ode 9
TRANSCRIPT
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C h a p t e r 9
C o l l o c a t i o n M e t h o d s
9 . 1 I n t r o d u c t i o n
L e t ' s c o n t i n u e t h e d i s c u s s i o n o f c o l l o c a t i o n m e t h o d s w i t h t h e s e c o n d - o r d e r l i n e a r B V P
L y = y
0 0
+ p ( x ) y
0
+ q ( x ) y = r ( x ) a < x < b ( 9 . 1 . 1 a )
y ( a ) = A y ( b ) = B : ( 9 . 1 . 1 b )
T o p i c k u p w h e r e w e l e f t o i n S e c t i o n 6 . 4 , c o n s i d e r a n a p p r o x i m a t e s o l u t i o n
Y ( x ) =
N
X
i = 0
c
i
3
i
( x ) + d
i
!
3
i
( x ) ] ( 9 . 1 . 2 a )
i n v o l v i n g t h e c u b i c H e r m i t e b a s i s
3
i
( x ) =
8
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a = x x x1 i-1 N 0
x = b
1
ii-1
i-1
i
xi
3
33
3
F i g u r e 9 . 1 . 1 : G e o m e t r y f o r t h e c o l l o c a t i o n s o l u t i o n o f ( 9 . 1 . 1 ) s h o w i n g t h e r e s t r i c t i o n o f
t h e c u b i c H e r m i t e p o l y n o m i a l b a s i s t o t h e s u b i n t e r v a l x
i ; 1
x
i
] .
I n o r d e r t o o b t a i n a p p r o x i m a t e s o l u t i o n s o f ( 9 . 1 . 1 ) u s i n g ( 9 . 1 . 2 ) , w e c o l l o c a t e a t t w o
p o i n t s
i 1
,
i 2
p e r s u b i n t e r v a l a n d s a t i s f y t h e b o u n d a r y c o n d i t i o n s . T h u s , t h e u n k n o w n
c o e c i e n t s c
i
, d
i
, i = 0 1 : : : N , a r e d e t e r m i n e d a s t h e s o l u t i o n o f
L Y (
i j
) = r (
i j
) j = 1 2 i = 1 2 : : : N ( 9 . 1 . 3 a )
Y ( a ) = A Y ( b ) = B : ( 9 . 1 . 3 b )
T h e r e s t r i c t i o n o f ( 9 . 1 . 2 a ) t o t h e s u b i n t e r v a l x
i ; 1
x
i
) i s
Y ( x ) = c
i ; 1
3
i ; 1
( x ) + d
i ; 1
!
3
i ; 1
( x ) + c
i
3
i
( x ) + d
i
!
3
i
( x ) : ( 9 . 1 . 4 )
S u b s t i t u t i n g ( 9 . 1 . 4 ) i n t o ( 9 . 1 . 3 a )
L Y (
i 1
)
LY (
i 2
)
= L
i
c
i ; 1
d
i ; 1
+ R
i
c
i
d
i
=
r (
i 1
)
r (
i 2
)
i = 1 2 : : : N ( 9 . 1 . 5 a )
w h e r e
L
i
=
L
3
i ; 1
(
i 1
) L !
3
i ; 1
(
i 1
)
L
3
i ; 1
(
i 2
)
L!
3
i ; 1
(
i 2
)
R
i
=
L
3
i
(
i 1
) L !
3
i
(
i 1
)
L
3
i
(
i 2
)
L!
3
i
(
i 2
)
: ( 9 . 1 . 5 b )
W i t h ( 9 . 1 . 4 ) , t h e b o u n d a r y c o n d i t i o n s ( 9 . 1 . 3 b ) b e c o m e
L
0
c
0
d
0
= A R
N + 1
c
N
d
N
= B ( 9 . 1 . 6 a )
2
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w h e r e
L
0
= R
N + 1
= 1 0 ] : ( 9 . 1 . 6 b )
C o m b i n i n g ( 9 . 1 . 5 ) a n d ( 9 . 1 . 6 ) , w e n d
2
6
6
6
6
6
4
L
0
L
1
R
1
.
.
.
.
.
.
L
N
R
N
R
N + 1
3
7
7
7
7
7
5
2
6
6
6
6
6
6
6
6
6
4
c
0
d
0
c
1
d
1
.
.
.
c
N
d
N
3
7
7
7
7
7
7
7
7
7
5
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
4
A
r (
1 1
)
r (
1 2
)
r (
2 1
)
r (
2 2
)
.
.
.
r (
N 1
)
r (
N 2
)
B
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
: ( 9 . 1 . 7 )
T h u s , t h e 2 ( N + 1 ) c o e c i e n t s c
i
, d
i
, i = 0 1 : : : N , a r e d e t e r m i n e d a s t h e s o l u t i o n o f a
b l o c k b i d i a g o n a l m a t r i x o f d i m e n s i o n 2 N w i t h 2 2 b l o c k s . T h i s s y s t e m m a y b e s o l v e d
b y t h e m e t h o d s o f S e c t i o n 8 . 5 .
W e c a n n o w a s k i f t h e r e i s a n o p t i m a l p l a c e m e n t o f t h e t w o c o l l o c a t i o n p o i n t s o n e a c h
s u b i n t e r v a l t h a t , e . g . , m i n i m i z e s t h e d i s c r e t i z a t i o n e r r o r y ( x ) ; Y ( x ) i n s o m e n o r m . T h i s
q u e s t i o n w a s a n s w e r e d i n a l a n d m a r k p a p e r b y d e B o o r a n d S w a r t z 3 ] a n d w e w i l l f o l l o w
t h e i r a n a l y s i s .
C o n s i d e r t h e i n n e r p r o d u c t
( v u ) =
Z
b
a
v ( x ) u ( x ) d x : ( 9 . 1 . 8 )
L e t u s a s s u m e t h a t u ( x ) a n d v ( x ) a r e s m o o t h e x c e p t , p e r h a p s f o r j u m p d i s c o n t i n u i t i e s
a t z
j
, j = 1 2 : : : M
;1 . A l s o l e t z
0
= a a n d z
M
= b . C o n s i d e r
( v L u ) =
Z
b
a
v u
0 0
+ p u
0
+ q u ] d x ( 9 . 1 . 9 )
w h e r e t h e i n t e g r a l i s i n t e r p r e t e d a s a s u m o f i n t e g r a l s o v e r t h e s u b i n t e r v a l s ( z
0
z
1
) ,
( z
1
z
2
) , : : : , ( z
M ; 1
z
M
) . F o r s i m p l i c i t y , w e ' l l a l s o a s s u m e t h a t A = B = 0 a n d t h a t u ( x )
a n d v ( x ) s a t i s f y t h e s e c o n d i t i o n s . U s i n g ( 9 . 1 . 1 ) , w e i n t e g r a t e ( 9 . 1 . 9 ) b y p a r t s t o o b t a i n
( v
LU ) =
Z
b
a
;v
0
u
0
;( p v )
0
u + q v u ] d x +
M
X
j = 1
fv ( x ) u
0
( x ) + p ( x ) v ( x ) u ( x )
g
z
j
z
j ; 1
:
3
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I n t e g r a t i n g t h e r s t t e r m i n t h e i n t e g r a n d b y p a r t s o n c e m o r e
( v L U ) =
Z
b
a
v
0 0
; ( p v )
0
+ q v ] u d x +
M
X
j = 1
f v ( x ) u
0
( x ) ; v
0
( x ) u ( x ) + p ( x ) v ( x ) u ( x ) g
z
j
z
j ; 1
:
W e c a n w r i t e t h i s r e s u l t i n a s i m p l e r f o r m b y u s i n g t h e i n n e r p r o d u c t n o t a t i o n ( 9 . 1 . 8 )
a n d b y d e n i n g t h e j u m p i n a f u n c t i o n q ( x ) a t a p o i n t z a s
q ( x ) ]
x = z
= l i m
! 0
q ( z + ) ; q ( z ; ) : ( 9 . 1 . 1 0 )
W i t h t h i s n o t a t i o n , w e h a v e
( v L u ) = ( L
T
v u ) ;
M ; 1
X
j = 1
v u
0
; v
0
u + p v u ]
z
j
( 9 . 1 . 1 1 a )
w h e r e L
T
i s c a l l e d t h e a d j o i n t o p e r a t o r a n d s a t i s e s
L
T
v = v
0 0
; ( p v )
0
+ q v : ( 9 . 1 . 1 1 b )
L e t u s s i m p l i f y m a t t e r s s o m e w h a t b y a s s u m i n g t h a t u 2 C
1
( a b ) a n d v 2 C
0
( a b ) .
T h e n , ( 9 . 1 . 1 1 a ) b e c o m e s
M ; 1
X
j = 1
v
0
u ]
z
j
= ( v L u ) ; ( L
T
v u ) : ( 9 . 1 . 1 1 c )
D e n i t i o n 9 . 1 . 1 . T h e G r e e n ' s f u n c t i o n G ( x ) f o r t h e o p e r a t o r L o f ( 9 . 1 . 1 ) s a t i s e s
G ( x ) 2 C
0
( a b ) ( a b ) ( 9 . 1 . 1 2 a )
L
T
G ( x ) = 0 ( a
;
) (
+
b )
= l i m
! 0
( 9 . 1 . 1 2 b )
G ( a ) = G ( b ) = 0 ( 9 . 1 . 1 2 c )
G
x
(
+
) ; G
x
(
;
) = 1 : ( 9 . 1 . 1 2 d )
W h e n v i e w e d a s a f u n c t i o n o f x , t h e G r e e n ' s f u n c t i o n h a s a u n i t j u m p i n i t s r s t
d e r i v a t i v e a t t h e p o i n t .
4
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N o w , i f w e c h o o s e v ( x ) i n ( 9 . 1 . 1 1 c ) a s t h e G r e e n ' s f u n c t i o n G ( x ) s o t h a t t h e o n l y
( M = 2 ) d i s c o n t i n u i t y o c c u r s a t z
1
= , w e h a v e
u ( ) = ( G ( ) L u ) : ( 9 . 1 . 1 3 a )
I f u ( x ) i s c h o s e n a s y ( x ) , t h e s o l u t i o n o f ( 9 . 1 . 1 ) , t h e n
y ( ) = ( G ( ) L y ) = ( G ( ) r ) : ( 9 . 1 . 1 3 b )
T h e r e l a t i o n ( 9 . 1 . 1 3 a ) h o l d s f o r a n y s m o o t h f u n c t i o n u ( x ) a n d n o t j u s t t h e s o l u t i o n o f
( 9 . 1 . 1 ) . S i n c e , f o r e x a m p l e , t h e c o l l o c a t i o n s o l u t i o n Y ( x ) 2 C
1
( a b ) , w e c a n r e p l a c e u i n
( 9 . 1 . 1 3 a ) b y Y t o o b t a i n
Y ( ) = ( G ( ) L Y ) : ( 9 . 1 . 1 3 c )
F i n a l l y , l e t t i n g
e ( x ) = y ( x ) ; Y ( x ) ( 9 . 1 . 1 4 a )
d e n o t e t h e d i s c r e t i z a t i o n e r r o r o f t h e c o l l o c a t i o n s o l u t i o n , w e s u b t r a c t ( 9 . 1 . 1 3 c ) f r o m
( 9 . 1 . 1 3 b ) t o o b t a i n
e ( ) = ( G ( ) L e ) =
Z
b
a
G ( x ) L e ( x ) d x : ( 9 . 1 . 1 4 b )
R e m a r k 1 . E a c h r e s u l t ( 9 . 1 . 1 3 b ) , ( 9 . 1 . 1 3 c ) , o r ( 9 . 1 . 1 4 b ) r e l a t e s a g l o b a l q u a n t i t y
( y Y e ) t o i t s l o c a l c o u n t e r p a r t ( L y L Y L e ) t h r o u g h t h e G r e e n ' s f u n c t i o n .
L e t u s w r i t e ( 9 . 1 . 1 4 b ) i n t h e m o r e e x p l i c i t f o r m
e ( ) =
N
X
i = 1
Z
x
i
x
i ; 1
G ( x ) L e ( x ) d x =
N
X
i = 1
e
i
( ) : ( 9 . 1 . 1 5 )
S u p p o s e t h a t =2 ( x
i ; 1
x
i
) s o t h a t G ( x ) i s s m o o t h f o r x 2 ( x
i ; 1
x
i
) . W r i t e
L e = L ( y ; Y ) = L y ; P r + P r ; L Y ( 9 . 1 . 1 6 a )
w h e r e P r i s a l i n e a r p o l y n o m i a l f o r x 2 ( x
i ; 1
x
i
) t h a t i n t e r p o l a t e s b o t h L y a n d L Y a t
t h e t w o c o l l o c a t i o n p o i n t s
i 1
a n d
i 2
o n t h i s s u b i n t e r v a l . T h u s ,
P r = r (
i 1
)
x ;
i 2
i 1
;
i 2
+ r (
i 2
)
x ;
i 1
i 2
;
i 1
: ( 9 . 1 . 1 6 b )
5
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Y
y
Pr
L
L
xi-1
xii,1 i,2
F i g u r e 9 . 1 . 2 : F u n c t i o n s L y , L Y , a n d P r o n a s u b i n t e r v a l ( x
i ; 1
x
i
) n o t c o n t a i n i n g t h e
p o i n t x = .
T h e f u n c t i o n s L y , L Y , a n d P r a r e i l l u s t r a t e d i n F i g u r e 9 . 1 . 2 . T h e d i e r e n c e s L y ; P r
a n d L Y ; P r c a n b e e s t i m a t e d u s i n g f o r m u l a s f o r t h e e r r o r i n l i n e a r i n t e r p o l a t i o n 2 ] a s
Ly
;P r =
1
2
( x
;
i 1
) ( x
;
i 2
) (
Ly )
0 0
(
i
) ( 9 . 1 . 1 7 a )
L Y ; P r =
1
2
( x ;
i 1
) ( x ;
i 2
) ( L Y )
0 0
(
i
) ( 9 . 1 . 1 7 b )
w h e r e
i
i
2 ( x
i ; 1
x
i
) .
U s i n g ( 9 . 1 . 1 7 ) i n ( 9 . 1 . 1 6 ) y i e l d s
e
i
( ) =
Z
x
i
x
i ; 1
( x ;
i 1
) ( x ;
i 2
) g ( x ) d x = 2 ( x
i ; 1
x
i
) ( 9 . 1 . 1 8 a )
w h e r e
g ( x ) =
1
2
G ( x ) ( L y )
0 0
(
i
) ; ( L Y )
0 0
(
i
) ] : ( 9 . 1 . 1 8 b )
w h e r e e
i
w a s d e n e d i n ( 9 . 1 . 1 5 ) . W e b o u n d ( 9 . 1 . 1 8 a ) a s
j e
i
( ) j
Z
x
i
x
i ; 1
j x ;
i 1
j j x ;
i 2
j j g ( x ) j d x :
S i n c e j x ;
i j
j h
i
, j = 1 2 , w e h a v e
j e
i
( ) j h
3
i
j j g ( ) j j
i 1
=2 ( x
i ; 1
x
i
) ( 9 . 1 . 1 9 a )
6
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xi-1
xii,1 i,2
GLy
GLY
GPr
F i g u r e 9 . 1 . 3 : F u n c t i o n s G L y , G L Y , a n d G P r o n a s u b i n t e r v a l ( x
i ; 1
x
i
) c o n t a i n i n g t h e
p o i n t x = .
w h e r e
j j f ( ) j j
i 1
= m a x
x
i ; 1
x x
i
j f ( x ) j : ( 9 . 1 . 1 9 b )
T y p i c a l l y , o n e s u b i n t e r v a l c o n t a i n s t h e p o i n t w h e r e G
x
i s d i s c o n t i n u o u s . T h e a n a l -
y s i s l e a d i n g t o ( 9 . 1 . 1 9 ) c a n n o t b e u s e d o n t h i s s u b i n t e r v a l s i n c e G ( x ) =2 C
1
( x
i ; 1
x
i
) .
T h e r e a r e s e v e r a l w a y s o f s h o w i n g t h a t j e
i
( ) j i n c r e a s e s f r o m O ( h
3
i
) t o O ( h
2
i
) o n t h i s
s u b i n t e r v a l . W e ' l l c h o o s e o n e w h i c h r e s t r i c t s t o l i e b e t w e e n
i 1
a n d
i 2
. I n t h i s c a s e ,
w e i n t e r p o l a t e G L y a n d G L Y b y p i e c e w i s e c o n s t a n t f u n c t i o n s
G ( x ) P r =
G (
i 1
) r (
i 1
) i f x
i ; 1
x <
G (
i 2
) r (
i 2
) i f x < x
i
( 9 . 1 . 2 0 )
a s s h o w n i n F i g u r e 9 . 1 . 3 . T h e e r r o r i n p i e c e w i s e c o n s t a n t i n t e r p o l a t i o n i s
G
Ly
;G P r =
( x ;
i 1
) ( G L y )
0
(
i 1
) i f x
i ; 1
x <
( x ;
i 2
) ( G L y )
0
(
i 2
) i f x < x
i
:
A s i m i l a r e x p r e s s i o n a p p l i e s f o r G
LY
;G P r . C o m b i n i n g t h e s e r e s u l t s i n t h e m a n n e r
u s e d t o o b t a i n ( 9 . 1 . 1 8 a ) y i e l d s
e
i
( ) =
Z
x
i ; 1
( x
;
i 1
) g ( x ) d x +
Z
x
i
( x
;
i 2
) g ( x ) d x
2( x
i ; 1
x
i
) ( 9 . 1 . 2 1 a )
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w h e r e
g ( x ) =
( G L y )
0
(
i 1
) ; ( G L Y )
0
(
i 1
) i f x
i ; 1
x <
( G L y )
0
(
i 2
) ; ( G L Y )
0
(
i 2
) i f x < x
i
: ( 9 . 1 . 2 1 b )
W e b o u n d ( 9 . 1 . 2 1 a ) a s
j e
i
( ) j
Z
x
i ; 1
j x ;
i 1
j j g ( x ) j d x +
Z
x
i
j x ;
i 2
j j g ( x ) j d x
o r
j e
i
( ) j h
2
i
k g ( ) k
i 1
2 ( x
i ; 1
x
i
) : ( 9 . 1 . 2 1 c )
W e ' l l u s e t h e s y m b o l k g ( ) k
i 1
w i t h t h e u n d e r s t a n d i n g t h a t t h e m a x i m u m i s c o m p u t e d
o n ( x
i ; 1
) ( x
i
) .
F i n a l l y , s u b s t i t u t i n g ( 9 . 1 . 1 9 a ) a n d ( 9 . 1 . 2 1 c ) i n t o ( 9 . 1 . 1 5 ) y i e l d s
j e ( ) j h
2
j
j j g ( ) j j
j 1
+
N
X
i = 1 i 6= j
h
3
i
k g ( ) k
i 1
2 ( x
j ; 1
x
j
)
o r
j e ( ) j h
2
j
+
N
X
i = 1 i 6= j
h
3
i
] k ~g ( ) k
1
2 ( x
j ; 1
x
j
)
w h e r e
k ~g ( ) k
1
= m a x ( k g ( ) k
1
k g ( ) k
1
) k f ( ) k
1
= m a x
1 i N
k f ( ) k
i 1
:
L e t t i n g
h = m a x
1 i N
j h
i
j ( 9 . 1 . 2 2 )
a n d o b s e r v i n g t h a t N h
( b
;a ) , w e h a v e
j e ( ) j h
2
+ ( N ; 1 ) h
3
] k ~g ( ) k
1
h
2
1 + ( b ; a ) ] k ~g ( ) k
1
] 2 ( x
j ; 1
x
j
)
o r
j e ( ) j C h
2
2 ( x
j ; 1
x
j
) ( 9 . 1 . 2 3 )
w h e r e
C = ( 1 + b ; a ) k ~g ( ) k
1
:
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I f = x
j
, j = 0 1 : : : N , t h e n t h e r e i s n o d i s c o n t i n u i t y i n t h e G r e e n ' s f u n c t i o n o n
a n y s u b i n t e r v a l a n d t h e e r r o r i s o b t a i n e d f r o m ( 9 . 1 . 1 9 a ) a n d ( 9 . 1 . 1 5 ) a s
j e ( x
j
) j
N
X
i = 1
h
3
i
k g ( x
j
) k
i 1
j = 0 1 : : : N :
F o l l o w i n g t h e s t e p s l e a d i n g t o ( 9 . 1 . 2 3 ) , w e a g a i n n d t h a t
je ( x
j
)
j C h
2
j = 0 1 : : : N : ( 9 . 1 . 2 4 )
T h u s , t h e g l o b a l a n d p o i n t w i s e e r r o r s a r e b o t h O ( h
2
) . T h i s o c c u r s b e c a u s e o f t h e l o w
p o l y n o m i a l d e g r e e a n d t h e a r b i t r a r y c h o i c e o f t h e c o l l o c a t i o n p o i n t s . W i t h e i t h e r h i g h e r -
d e g r e e p o l y n o m i a l s o r a s p e c i a l c h o i c e o f c o l l o c a t i o n p o i n t s w e c a n r e d u c e t h e p o i n t w i s e
e r r o r r e l a t i v e t o t h e g l o b a l e r r o r . T h i s p h e n o m e n o n i s c a l l e d n o d a l s u p e r c o n v e r g e n c e .
D e n i t i o n 9 . 1 . 2 . N o d a l s u p e r c o n v e r g e n c e i m p l i e s t h a t t h e c o l l o c a t i o n s o l u t i o n o n t h e
m e s h f a = x
0
< x
1
-
8/2/2019 Ode 9
10/25
A s s u m i n g t h a t t h i s w e r e p o s s i b l e , f o r t h e m o m e n t , t h e n ( 9 . 1 . 2 5 ) w o u l d b e c o m e
e
i
( x
j
) =
Z
x
i
x
i ; 1
1
2
( x ;
i 1
) ( x ;
i 2
) g
x x
( x
j
i
) ( x ; x
i ; 1 = 2
)
2
] d x :
W e b o u n d t h i s a s
j e
i
( x
j
) j
1
2
Z
x
i
x
i ; 1
j x ;
i 1
j j x ;
i 2
j j g
x x
( x
j
i
) j j x ; x
i ; 1 = 2
j
2
d x
o r
j e
i
( x
j
) j
1
2
h
5
i
k g
x x
( x
j
) k
i 1
( 9 . 1 . 2 7 )
S u b s t i t u t i n g t h i s r e s u l t i n t o ( 9 . 1 . 1 5 ) y i e l d s
j e ( x
j
) j
1
2
N
X
i = 1
h
5
i
k g
x x
( x
j
) k
i 1
j = 0 1 : : : N
o r
j e ( x
j
) j
1
2
k g
x x
( x
j
) k
1
N
X
i = 1
h
5
i
=
1
2
k g
x x
( x
j
) k
1
N h
5
:
T h u s ,
je ( x
j
)
j C h
4
j = 0 1 : : : N ( 9 . 1 . 2 8 a )
w h e r e
C =
1
2
( b ; a ) k g
x x
( x
j
) k
1
: ( 9 . 1 . 2 8 b )
T h e p o i n t w i s e e r r o r h a s b e e n i n c r e a s e d b y t w o o r d e r s w i t h t h e s p e c i a l c h o i c e o f c o l l o c a t i o n
p o i n t s d i c t a t e d b y ( 9 . 1 . 2 6 ) . T h i s i s m o s t d e n i t e l y a n e x a m p l e o f n o d a l s u p e r c o n v e r g e n c e
s i n c e t h e g l o b a l e r r o r i s s t i l l O ( h
2
) .
I t r e m a i n s t o d e t e r m i n e t h e c o l l o c a t i o n p o i n t s t h a t s a t i s f y ( 9 . 1 . 2 6 ) . L e t u s b e g i n b y
t r a n s f o r m i n g t h e s e i n t e g r a l s t o ; 1 1 ] u s i n g t h e m a p p i n g
x = x
i ; 1 = 2
+
h
i
2
; 1 1 : ( 9 . 1 . 2 9 a )
A l s o l e t
i 1
= x
i ; 1 = 2
;
1
h
i
i 2
= x
i ; 1 = 2
+
2
h
i
: ( 9 . 1 . 2 9 b )
1 0
-
8/2/2019 Ode 9
11/25
T h e n
x ;
i 1
=
h
i
2
( + 2
1
) x ;
i 2
=
h
i
2
( ; 2
2
) : ( 9 . 1 . 2 9 c )
C o n d i t i o n s ( 9 . 1 . 2 6 a , b ) b e c o m e
(
h
i
2
)
3
Z
1
; 1
( + 2
1
) ( ; 2
2
) d = 0 (
h
i
2
)
4
Z
1
; 1
( + 2
1
) ( ; 2
2
) d = 0 :
T h u s , i t s u c e s t o d e t e r m i n e
1
a n d
2
s u c h t h a t
Z
1
; 1
( + 2
1
) ( ; 2
2
) P ( ) d = 0 ( 9 . 1 . 3 0 )
f o r a l l l i n e a r p o l y n o m i a l s P ( ) .
S i n c e t h e i n t e g r a n d i s a c u b i c p o l y n o m i a l i t c a n b e e v a l u a t e d e x a c t l y b y t h e t w o - p o i n t
G a u s s - L e g e n d r e q u a d r a t u r e f o r m u l a ( c f . 5 ] , C h a p t e r 7 )
Z
1
; 1
f ( ) d = f ( ;
1
p
3
) + f (
1
p
3
) + E ( 9 . 1 . 3 1 a )
w h e r e t h e d i s c r e t i z a t i o n e r r o r E i s g i v e n b y
E =
1
1 3 5
f
i v
( ) 2 ( ; 1 1 ) : ( 9 . 1 . 3 1 b )
T h u s , a p p l y i n g ( 9 . 1 . 3 1 ) t o ( 9 . 1 . 3 0 ) , w e h a v e
( ;
1
p
3
+ 2
1
) ( ;
1
p
3
; 2
2
) P ( ;
1
p
3
) + (
1
p
3
+ 2
1
) (
1
p
3
; 2
2
) P (
1
p
3
) = 0 :
O n c e a g a i n , t h e i n t e g r a n d i n ( 9 . 1 . 3 0 ) i s a c u b i c p o l y n o m i a l s o E = 0 . W e s e e t h a t w e
c a n s a t i s f y t h e a b o v e c o n d i t i o n b y c h o o s i n g
1
=
2
=
1
2
p
3
: ( 9 . 1 . 3 2 a )
E x p r e s s e d i n t e r m s o f t h e o r i g i n a l v a r i a b l e s t h r o u g h ( 9 . 1 . 2 9 b ) , t h e c o l l o c a t i o n p o i n t s a r e
i 1
= x
i ; 1 = 2
;
h
i
2
p
3
i 2
= x
i ; 1 = 2
+
h
i
2
p
3
: ( 9 . 1 . 3 2 b )
R e g a r d l e s s o f h o w t h e r e s u l t i s e x p r e s s e d , t h e k e y i s t o p e r f o r m c o l l o c a t i o n a t t h e G a u s s -
L e g e n d r e p o i n t s m a p p e d t o t h e a p p r o p r i a t e i n t e r v a l .
1 1
-
8/2/2019 Ode 9
12/25
N M i d p o i n t R u l e C o l l o c a t i o n
2 0 : 2 0
1 0
; 3
5 0 : 6 4 1 0
; 5
1 0 0 : 1 0 1 0
; 4
0 : 4 6 1 0
; 6
2 0 0 : 2 6 1 0
; 5
0 : 3 3 1 0
; 7
4 0 0 : 6 5 1 0
; 6
0 : 2 3 1 0
; 8
8 0 0 : 1 6 1 0
; 6
0 : 1 6 1 0
; 9
T a b l e 9 . 1 . 1 : M a x i m u m p o i n t w i s e e r r o r s i n t h e s o l u t i o n o f E x a m p l e 9 . 1 . 1 u s i n g t h e m i d -
p o i n t r u l e a n d c o l l o c a t i o n a t t w o G a u s s - L e g e n d r e p o i n t s .
T h e e r r o r f o r m u l a ( 9 . 1 . 3 1 b ) c a n b e u s e d t o o b t a i n a m o r e p r e c i s e e s t i m a t e o f t h e
c o l l o c a t i o n e r r o r t h a n g i v e n b y ( 9 . 1 . 2 8 ) .
L e t u s c o n c l u d e t h i s s e c t i o n w i t h t w o e x a m p l e s .
E x a m p l e 9 . 1 . 1 ( c f . 1 ] , C h a p t e r 5 ) . C o n s i d e r t h e p r o b l e m
y
0 0
+
y
0
x
= (
8
8 ; x
2
)
2
0 < x
-
8/2/2019 Ode 9
13/25
N E r r o r
4 0 : 4 4
1 0
; 4
8 0 : 3 7 1 0
; 4
1 6 0 : 5 8 1 0
; 8
3 2 0 : 6 0 1 0
; 9
6 4 0 : 1 9 1 0
; 1 0
1 2 8 0 : 6 4 1 0
; 1 2
T a b l e 9 . 1 . 2 : M a x i m u m p o i n t w i s e e r r o r s i n t h e s o l u t i o n o f E x a m p l e 9 . 1 . 2 u s i n g c o l l o c a t i o n
a t F l a h e r t y a n d M a t h o n ' s 4 ] p o i n t s .
w h i c h h a s t h e e x a c t s o l u t i o n
y ( x ) = e
; x = 2
; e
; x ( x
2
+ 3 x + 3 ) = 3
:
F o r 0 < 1 t h e r e i s a b o u n d a r y l a y e r o f w i d t h O ( ) a t x = 0 .
C o l l o c a t i o n u s i n g c u b i c p o l y n o m i a l s w a s p e r f o r m e d a t t h e G a u s s - L e g e n d r e p o i n t s a n d
a t t h e p o i n t s
x
i 1
= x
i ; 1 = 2
; h
i
x
i 2
= x
i ; 1 = 2
+ h
i
w h e r e
=
1
2
;
2
i
! (
i
= 2 )
1 +
p
1 ; 4 ! (
i
= 2 ) =
i
i
=
p ( x
k
) ;
q ( x
k
)
p ( x
k
)
k =
i ; 1 i f
p ( x
i ; 1
) + p ( x
i
)
2
-
8/2/2019 Ode 9
14/25
f o r p o l y n o m i a l s P ( ) o f a s h i g h a d e g r e e a s p o s s i b l e .
9 . 2 C o l l o c a t i o n f o r F i r s t - O r d e r S y s t e m s
L e t u s e x t e n d t h e c o l l o c a t i o n m e t h o d s t o r s t - o r d e r v e c t o r B V P s o f t h e u s u a l f o r m
y
0
= f ( x y ) a < x < b ( 9 . 2 . 1 a )
g
L
( y ( a ) ) = g
R
( y ( b ) ) = 0 : ( 9 . 2 . 1 b )
W e ' l l f o l l o w a d i e r e n t a p p r o a c h t h a n t h e o n e u s e d i n S e c t i o n 9 . 1 a n d i n t e g r a t e ( 9 . 2 . 1 a )
o n a s u b i n t e r v a l ( x
i ; 1
x
i
) t o o b t a i n
y ( x ) = y ( x
i ; 1
) +
Z
x
x
i ; 1
y
0
( x ) d x = y ( x
i ; 1
) +
Z
x
x
i ; 1
f ( x y ) d x : ( 9 . 2 . 2 )
A s w i t h i n i t i a l v a l u e m e t h o d s , w e ' l l c o n s t r u c t a n u m e r i c a l m e t h o d b y a p p r o x i m a t i n g y
0
( o r f ) b y a p o l y n o m i a l a n d i n t e g r a t i n g t h e r e s u l t . A n y p o l y n o m i a l b a s i s m a y b e u s e d ,
b u t l e t u s c o n c e n t r a t e o n t h e L a g r a n g e f o r m o f t h e i n t e r p o l a t i n g p o l y n o m i a l
y
0
( x ) =
J
X
k = 1
y
0
(
i k
) L
k
( ) + R ( ) ( 9 . 2 . 3 a )
w h e r e
L
k
( ) =
J
Y
j = 1 j 6= k
;
j
k
;
j
=
( ;
1
) ( ;
2
) : : : ( ;
k ; 1
) ( ;
k + 1
) : : : ( ;
J
)
(
k
;
1
) (
k
;
2
) : : : (
k
;
k ; 1
) (
k
;
k + 1
) : : : (
k
;
J
)
( 9 . 2 . 3 b )
R = y
0
i 1
i 2
: : :
i J
]
J
Y
j = 1
( ;
j
) ( 9 . 2 . 3 c )
x = x
i ; 1
+ h
i
0 1 ( 9 . 2 . 3 d )
i k
= x
i ; 1
+
k
h
i
: ( 9 . 2 . 3 e )
T h e i m a g e o f t h e c o l l o c a t i o n p o i n t s
k
, k = 1 2 : : : J , a r e o r d e r e d s u c h t h a t
0
1
<
2
-
8/2/2019 Ode 9
15/25
T h e d i v i d e d d i e r e n c e y
0
i 1
i 2
: : : :
i J
] w i l l b e d e n e d s h o r t l y .
S u b s t i t u t i n g ( 9 . 2 . 3 a ) i n t o ( 9 . 2 . 2 ) y i e l d s
y ( x ) = y ( x
i ; 1
) + h
i
J
X
k = 1
y
0
(
i k
)
Z
0
L
k
( ) d + h
i
Z
0
R ( ) d : ( 9 . 2 . 4 )
E v a l u a t i n g ( 9 . 2 . 4 ) a t t h e c o l l o c a t i o n p o i n t s
y (
i j
) = y ( x
i ; 1
) + h
i
J
X
k = 1
y
0
(
i k
)
Z
j
0
L
k
( ) d + h
i
Z
j
0
R ( ) d :
L e t
a
j k
=
Z
j
0
L
k
( ) d ( 9 . 2 . 5 a )
a n d
E
j
=
Z
j
0
R ( ) d : ( 9 . 2 . 5 b )
T h e n
y (
i j
) = y ( x
i ; 1
) + h
i
J
X
k = 1
y
0
(
i k
) a
j k
+ h
i
E
j
: ( 9 . 2 . 5 c )
E v a l u a t i n g ( 9 . 2 . 4 ) a t x = x
i
y i e l d s
y ( x
i
) = y ( x
i ; 1
) + h
i
J
X
k = 1
y
0
(
i k
)
Z
1
0
L
k
( ) d + h
i
Z
1
0
R ( ) d :
L e t t i n g
b
k
=
Z
1
0
L
k
( ) d ( 9 . 2 . 6 a )
a n d
E =
Z
1
0
R ( ) d : ( 9 . 2 . 6 b )
w e h a v e
y ( x
i
) = y ( x
i ; 1
) + h
i
J
X
k = 1
y
0
(
i k
) b
k
+ h
i
E : ( 9 . 2 . 6 c )
1 5
-
8/2/2019 Ode 9
16/25
I f t h e e r r o r s E
j
, j = 1 2 : : : J , a n d E a r e n e g l e c t e d , w e s e e t h a t t h e c o l l o c a t i o n
s o l u t i o n i s o b t a i n e d f r o m a J s t a g e i m p l i c i t R u n g e - K u t t a m e t h o d . T h u s , l e t t i n g Y ( x )
d e n o t e t h e a p p r o x i m a t e s o l u t i o n , w e h a v e
Y ( x
i
) = Y ( x
i ; 1
) + h
i
J
X
k = 1
b
k
k
i k
( 9 . 2 . 7 a )
w h e r e
k
i k
= Y
0
(
i k
) = f ( x
i ; 1
+
k
h
i
Y ( x
i ; 1
) + h
i
J
X
j = 1
a
k j
k
i j
) k = 1 2 : : : J : ( 9 . 2 . 7 b )
S o l u t i o n c o n t i n u i t y , r e q u i r e d f o r r s t - o r d e r B V P s ,
y ( x
;
i
) = y ( x
+
i
) i = 1 2 : : : N ; 1
i s a u t o m a t i c a l l y s a t i s e d .
S o l u t i o n s a t a n y p o i n t x
2 x
i ; 1
x
i
] a r e d e n e d b y t h e i n t e r p o l a t i o n p o l y n o m i a l ( 9 . 2 . 4 )
a n d ( 9 . 2 . 3 a ) a s
Y ( x ) = Y ( x
i ; 1
) + h
i
J
X
k = 1
Y
0
(
i k
)
Z
0
L
k
( ) d : ( 9 . 2 . 7 c )
S o m e c o m m o n c h o i c e s o f t h e c o l l o c a t i o n p o i n t s a r e :
1 . G a u s s - L e g e n d r e p o i n t s . T h e p o i n t s
k
, k = 1 2 : : : J , a r e t h e r o o t s o f t h e L e g e n d r e
p o l y n o m i a l o f d e g r e e J m a p p e d t o t h e i n t e r v a l ( 0 1 ) . T h e r o o t s o f t h e L e g e n d r e
p o l y n o m i a l a r e n o r m a l l y p r e s c r i b e d o n (
;1 1 ) . T h i s m a y b e d o n e b y t h e l i n e a r
t r a n s f o r m a t i o n = ( 1 + ) = 2 , 2 ; 1 1 ] . F o r a l l J ,
1
> 0 a n d
J
-
8/2/2019 Ode 9
17/25
J P
J
( )
j
, j = 1 2 : : : J
0 1
1 0
2
1
2
( 3
2
; 1 )
1
p
3
3
2
( 5
2
; 3 ) 0 ,
q
3
5
4
1
8
( 3 5
4
; 3 0
2
+ 3 ) 0 : 3 3 9 9 8 1 0 4 3 6 , 0 : 8 6 1 1 3 6 3 1 1 6
T a b l e 9 . 2 . 1 : L e g e n d r e p o l y n o m i a l s P
J
( ) a n d t h e i r r o o t s f o r J = 0 1 : : : 4 .
w h e r e P
J
( ) i s t h e L e g e n d r e p o l y n o m i a l o f d e g r e e J . T h e n e g a t i v e s i g n i s c h o s e n
w i t h
J
= 1 a n d t h e p o s i t i v e s i g n i s s e l e c t e d w i t h
1
= 0 . T h e s i m p l e s t s c h e m e ( J =
1 ) i s t h e f o r w a r d o r b a c k w a r d E u l e r m e t h o d w h e n
1
= 0 o r
J
= 1 , r e s p e c t i v e l y .
J u d g i n g f r o m o u r e x p e r i e n c e w i t h s t i I V P s , R a d a u s c h e m e s s h o u l d b e s u i t a b l e
f o r s i n g u l a r l y p e r t u r b e d B V P s . T h i s w i l l d e t e r m i n e t h e p r o p e r c h o i c e o f t h e x e d
e n d p o i n t , a s d e s c r i b e d i n C h a p t e r 1 0 . T h e p o i n t w i s e a c c u r a c y o f a J p o i n t R a d a u
s c h e m e i s O ( h
2 J ; 1
) .
3 . L o b a t t o p o i n t s . T h e p o i n t s
1
= 0 ,
J
= 1 , a n d t h e r e m a i n i n g p o i n t s
k
, k =
2 3 : : : J ; 1 , a r e s e l e c t e d a s t h e r o o t s o f t h e L e g e n d r e p o l y n o m i a l o f d e g r e e J ; 2 .
T h e s i m p l e s t s c h e m e ( J = 2 ) i s t h e t r a p e z o i d a l r u l e . T h e p o i n t w i s e a c c u r a c y o f a
J p o i n t s c h e m e i s O ( h
2 ( J ; 1 )
) .
E x a m p l e 9 . 1 . 1 ( 1 ] , C h a p t e r 5 ) . A s i n E x a m p l e 9 . 1 . 1 , c o n s i d e r
y
0 0
+
y
0
x
= (
8
8 ; x
2
)
2
0 < x
-
8/2/2019 Ode 9
18/25
N 2 - P o i n t G a u s s 3 - P o i n t L o b a t t o 3 - P o i n t G a u s s
2 0 : 2 0
1 0
; 3
0 : 1 7
1 0
; 4
0 : 1 4
1 0
; 6
5 0 : 6 4 1 0
; 5
0 : 5 7 1 0
; 6
0 : 7 0 1 0
; 9
1 0 0 : 4 6 1 0
; 6
0 : 3 7 1 0
; 7
0 : 1 3 1 0
; 1 0
2 0 0 : 3 3 1 0
; 7
0 : 2 3 1 0
; 8
0 : 2 7 1 0
; 1 2
4 0 0 : 2 3 1 0
; 8
0 : 1 5 1 0
; 9
0 : 6 0 1 0
; 1 4
8 0 0 : 1 6 1 0
; 9
0 : 9 1 1 0
; 1 1
0 : 1 3 1 0
; 1 4
T a b l e 9 . 2 . 2 : M a x i m u m p o i n t w i s e e r r o r s i n t h e s o l u t i o n o f E x a m p l e 9 . 2 . 1 u s i n g c o l l o c a t i o n
a t t w o G a u s s - L e g e n d r e p o i n t s , t h r e e L o b a t t o p o i n t s , a n d t h r e e G a u s s - L e g e n d r e p o i n t s .
t h r e e - p o i n t G a u s s - L e g e n d r e c o l l o c a t i o n a t
1
=
1
2
( 1 ;
r
3
5
)
2
= 1 = 2
3
=
1
2
( 1 +
r
3
5
) :
T h e G a u s s - L e g e n d r e m e t h o d s d o n o t n e e d f u n c t i o n e v a l u a t i o n s a t t h e e n d p o i n t s o f
s u b i n t e r v a l s a n d , t h u s , t h e s i n g u l a r c o e c i e n t i n t h e d i e r e n t i a l e q u a t i o n a t x = 0
p o s e s n o p r o b l e m . H o w e v e r , t h e L o b a t t o m e t h o d m u s t b e m o d i e d t o a c c o u n t f o r t h e
s i n g u l a r i t y . U s i n g L ' H o p i t a l ' s r u l e ,
l i m
x ! 0
y
0
( x )
x
= y
0 0
( 0 ) :
U s i n g t h i s r e s u l t i n t h e d i e r e n t i a l e q u a t i o n y i e l d s y
0 0
( 0 ) = 1 = 2 t h u s ,
l i m
x ! 0
y
0
( x )
x
=
1
2
:
T h e m a x i m u m p o i n t w i s e e r r o r s i n y f o r t h e t h r e e m e t h o d s a r e s h o w n i n T a b l e 9 . 2 . 2 .
T h e t w o - p o i n t G a u s s - L e g e n d r e , t h r e e - p o i n t L o b a t t o , a n d t h r e e - p o i n t G a u s s - L e g e n d r e a r e
c o n v e r g i n g a t t h e i r e x p e c t e d r a t e s o f O ( N
; 4
) , O ( N
; 4
) , a n d O ( N
; 6
) , r e s p e c t i v e l y .
T h e i m p l e m e n t a t i o n o f t h e c o l l o c a t i o n s c h e m e i s u s u a l l y d o n e b y e l i m i n a t i n g t h e
u n k n o w n s k
i k
, k = 1 2 : : : J , a p p e a r i n g i n ( 9 . 2 . 7 b ) o n e a c h s u b i n t e r v a l a n d t h e n s o l v i n g
f o r t h e n o d a l v a l u e s Y
i
= Y ( x
i
) , i = 0 1 : : : N . W e ' l l i l l u s t r a t e t h i s f o r a l i n e a r s y s t e m
f ( x y ) = A ( x ) y + b ( x ) ( 9 . 2 . 8 a )
g
L
( y ( a ) ) = L y ( a ) ; l g
R
( y ( b ) ) = R y ( b ) ; r : ( 9 . 2 . 8 b )
1 8
-
8/2/2019 Ode 9
19/25
N o n l i n e a r p r o b l e m s a r e s o l v e d u s i n g N e w t o n ' s m e t h o d t o l i n e a r i z e t h e m t o t h e f o r m o f
( 9 . 2 . 8 ) .
F o r a l i n e a r p r o b l e m , ( 9 . 2 . 7 b ) b e c o m e s
k
i k
= A (
i k
) Y
i ; 1
+ h
i
J
X
j = 1
a
k j
k
i j
] + b (
i k
) k = 1 2 : : : J : ( 9 . 2 . 9 a )
T h i s s y s t e m c a n b e w r i t t e n i n m a t r i x f o r m a s
W
i
k
i
= V
i
Y
i ; 1
+ q
i
( 9 . 2 . 9 b )
w h e r e
W
i
= I ; h
i
2
6
6
6
4
a
1 1
A (
i 1
) a
1 2
A (
i 1
) a
1 J
A (
i 1
)
a
2 1
A (
i 2
) a
2 2
A (
i 2
) a
2 J
A (
i 2
)
.
.
.
.
.
.
.
.
.
.
.
.
a
J 1
A (
i J
) a
J 2
A (
i J
) a
J J
A (
i J
)
3
7
7
7
5
( 9 . 2 . 9 c )
k
i
=
2
6
6
6
4
k
i 1
k
i 2
.
.
.
k
i J
3
7
7
7
5
V
i
=
2
6
6
6
4
A (
i 1
)
A (
i 2
)
.
.
.
A (
i J
)
3
7
7
7
5
q
i
=
2
6
6
6
4
b (
i 1
)
b (
i 2
)
.
.
.
b (
i J
)
3
7
7
7
5
: ( 9 . 2 . 9 d )
L e t u s w r i t e ( 9 . 2 . 7 a ) i n t h e f o r m
Y
i
= Y
i ; 1
+ h
i
D k
i
( 9 . 2 . 1 0 a )
w h e r e
D =
2
6
6
6
4
b
1
I
b
2
I
.
.
.
b
J
I
3
7
7
7
5
: ( 9 . 2 . 1 0 b )
E l i m i n a t i n g k
i
i n ( 9 . 2 . 1 0 a ) u s i n g ( 9 . 2 . 9 b ) y i e l d s
Y
i
= ;
i
Y
i ; 1
+ g
i
i = 1 2 : : : J ( 9 . 2 . 1 1 a )
w h e r e
;
i
= I + h
i
D
i
W
; 1
i
V
i
( 9 . 2 . 1 1 b )
1 9
-
8/2/2019 Ode 9
20/25
g
i
= h
i
D W
; 1
i
q
i
: ( 9 . 2 . 1 1 c )
T h e r e s u l t i n g l i n e a r a l g e b r a i c s y s t e m i s
2
6
6
6
6
6
6
6
4
L
; ;
1
I
; ;
2
.
.
.
; ;
J
I
R
3
7
7
7
7
7
7
7
5
2
6
6
6
4
Y
0
Y
1
.
.
.
Y
J
3
7
7
7
5
=
2
6
6
6
6
6
4
l
g
1
.
.
.
g
J
r
3
7
7
7
7
7
5
: ( 9 . 2 . 1 1 d )
T h u s , o n c e a g a i n , t h e a l g e b r a i c s y s t e m i s b l o c k b i d i a g o n a l a n d m a y b e s o l v e d b y t h e
m e t h o d s o f S e c t i o n 8 . 5 .
9 . 3 C o n v e r g e n c e a n d S t a b i l i t y
L e t u s c o n s i d e r t h e l i n e a r r s t - o r d e r B V P
L y = y
0
; A ( x ) y = b ( x ) a < x < b ( 9 . 3 . 1 a )
L y ( a ) = l R y ( b ) = r : ( 9 . 3 . 1 b )
U s i n g ( 9 . 2 . 4 - 9 . 2 . 6 ) w e h a v e
y (
i j
) = y ( x
i ; 1
) + h
i
J
X
k = 1
a
j k
y
0
(
i k
) + h
i
E
j
( 9 . 3 . 2 a )
y ( x
i
) = y ( x
i ; 1
) + h
i
J
X
k = 1
b
k
y
0
(
i k
) + h
i
E ( 9 . 3 . 2 b )
w h e r e
E
j
=
Z
j
0
R ( ) d ( 9 . 3 . 2 c )
a n d
E =
Z
1
0
R ( ) d : ( 9 . 3 . 2 d )
2 0
-
8/2/2019 Ode 9
21/25
T h e f u n c t i o n R ( x ) i s t h e i n t e r p o l a t i o n e r r o r . I n g e n e r a l , i f w e i n t e r p o l a t e a f u n c t i o n f ( x )
a t J d i s t i n c t p o i n t s
f
1
<
2
0
: ( 9 . 3 . 4 )
R e c a l l ( L e m m a 5 . 2 . 1 ) t h a t d i v i d e d d i e r e n c e s a n d d e r i v a t i v e s a r e r e l a t e d i n t h a t t h e r e
e x i s t s a p o i n t
2(
1
J
) ,
1
<
2
-
8/2/2019 Ode 9
22/25
T h e n o t a t i o n i s s l i g h t l y c o n f u s i n g s i n c e ( )
0
d e n o t e s a n x d e r i v a t i v e a n d t h e r e m a i n i n g
d e r i v a t i v e s a r e t a k e n w i t h r e s p e c t t o . W e ' l l u s e ( 9 . 2 . 3 d ) t o t r a n s f e r a l l d e r i v a t i v e s t o
t h e p h y s i c a l d o m a i n , i . e . ,
d
d
= h
j
d
d x
:
T h e n
y
0
i 1
i 2
: : :
i J
] =
h
J
j
J !
y
( J + 1 )
( ) :
T a k i n g a v e c t o r n o r m , w e h a v e
j R ( x ) j C h
J
j
w h i c h p r o v e s t h e r e s u l t .
H a v i n g t h e s e p r e l i m i n a r y r e s u l t s , w e e s t a b l i s h a b a s i c s t a b i l i t y r e s u l t .
T h e o r e m 9 . 3 . 1 . T h e J - s t a g e c o l l o c a t i o n s o l u t i o n ( 9 . 2 . 7 ) o f l i n e a r B V P s ( 9 . 2 . 8 ) e x i s t s
a n d i s s t a b l e . T h e d i s c r e t i z a t i o n e r r o r
e ( x ) = y ( x )
;y ( x ) ( 9 . 3 . 7 a )
s a t i s e s
m a x
x
i ; 1
x x
i
k e
( j )
( x ) k = O ( h
J ; j
)
j ; 1
i
j = 0 1 : : : J ( 9 . 3 . 7 b )
w h e r e
i
=
h
h
i
: ( 9 . 3 . 7 c )
P r o o f . F o l l o w i n g t h e s t e p s l e a d i n g t o ( 9 . 2 . 1 1 ) , w e u s e ( 9 . 3 . 2 ) t o s h o w t h a t
y ( x
i
) = ;
i
y ( x
i ; 1
) + g
i
+ h
i
i
: ( 9 . 3 . 8 )
T h e t e r m
i
a r i s e s f r o m t h e E a n d E
j
t e r m s i n ( 9 . 3 . 2 ) h e n c e , u s i n g ( 9 . 3 . 6 ) i t i s O ( h
J
i
) .
I f J
1 , t h e o n e - s t e p s c h e m e ( 9 . 2 . 7 ) i s c o n s i s t e n t . A c o n s i s t e n t o n e - s t e p s c h e m e i s
a l s o s t a b l e a n d c o n v e r g e n t ( T h e o r e m s 3 . 4 . 1 , 2 ) . T h u s , s o l u t i o n s o f ( 9 . 2 . 7 ) e x i s t a n d , b y
s u b t a c t i n g ( 9 . 2 . 1 1 a ) f r o m ( 9 . 3 . 8 ) , s a t i s f y
m a x
0 i N
j y ( x
i
) ; Y
i
j = O ( h
J
) : ( 9 . 3 . 9 )
2 2
-
8/2/2019 Ode 9
23/25
W e c a n g e t m o r e d e t a i l e d i n f o r m a t i o n a b o u t e r r o r s a t t h e c o l l o c a t i o n p o i n t s b y s u b t r a c t -
i n g ( 9 . 2 . 7 b ) f r o m ( 9 . 3 . 2 a ) w h i l e u s i n g ( 9 . 3 . 1 a )
e (
i j
) = e ( x
i ; 1
) + h
i
J
X
k = 1
a
j k
A (
i k
) e (
i k
) + h
i
E
j
T a k i n g a m a t r i x n o r m
j e (
i j
) j ( 1 + C
1
h
i
) j e ( x
i ; 1
) j + C
2
h
J + 1
i
:
I n o b t a i n i n g t h i s r e l a t i o n s h i p , w e b o u n d e d t h e R u n g e - K u t t a c o e c i e n t s a n d j A j b y t h e i r
m a x i m a l v a l u e s a n d u s e d c o n s i s t e n c y t o i n f e r t h a t e (
i k
) d o e s n o t d i e r f r o m e ( x
i ; 1
) b y
m o r e t h a n O ( h
i
) . T h e l a s t t e r m a b o v e w a s b o u n d e d u s i n g ( 9 . 3 . 6 ) .
S i n c e ( 9 . 3 . 9 ) i m p l i e s t h a t
je ( x
j
)
j= O ( h
J
) , j = 0 1 : : : J , w e h a v e
m a x
0 i N 1 j J
j e (
i j
) j = O ( h
J
) : ( 9 . 3 . 1 0 )
F u r t h e r m o r e , s i n c e t h e e x a c t a n d n u m e r i c a l s o l u t i o n s s a t i s f y t h e d i e r e n t i a l e q u a t i o n
( 9 . 3 . 1 ) a t t h e c o l l o c a t i o n p o i n t s , w e h a v e
e
0
(
i j
) = A (
i j
) e (
i j
) :
T a k i n g a v e c t o r n o r m a n d u s i n g ( 9 . 3 . 1 0 )
m a x
0 i N 1 j J
j e
0
(
i j
) j = O ( h
J
) : ( 9 . 3 . 1 1 )
F i n a l l y , a p p l y i n g t h e i n t e r p o l a t i o n f o r m u l a ( 3 ) t o y
0
a n d u s i n g ( 9 . 2 . 3 a ) , w e h a v e
e
0
( x ) =
J
X
k = 1
e
0
(
i k
) L
k
( ) + R ( ) : ( 9 . 3 . 1 2 )
D i e r e n t i a t i n g a n d u s i n g ( 9 . 3 . 6 ) y i e l d s t h e r e s u l t ( 9 . 3 . 7 b ) .
T h i s r e s u l t i s n o t a s s h a r p a s i t c o u l d b e a s d e s c r i b e d b y t h e n e x t T h e o r e m .
T h e o r e m 9 . 3 . 2 . S u p p o s e t h a t t h e c o l l o c a t i o n p o i n t s a r e d i s t i n c t w i t h
i 1
<
i 2
-
8/2/2019 Ode 9
24/25
R e m a r k . I f p > J c o n v e r g e n c e a t t h e n o d e s a n d c o l l o c a t i o n p o i n t s i s a t a h i g h e r r a t e
t h a n i m p l i e d b y T h e o r e m 9 . 3 . 1 . T h i s i s t h e p h e n o m e n o n o f s u p e r c o n v e r g e n c e t h a t w e
i l l u s t r a t e d i n S e c t i o n 9 . 1 .
P r o o f . A s i n S e c t i o n 9 . 1 , i n t r o d u c e t h e G r e e n ' s f u n c t i o n
e ( ) =
Z
b
a
G ( x ) L e ( x ) d x =
J
X
j = 1
Z
x
j
x
j ; 1
G ( x ) L e ( x ) d x : ( 9 . 3 . 1 4 )
L e t u s a s s u m e t h a t =2 ( x
j ; 1
x
j
) a n d , f o l l o w i n g t h e l o g i c i n t r o d u c e d i n S e c t i o n 9 . 1 , w r i t e
G ( x ) L e ( x ) = w ( x )
J
Y
j = 1
( x ;
i j
)
T h e f u n c t i o n w ( x ) i n v o l v e s t h e J t h d e r i v a t i v e o f L e . U s i n g t h i s a n d ( 9 . 3 . 7 b ) , w e m a y
e x p e c t w ( x ) t o h a v e p ; J b o u n d e d d e r i v a t i v e s . T h u s , e x p a n d w ( x ) i n a T a y l o r ' s s e r i e s
o f t h e f o r m
w ( x ) = P
p ; J ; 1
( x ) + O ( h
p ; J
i
)
w h e r e P
p ; J ; 1
( x ) i s a p o l y n o m i a l o f d e g r e e p ; J ; 1 .
I f t h e o n e - s t e p m e t h o d i s a c c u r a t e t o o r d e r p t h e n
Z
x
j
x
j ; 1
P
p ; J ; 1
( x )
J
Y
j = 1
( x ;
i j
) d x = 0 :
S i n c e
J
Y
j = 1
( x ;
i j
) = O ( h
J
i
)
w e h a v e
Z
x
j
x
j ; 1
G ( x ) L e ( x ) d x = O ( h
P ; J
i
) O ( h
J
i
) h
i
=2 ( x
j ; 1
x
j
) : ( 9 . 3 . 1 5 a )
T h e r e s u l t ( 9 . 3 . 1 3 a ) i s o b t a i n e d b y s u m m i n g t h e a b o v e r e l a t i o n o v e r t h e s u b i n t e r v a l s .
W h e n 2 ( x
j ; 1
x
j
) t h e n w e a r e o n l y a b l e t o s h o w t h a t
Z
x
j
x
j ; 1
G ( x ) L e ( x ) d x = O ( h
J + 1
i
) 2 ( x
j ; 1
x
j
) : ( 9 . 3 . 1 5 b )
S u m m i n g ( 9 . 3 . 1 5 a , b ) y i e l d s ( 9 . 3 . 1 3 b ) .
S u p e r c o n v e r g e n c e o c c u r s w h e n e v e r p > J + 1 .
2 4
-
8/2/2019 Ode 9
25/25
B i b l i o g r a p h y
1 ] U . M . A s c h e r , R . M a t t h e i j , a n d R . R u s s e l l . N u m e r i c a l S o l u t i o n o f B o u n d a r y V a l u e
P r o b l e m s f o r O r d i n a r y D i e r e n t i a l E q u a t i o n s . S I A M , P h i l a d e l p h i a , s e c o n d e d i t i o n ,
1 9 9 5 .
2 ] C . d e B o o r . A P r a c t i c a l G u i d e t o S p l i n e s . S p r i n g e r - V e r l a g , N e w Y o r k , 1 9 7 8 .
3 ] C . d e B o o r a n d B . S w a r t z . C o l l o c a t i o n a t g a u s s i a n p o i n t s . S I A M J . N u m e r . A n a l . ,
1 0 : 5 8 2 { 6 8 7 , 1 9 7 3 .
4 ] J . E . F l a h e r t y a n d W . M a t h o n . C o l l o c a t i o n w i t h p o l y n o m i a l a n d t e n s i o n s p l i n e s f o r
s i n g u l a r l y p e r t u r b e d b o u n d a r y v a l u e p r o b l e m s . S I A M J . S c i . S t a t . C o m p u t , 1 : 2 6 0 {
2 8 9 , 1 9 8 0 .
5 ] E . I s a a c s o n a n d H . B . K e l l e r . A n a l y s i s o f N u m e r i c a l M e t h o d s . J o h n W i l e y a n d S o n s ,
N e w Y o r k , 1 9 6 6 .
2 5