
IMSL and NAG libraries.
Runge-Kutta
methods are a continuing active area of theoretical research and
program development, and a very general development
is
given in Butcher (1987).
New methods are being developed for nonstiff problems; for example, see
Shampine (1986) and Shampine and Baca (1986). There
is
also great interest in
implicit
Runge-Kutta
methods, for
use
in solving stiff differential equations.
For
a survey of the latter, see Aiken
(1985,
pp. 70-92). An important competitor to
the code RKF45
is
the code DVERK described in Hull et al. (1976).
It
is
based
on a Fehlberg-type scheme, with a pair of formulas of orders
5 and
6.
A third class of methods has been ignored in our presentation, those based on
extrapolation.
Current work in this area began with Gragg (1965) and Bulirsch
and
Stoer (1966). The main idea
is
to perform repeated extrapolation on some
simple method, to obtain methods of increasingly higher order. In effect, this
gives another way to produce variable-order methods. These methods have
performed fairly
well
in the tests of Enright and Hull (1976) and Shampine et al.
(1976),
but
they were judged to not be as advanced in their practical and
theoretical development as are the multistep
and
Runge-Kutta methods.
For
a
recent survey of the area, see Deuflhard (1985). Also, see Shampine and Baca
(1986), in which extrapolation methods are discussed as one example of variable
order
Runge-Kutta
methods.
Global error estimation
is
an area in which comparatively little has been
published.
For
a general survey,
see
Skeel (1986). To our knowledge, the only
'
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446 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS
running computer code
is
GERK
from Shampine and Watts (1976a). Additional
work
is
needed
in
this area. Many users of automatic packages are under the
mistaken impression that the automatic codes they are using are controlling the
global error, but such global error control
is
not possible in a practical sense. In
many cases, it would seem to be important to have some idea of the actual
magnitude of the global error present in the numerical solution.
Boundary value problems for ordinary differential equations are another
important topic; but both their general theory and their numerical analysis are
much more sophisticated than for the initial value problem. Important texts are
Keller (1968) and (1976), and the proceedings
of
Child et
al.
(1979) gives many
important papers on producing computer codes. For additional papers, see the
collections of Aziz (1975), Hall and Watt (1976), and Ascher and Russell (1985).
This area
is
still comparatively young relative to the initial value problem for
nonstiff equations. The development of computer codes
is
proceeding in a
number of directions, and some quite good codes have been produced in recent
years. More work has been done on codes using the shooting method, but there
are also excellent codes being produced for collocation and finite difference
methods. For some discussion of such codes, see Enright (1985) and Gladwell
and
Sayers (1980, pp. 273-303). Some boundary value codes are given in the
IMSL and NAG libraries.
Stiff differential equations
is
one of several special areas that have become
much more important in the past ten years. The best general survey of this area
is
given in Aiken
(1985).
It
gives
examples of how such problems arise, the theory
of numerical methods for solving stiff problems, and a survey of computer codes
that exist for their solution. Many of the other texts in our bibliography also
address the problem of stiff differential equations. We also recommend the paper
of Shampine and Gear (1979).
Equations with a highly oscillatory solution occur in a number of applications.
For
some discussion of this, see Aiken (1985, pp. 111-123). The method of lines
for solving time-dependent partial differential equations is a classical procedure
that has become more popular in recent years.
It
is discussed in Aiken (1985, pp.
124-138), Sincovec and Madsen (1975), and Melgaard and Sincovec (1981).
Yet another area of interest
is
the solution of mixed systems of differential and
algebraic equations (DAEs). This refers to systems in which there are
n un-
knowns, m
< n differential equations, and n - m algebraic equations, involving
the
n unknown functions. Such problems occur in many areas of applications.
One such area of much interest in recent years
is
that of computer aided design
(CAD).
For
papers applicable to such problems and to other DAEs, see
Rheinboldt (1984) and (1986).
Because of the creation of a number of automatic programs for solving
differential equations, several empirical studies have been made to assess their
I
performance and to make comparisons between programs. Some of the major
comparisons are given in Enright and Hull (1976), Enright et
al.
(1975), and
Shampine et al. (1976).
It
is
clear from their work that programs must be
compared, as
well
as methods. Different program implementations of the same
method can vary widely in their performance.
No
similar results are known for
comparisons of boundary value codes.
i
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BIBLIOGRAPHY 447
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Problems
1.
Draw the direction
field
for
y'
=
x-
y
2
,
and then draw in some sample
solution curves. Attempt to guess the behavior of the solutions
Y(x)
as
X~
00.
2.
Determine Lipschitz constants for the following functions,
as
in (6.1.2).
(a)
f(x,
y)
=
2yjx,
x
~
1
(b)
f(x,
y)
=
tan-
1
(y)
(c)
f(x,
y)
=
(x
3
-
2)
27
/(17x
2
+ 4)
(d)
f(x,
y)
= x - y
2
,
IYI
~
10
3. Convert the following problems to first-order systems.
(a)
y"
-
3y'
+
2y
=
0,
y(O) = 1, y'(O) = 1