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TRANSCRIPT
Chapter �
Collocation Methods
��� Introduction
Let�s continue the discussion of collocation methods with the second�order linear BVP
Ly � y�� � p�x�y� � q�x�y � r�x�� a � x � b� ������a�
y�a� � A� y�b� � B� ������b�
To pick up where we left o in Section ��� consider an approximate solution
Y �x� �NXi��
ci��i �x� � di�
�i �x�� ������a�
involving the cubic Hermite basis
��i �x� �
���
�� ��x�xihi
�� � ��x�xihi
��� if xi�� � x � xi�� ��x�xi
hi���� � ��x�xi
hi����� if xi � x � xi��
�� otherwise
� ������b�
��i �x� �
���
�x� xi� � �x�xihi
��� if xi�� � x � xi�x� xi� �� x�xi
hi����� if xi � x � xi��
�� otherwise
� ������c�
Recall� that a� b� is divided into N subintervals with each having contributions from four
non�trivial basis functions �Figure ������� In particular� the restriction of the basis to
xi��� xi� involves non�trivial contributions from ��i���x�� �
�i���x�� �
�i �x�� and ��
i �x�
�
a = x x x1 i-1 N0
x = b
1 φ φi
i-1
ωi-1
ωi
xi
3
33
3
Figure ������ Geometry for the collocation solution of ������� showing the restriction ofthe cubic Hermite polynomial basis to the subinterval xi��� xi��
In order to obtain approximate solutions of ������� using �������� we collocate at two
points �i�� �i� per subinterval and satisfy the boundary conditions� Thus� the unknown
coe�cients ci� di� i � �� �� ���� N � are determined as the solution of
LY ��ij� � r��ij�� j � �� �� i � �� �� � � � � N� ������a�
Y �a� � A� Y �b� � B� ������b�
The restriction of ������a� to the subinterval xi��� xi� is
Y �x� � ci����i���x� � di���
�i���x� � ci�
�i �x� � di�
�i �x�� �������
Substituting ������� into ������a�� LY ��i���LY ��i���
�� Li
�ci��di��
��Ri
�cidi
��
�r��i���r��i���
�� i � �� �� � � � � N� ������a�
where
Li �
� L��i����i��� L��
i����i���L��
i����i��� L��i����i���
�� Ri �
� L��i ��i��� L��
i ��i���L��
i ��i��� L��i ��i���
�� ������b�
With �������� the boundary conditions ������b� become
L�
�c�d�
�� A� RN��
�cNdN
�� B� �����a�
�
where
L� � RN�� � � ��� �����b�
Combining ������� and ������� we �nd
�������
L�
L� R�
� � � � � �
LN RN
RN��
�
�����������
c�d�c�d����cNdN
��
���������������
Ar������r������r������r������
���r��N���r��N���B
�� �������
Thus� the ��N ��� coe�cients ci� di� i � �� �� � � � � N � are determined as the solution of a
block bidiagonal matrix of dimension �N with �� � blocks� This system may be solved
by the methods of Section ����
We can now ask if there is an optimal placement of the two collocation points on each
subinterval that� e�g�� minimizes the discretization error y�x��Y �x� in some norm� This
question was answered in a landmark paper by de Boor and Swartz �� and we will follow
their analysis�
Consider the inner product
�v� u� �
Z b
a
v�x�u�x�dx� �������
Let us assume that u�x� and v�x� are smooth except� perhaps for jump discontinuities
at zj� j � �� �� � � � �M � �� Also let z� � a and zM � b� Consider
�v�Lu� �Z b
a
v u�� � pu� � qu�dx �������
where the integral is interpreted as a sum of integrals over the subintervals �z�� z���
�z�� z��� � � � � �zM��� zM�� For simplicity� we�ll also assume that A � B � � and that u�x�
and v�x� satisfy these conditions� Using �������� we integrate ������� by parts to obtain
�v�LU� �Z b
a
�v�u� � �pv��u� qvu�dx�MXj��
fv�x�u��x� � p�x�v�x�u�x�gzjzj���
�
Integrating the �rst term in the integrand by parts once more
�v�LU� �Z b
a
v�� � �pv�� � qv�udx�MXj��
fv�x�u��x�� v��x�u�x� � p�x�v�x�u�x�gzjzj�� �
We can write this result in a simpler form by using the inner product notation �������
and by de�ning the jump in a function q�x� at a point z as
q�x��x�z � lim���
q�z � ��� q�z � ��� ��������
With this notation� we have
�v�Lu� � �LTv� u��M��Xj��
vu� � v�u� pvu�zj �������a�
where LT is called the adjoint operator and satis�es
LTv � v�� � �pv�� � qv� �������b�
Let us simplify matters somewhat by assuming that u � C��a� b� and v � C��a� b��
Then� �������a� becomes
M��Xj��
v�u�zj � �v�Lu�� �LTv� u�� �������c�
De�nition ������ The Green�s function G��� x� for the operator L of ������� satis�es
G��� x� � C��a� b�� �a� b�� �������a�
LTG��� x� � �� �a� ��� � ���� b�� �� � lim���
� � �� �������b�
G��� a� � G��� b� � �� �������c�
Gx��� ����Gx��� �
�� � �� �������d�
When viewed as a function of x� the Green�s function has a unit jump in its �rst
derivative at the point ��
�
Now� if we choose v�x� in �������c� as the Green�s function G��� x� so that the only
�M � �� discontinuity occurs at z� � �� we have
u��� � �G��� ���Lu�� �������a�
If u�x� is chosen as y�x�� the solution of �������� then
y��� � �G��� ���Ly� � �G��� ��� r�� �������b�
The relation �������a� holds for any smooth function u�x� and not just the solution of
�������� Since� for example� the collocation solution Y �x� � C��a� b�� we can replace u in
�������a� by Y to obtain
Y ��� � �G��� ���LY �� �������c�
Finally� letting
e�x� � y�x�� Y �x� �������a�
denote the discretization error of the collocation solution� we subtract �������c� from
�������b� to obtain
e��� � �G��� ���Le� �Z b
a
G��� x�Le�x�dx� �������b�
Remark �� Each result �������b�� �������c�� or �������b� relates a global quantity
�y� Y� e� to its local counterpart �Ly�LY�Le� through the Green�s function�
Let us write �������b� in the more explicit form
e��� �NXi��
Z xi
xi��
G��� x�Le�x�dx �NXi��
ei���� ��������
Suppose that � �� �xi��� xi� so that G��� x� is smooth for x � �xi��� xi�� Write
Le � L�y � Y � � Ly � Pr � Pr � LY ������a�
where Pr is a linear polynomial for x � �xi��� xi� that interpolates both Ly and LY at
the two collocation points �i�� and �i�� on this subinterval� Thus�
Pr � r��i���x� �i���i�� � �i��
� r��i���x� �i���i�� � �i��
� ������b�
�
Y
y
Pr
L
L
xi-1
ξ ξ xii,1 i,2
Figure ������ Functions Ly� LY � and Pr on a subinterval �xi��� xi� not containing thepoint x � ��
The functions Ly� LY � and Pr are illustrated in Figure ������ The dierences Ly � Pr
and LY �Pr can be estimated using formulas for the error in linear interpolation �� as
Ly � Pr ��
��x� �i����x� �i����Ly�����i� �������a�
LY � Pr ��
��x� �i����x� �i����LY ����i� �������b�
where �i� i � �xi��� xi��
Using �������� in ������� yields
ei��� �
Z xi
xi��
�x� �i����x� �i���g��� x�dx� � �� �xi��� xi�� �������a�
where
g��� x� ��
�G��� x� �Ly�����i�� �LY ����i��� �������b�
where ei was de�ned in ��������� We bound �������a� as
jei���j �Z xi
xi��
jx� �i��jjx� �i��jjg��� x�jdx�
Since jx� �i�jj � hi� j � �� �� we have
jei���j � h�i jjg��� ��jji��� � �� �xi��� xi�� �������a�
xi-1
ξ ξ xii,1 i,2
GL y
GL Y
ξ
GPr
Figure ������ Functions GLy� GLY � and GPr on a subinterval �xi��� xi� containing thepoint x � ��
where
jjf���jji�� � maxxi���x�xi
jf�x�j� �������b�
Typically� one subinterval contains the point � where Gx is discontinuous� The anal�
ysis leading to �������� cannot be used on this subinterval since G��� x� �� C��xi��� xi��
There are several ways of showing that jei���j increases from O�h�i � to O�h�i � on this
subinterval� We�ll choose one which restricts � to lie between �i�� and �i��� In this case�
we interpolate GLy and GLY by piecewise constant functions
G��� x�Pr �
�G��� �i���r��i���� if xi�� � x � �G��� �i���r��i���� if � � x � xi
��������
as shown in Figure ������ The error in piecewise constant interpolation is
GLy �GPr �
��x� �i����GLy����i���� if xi�� � x � ��x� �i����GLy����i���� if � � x � xi
�
A similar expression applies for GLY � GPr� Combining these results in the manner
used to obtain �������a� yields
ei��� �
Z �
xi��
�x� �i����g��� x�dx�
Z xi
�
�x� �i����g��� x�dx� � � �xi��� xi�� �������a�
�
where
�g��� x� �
��GLy����i���� �GLY ���i���� if xi�� � x � ��GLy����i���� �GLY ���i���� if � � x � xi
� �������b�
We bound �������a� as
jei���j �Z �
xi��
jx� �i��jj�g��� x�jdx�Z xi
�
jx� �i��jj�g��� x�jdx
or
jei���j � h�i k�g��� ��ki�� � � �xi��� xi�� �������c�
We�ll use the symbol k�g��� ��ki�� with the understanding that the maximum is computed
on �xi��� �� � ��� xi��
Finally� substituting �������a� and �������c� into �������� yields
je���j � h�j jj�g��� ��jjj�� �NX
i���i��j
h�ikg��� ��ki��� � � �xj��� xj��
or
je���j � h�j �NX
i���i��j
h�i �k�g��� ��k�� � � �xj��� xj��
where
k�g��� ��k� � max�k�g��� ��k�� k�g��� ��k��� kf���k� � max��i�N
kf���ki���
Letting
h � max��i�N
jhij ��������
and observing that Nh � �b� a�� we have
je���j � h� � �N � ��h��k�g��� ��k� � h� � � �b� a��k�g��� ��k��� � � �xj��� xj��
or
je���j � Ch�� � � �xj��� xj�� ��������
where
C � �� � b� a�k�g��� ��k��
�
If � � xj� j � �� �� � � � � N � then there is no discontinuity in the Green�s function on
any subinterval and the error is obtained from �������a� and �������� as
je�xj�j �NXi��
h�i kg�xj� ��ki��� j � �� �� � � � � N�
Following the steps leading to ��������� we again �nd that
je�xj�j � Ch�� j � �� �� � � � � N� ��������
Thus� the global and pointwise errors are both O�h��� This occurs because of the low
polynomial degree and the arbitrary choice of the collocation points� With either higher�
degree polynomials or a special choice of collocation points we can reduce the pointwise
error relative to the global error� This phenomenon is called nodal superconvergence�
De�nition ������ Nodal superconvergence implies that the collocation solution on the
mesh fa � x� � x� � � � � � xN � bg converges to a higher order than it does globally�
Now� let us resume the search for special collocation points� Consider the case when
� � xj� j � �� �� � � � � N � so that g�xj� x� is smooth and expand it in a Taylor�s series on
the subinterval �xi��� xi� to obtain
g�xj� x� � g�xj� xi����� � gx�xj� xi������x� xi����� ��
�gxx�xj� i��x� xi�����
��
i � �xi��� xi��
Substitute this expansion into �������a� to obtain
ei�xj� �
Z xi
xi��
�x� �i����x� �i��� g�xj� xi����� � gx�xj� xi������x� xi������
�
�gxx�xj� i��x� xi�����
��dx� ��������
The choice of �i�� and �i�� is now clear� We should select them so thatZ xi
xi��
�x� �i����x� �i���dx � � ������a�
and Z xi
xi��
�x� �i����x� �i����x� xi�����dx � �� ������b�
�
Assuming that this were possible� for the moment� then �������� would become
ei�xj� �
Z xi
xi��
�
��x� �i����x� �i���gxx�xj� i��x� xi�����
��dx�
We bound this as
jei�xj�j � �
�
Z xi
xi��
jx� �i��jjx� �i��jjgxx�xj� i�jjx� xi����j�dx
or
jei�xj�j � �
�h�ikgxx�xj� ��ki�� ��������
Substituting this result into �������� yields
je�xj�j � �
�
NXi��
h�i kgxx�xj� ��ki��� j � �� �� � � � � N�
or
je�xj�j � �
�kgxx�xj� ��k�
NXi��
h�i ��
�kgxx�xj� ��k�Nh��
Thus�
je�xj�j � Ch�� j � �� �� � � � � N� �������a�
where
C ��
��b� a�kgxx�xj� ��k�� �������b�
The pointwise error has been increased by two orders with the special choice of collocation
points dictated by �������� This is most de�nitely an example of nodal superconvergence
since the global error is still O�h���
It remains to determine the collocation points that satisfy �������� Let us begin by
transforming these integrals to ��� �� using the mapping
x � xi���� � hi�� �� � � �� �������a�
Also let
�i�� � xi���� � ��hi� �i�� � xi���� � ��hi� �������b�
��
Then
x� �i�� �hi�� � ����� x� �i�� �
hi�� � ����� �������c�
Conditions ������a�b� become
�hi���Z �
��� � ����� � ����d � �� �
hi���Z �
��� � ����� � ����d � ��
Thus� it su�ces to determine �� and �� such that
Z �
��� � ����� � ����P ��d � � ��������
for all linear polynomials P ���
Since the integrand is a cubic polynomial it can be evaluated exactly by the two�point
Gauss�Legendre quadrature formula �cf� ��� Chapter ��
Z �
��f��d � f�� �p
�� � f�
�p�� � E �������a�
where the discretization error E is given by
E ��
���f iv��� � ���� ��� �������b�
Thus� applying �������� to ��������� we have
�� �p�� ������ �p
�� ����P �� �p
�� � �
�p�� �����
�p�� ����P �
�p�� � ��
Once again� the integrand in �������� is a cubic polynomial so E � �� We see that we
can satisfy the above condition by choosing
�� � �� ��
�p�� �������a�
Expressed in terms of the original variables through �������b�� the collocation points are
�i�� � xi���� � hi
�p�� �i�� � xi���� �
hi
�p�� �������b�
Regardless of how the result is expressed� the key is to perform collocation at the Gauss�
Legendre points mapped to the appropriate interval�
��
N Midpoint Rule Collocation
� ����� ����
� ���� ����
�� ����� ���� ���� ����
�� ���� ���� ����� ���
�� ���� ���� ����� ���
�� ���� ���� ���� ����
Table ������ Maximum pointwise errors in the solution of Example ����� using the mid�point rule and collocation at two Gauss�Legendre points�
The error formula �������b� can be used to obtain a more precise estimate of the
collocation error than given by ���������
Let us conclude this section with two examples�
Example ����� �cf� ��� Chapter ��� Consider the problem
y�� �y�
x� �
�
�� x���� � � x � �� y���� � y��� � ��
which has the exact solution
y�x� � � ln��
�� x���
The solution is smooth on � � x � �� but the coe�cient p�x� � ��x is unbounded
at x � �� This would not be a problem when using collocation at the Gauss�Legendre
points or� e�g�� the midpoint rule since functions need not be evaluated at the endpoints�
Formulas� such as the trapezoidal rule� would have to be modi�ed to account for the
singularity in p�x�� We�ll avoid this and present results using both the midpoint rule and
collocation ������� at the two Gauss�Legendre points ��������� The maximum pointwise
errors are presented in Table ������
A simple calculation will verify that the midpoint rule and collocation solutions are
converging at their expected rates of O�N��� and O�N���� respectively� The advantages
of the higher�order collocation method on this problem are apparent�
Example ����� ��� Consider
�y�� � �� � x��y�� �e�x��
� �� � x���� x� �
�
��� � � x � ��
y��� � �� y��� � e���� � e�����
��
N Error
� ����� ����
� ����� ����
� ����� ���
�� ���� ����
� ����� �����
��� ���� �����
Table ������ Maximum pointwise errors in the solution of Example ����� using collocationat Flaherty and Mathon�s �� points�
which has the exact solution
y�x� � e�x�� � e�x�x���x�� ����
For � � �� � there is a boundary layer of width O��� at x � ��
Collocation using cubic polynomials was performed at the Gauss�Legendre points and
at the points
xi�� � xi���� � �hi� xi�� � xi���� � �hi�
where
� ��
�� �
�i
���i���
� �p�� ����i�����i
�
�i �
����p�xk�� q�xk�
p�xk�
���� � k �
�i� �� if p�xi�� �p�xi
�� �
i� otherwise�
��z� � coth z � �
z�
With � � ���� and N � �� collocation at the Gauss�Legendre points produced a solu�
tion with a maximum pointwise error of approximately ��� The results using collocation
at Flaherty and Mathon�s �� points are shown in Table ������
Following the logic introduced in ��������� Flaherty and Mathon selected their col�
location points to re�ect the boundary layer in the Green�s function� Speci�cally� they
chose points
�i�� � xi���� � �hi� �i�� � xi���� � �hi�
symmetrically disposed with respect to the center of each subinterval so thatZ �
��e�
������� �� � �� ��P ��d � �
��
for polynomials P �� of as high a degree as possible�
��� Collocation for First�Order Systems
Let us extend the collocation methods to �rst�order vector BVPs of the usual form
y� � f�x�y�� a � x � b� ������a�
gL�y�a�� � gR�y�b�� � �� ������b�
We�ll follow a dierent approach than the one used in Section ��� and integrate ������a�
on a subinterval �xi��� xi� to obtain
y�x� � y�xi��� �Z x
xi��
y��x�dx � y�xi��� �Z x
xi��
f�x�y�dx� �������
As with initial value methods� we�ll construct a numerical method by approximating y�
�or f� by a polynomial and integrating the result� Any polynomial basis may be used�
but let us concentrate on the Lagrange form of the interpolating polynomial
y��x� �JX
k��
y���i�k�Lk�� �R�� ������a�
where
Lk�� �JY
j���j ��k
� �j�k � �j
�� � ���� � ��� � � � � � �k���� � �k��� � � � � � �J�
��k � �����k � ��� � � � ��k � �k�����k � �k��� � � � ��k � �J��
������b�
R � y� �i��� �i��� � � � � �i�J�JY
j��
� � �j� ������c�
x � xi�� � hi� � � � �� ������d�
�i�k � xi�� � �khi� ������e�
The image of the collocation points �k� k � �� �� � � � � J � are ordered such that
� � �� � �� � � � � � �J � �� ������f�
��
The divided dierence y� �i��� �i��� � � � � �i�J � will be de�ned shortly�
Substituting ������a� into ������� yields
y�x� � y�xi��� � hi
JXk��
y���i�k�Z �
�
Lk���d� � hi
Z �
�
R���d�� �������
Evaluating ������� at the collocation points
y��ij� � y�xi��� � hi
JXk��
y���i�k�Z �j
�
Lk���d� � hi
Z �j
�
R���d��
Let
ajk �
Z �j
�
Lk���d� ������a�
and
Ej �
Z �j
�
R���d�� ������b�
Then
y��i�j� � y�xi��� � hi
JXk��
y���i�k�ajk � hiEj� ������c�
Evaluating ������� at x � xi yields
y�xi� � y�xi��� � hi
JXk��
y���i�k�Z �
�
Lk���d� � hi
Z �
�
R���d��
Letting
bk �
Z �
�
Lk���d�� �����a�
and
E �
Z �
�
R���d�� �����b�
we have
y�xi� � y�xi��� � hi
JXk��
y���i�k�bk � hiE� �����c�
��
If the errors Ej� j � �� �� � � � � J � and E are neglected� we see that the collocation
solution is obtained from a J stage implicit Runge�Kutta method� Thus� letting Y�x�
denote the approximate solution� we have
Y�xi� � Y�xi��� � hi
JXk��
bkki�k ������a�
where
ki�k � Y���i�k� � f�xi�� � �khi�Y�xi��� � hi
JXj��
akjki�j�� k � �� �� � � � � J� ������b�
Solution continuity� required for �rst�order BVPs�
y�x�i � � y�x�i �� i � �� �� � � � � N � ��
is automatically satis�ed�
Solutions at any point x � xi��� xi� are de�ned by the interpolation polynomial �������
and ������a� as
Y�x� � Y�xi��� � hi
JXk��
Y���i�k�Z �
�
Lk���d�� ������c�
Some common choices of the collocation points are�
�� Gauss�Legendre points� The points �k� k � �� �� � � � � J � are the roots of the Legendre
polynomial of degree J mapped to the interval ��� ��� The roots of the Legendre
polynomial are normally prescribed on ���� ��� This may be done by the linear
transformation � �� � ���� � ��� ��� For all J � �� � � and �J � �� thus�
subinterval endpoints are not collocation points� The �rst �ve Legendre polynomi�
als are shown in Table ������ The simplest scheme �J � �� is the midpoint rule�
The pointwise accuracy of a J�point scheme is O�h�J��
�� Radau points� As described in Section ���� either �� � � or �J � � and the remaining
points �k� k � �� �� � � � J � or k � �� �� � � � � J � �� respectively� are selected to achieve
maximal order� This involves selecting the points on ��� �� as the roots of the
Radau polynomial of degree J
RJ� � � PJ� �� J
�J � �PJ��� �
�
J PJ� � j� j � �� �� � � � � J
� �� �� �
��� � � �� � �p
�
� ���� � � �� �� �
q��
� ���� � � �� � � �� ������������� ����������
Table ������ Legendre polynomials PJ� � and their roots for J � �� �� � � � � ��
where PJ� � is the Legendre polynomial of degree J � The negative sign is chosen
with �J � � and the positive sign is selected with �� � �� The simplest scheme �J �
�� is the forward or backward Euler method when �� � � or �J � �� respectively�
Judging from our experience with sti IVPs� Radau schemes should be suitable
for singularly perturbed BVPs� This will determine the proper choice of the �xed
endpoint� as described in Chapter ��� The pointwise accuracy of a J point Radau
scheme is O�h�J����
�� Lobatto points� The points �� � �� �J � �� and the remaining points �k� k �
�� �� � � � � J��� are selected as the roots of the Legendre polynomial of degree J���
The simplest scheme �J � �� is the trapezoidal rule� The pointwise accuracy of a
J point scheme is O�h��J�� ��
Example ����� � ��� Chapter ��� As in Example ������ consider
y�� �y�
x� �
�
�� x���� � � x � �� y���� � y��� � ��
Ascher et al� �� solve this problem using
two�point Gauss�Legendre collocation at
�� ��
���� �p
��� �� �
�
��� �
�p���
three�point Lobatto collocation at
�� � �� �� � ���� �� � ��
and
��
N ��Point Gauss ��Point Lobatto ��Point Gauss
� ����� ���� ����� ���� ����� ����
� ���� ���� ����� ���� ����� ����
�� ���� ���� ����� ��� ����� �����
�� ����� ��� ����� ��� ����� �����
�� ����� ��� ����� ���� ���� �����
�� ���� ���� ����� ����� ����� �����
Table ������ Maximum pointwise errors in the solution of Example ����� using collocationat two Gauss�Legendre points� three Lobatto points� and three Gauss�Legendre points�
three�point Gauss�Legendre collocation at
�� ��
����
r�
��� �� � ���� �� �
�
��� �
r�
���
The Gauss�Legendre methods do not need function evaluations at the endpoints of
subintervals and� thus� the singular coe�cient in the dierential equation at x � �
poses no problem� However� the Lobatto method must be modi�ed to account for the
singularity� Using L�Hopital�s rule�
limx��
y��x�x
� y������
Using this result in the dierential equation yields y����� � ���� thus�
limx��
y��x�x
��
��
The maximum pointwise errors in y for the three methods are shown in Table ������
The two�point Gauss�Legendre� three�point Lobatto� and three�point Gauss�Legendre are
converging at their expected rates of O�N���� O�N���� and O�N���� respectively�
The implementation of the collocation scheme is usually done by eliminating the
unknowns ki�k� k � �� �� � � � � J � appearing in ������b� on each subinterval and then solving
for the nodal values Yi � Y�xi�� i � �� �� � � � � N � We�ll illustrate this for a linear system
f�x�y� � A�x�y � b�x�� ������a�
gL�y�a�� � Ly�a�� l� gR�y�b�� � Ry�b�� r� ������b�
��
Nonlinear problems are solved using Newton�s method to linearize them to the form of
��������
For a linear problem� ������b� becomes
ki�k � A��i�k� Yi�� � hi
JXj��
akjki�j� � b��i�k�� k � �� �� � � � � J� ������a�
This system can be written in matrix form as
Wiki � ViYi�� � qi ������b�
where
Wi � I� hi
�����
a���A��i��� a���A��i��� a��JA��i���a���A��i��� a���A��i��� a��JA��i���
������
� � ����
aJ��A��i�J� aJ��A��i�J� aJ�JA��i�J�
� � ������c�
ki �
�����ki��ki�����
ki�J
� � Vi �
�����A��i���A��i���
���A��i�J�
� � qi �
�����b��i���b��i���
���b��i�J�
� � ������d�
Let us write ������a� in the form
Yi � Yi�� � hiDki �������a�
where
D �
�����b�I
b�I� � �
bJI
� � �������b�
Eliminating ki in �������a� using ������b� yields
Yi � �iYi�� � gi� i � �� �� � � � � J� �������a�
where
�i � I� hiDiW��i Vi� �������b�
��
gi � hiDW��i qi� �������c�
The resulting linear algebraic system is���������
L
��� I
���
� � �
��J I
R
�
�����Y�
Y����YJ
� �
�������
l
g����gJr
� � �������d�
Thus� once again� the algebraic system is block bidiagonal and may be solved by the
methods of Section ����
��� Convergence and Stability
Let us consider the linear �rst�order BVP
Ly � y� �A�x�y � b�x�� a � x � b� ������a�
Ly�a� � l� Ry�b� � r� ������b�
Using ������ � ����� we have
y��ij� � y�xi��� � hi
JXk��
ajky���i�k� � hiEj ������a�
y�xi� � y�xi��� � hi
JXk��
bky���i�k� � hiE ������b�
where
Ej �
Z �j
�
R���d� ������c�
and
E �
Z �
�
R���d�� ������d�
��
The function R�x� is the interpolation error� In general� if we interpolate a function f�x�
at J distinct points f�� � �� � � � � � �Jg then
R�x� � f�x�� P �x� � f ��� ��� � � � � �J �JY
j��
�x� �j� �������
where P �x� is an interpolating polynomial having the form of ������a� �Section ��� or ���
Chapter ��� Recall� that the divided dierence f ��� ��� � � � � �J � is de�ned recursively as
f �l� �l��� � � � � �l�k� �
f��l�� if k � �f ��l��������l�k��f ��l������l�k���
�l�k��l � if k � �� �������
Recall �Lemma ������ that divided dierences and derivatives are related in that there
exists a point � � ���� �J�� �� � �� � � � � � �J � such that
f �l� �l��� � � � � �l�k� �f �J ���
J ��������
when f�x� � CJ���� �J�� We can specialize this result to the problem at hand�
Lemma ������ Suppose y � CJ���a� b� then the error �����c� in the interpolation
����� satis�es
jR�x�j � O�hJj �� x � �xj��� xj�� j � �� �� � � � � J� �����a�
where
hj � xj � xj�� �����b�
and j � j denotes a vector norm�
Proof� Using ������c�
R � y� �i��� �i��� � � � � �i�J �JY
j��
� � �j��
Since and �j� j � �� �� � � � � J � are on �� ��� the product appearing above has at most
unit magnitude� Applying ������� in this case implies
y� �i��� �i��� � � � � �i�J � ��
J �
dJ
dJy�����
��
The notation is slightly confusing since � �� denotes an x derivative and the remaining
derivatives are taken with respect to � We�ll use ������d� to transfer all derivatives to
the physical domain� i�e��d
d� hj
d
dx�
Then
y� �i��� �i��� � � � � �i�J � �hJjJ �y�J�� ����
Taking a vector norm� we have
jR�x�j � ChJj
which proves the result�
Having these preliminary results� we establish a basic stability result�
Theorem ������ The J�stage collocation solution ������� of linear BVPs ������� exists
and is stable� The discretization error
e�x� � y�x�� y�x� ������a�
satis�es
maxxi���x�xi
ke�j �x�k � O�hJ�j�j��i � j � �� �� � � � � J� ������b�
where
i �h
hi� ������c�
Proof� Following the steps leading to ��������� we use ������� to show that
y�xi� � �iy�xi��� � gi � hi�i� �������
The term �i arises from the E and Ej terms in �������� hence� using ������ it is O�hJi ��
If J �� the one�step scheme ������� is consistent� A consistent one�step scheme is
also stable and convergent �Theorems ������ ��� Thus� solutions of ������� exist and� by
subtacting �������a� from �������� satisfy
max��i�N
jy�xi��Yij � O�hJ�� �������
��
We can get more detailed information about errors at the collocation points by subtract�
ing ������b� from ������a� while using ������a�
e��ij� � e�xi��� � hi
JXk��
ajkA��i�k�e��i�k� � hiEj
Taking a matrix norm
je��ij�j � �� � C�hi�je�xi���j� C�hJ��i �
In obtaining this relationship� we bounded the Runge�Kutta coe�cients and jAj by their
maximal values and used consistency to infer that e��i�k� does not dier from e�xi��� by
more than O�hi�� The last term above was bounded using �������
Since ������� implies that je�xj�j � O�hJ�� j � �� �� � � � � J � we have
max��i�N���j�J
je��ij�j � O�hJ�� ��������
Furthermore� since the exact and numerical solutions satisfy the dierential equation
������� at the collocation points� we have
e���ij� � A��ij�e��ij��
Taking a vector norm and using ��������
max��i�N���j�J
je���ij�j � O�hJ�� ��������
Finally� applying the interpolation formula ��� to y� and using ������a�� we have
e��x� �JX
k��
e���i�k�Lk�� �R��� ��������
Dierentiating and using ������ yields the result ������b��
This result is not as sharp as it could be as described by the next Theorem�
Theorem ������ Suppose that the collocation points are distinct with �i�� � �i�� � � � � ��i�J i � �� �� � � � � N the one�step method ������b� is accurate to O�hp� and A�x��b�x� �Cp�a� b�� Then
jY�xi�� y�xi�j � O�hp�� i � �� �� � � � � N� �������a�
jY��ij�� y��ij�j � O�hJ��i � �O�hp�� j � �� �� � � � � J� i � �� �� � � � � N� �������b�
��
Remark� If p � J convergence at the nodes and collocation points is at a higher rate
than implied by Theorem ������ This is the phenomenon of superconvergence that we
illustrated in Section ����
Proof� As in Section ���� introduce the Green�s function
e��� �
Z b
a
G��� x�Le�x�dx �JX
j��
Z xj
xj��
G��� x�Le�x�dx� ��������
Let us assume that � �� �xj��� xj� and� following the logic introduced in Section ���� write
G��� x�Le�x� � w�x�JY
j��
�x� �ij�
The function w�x� involves the J th derivative of Le� Using this and ������b�� we may
expect w�x� to have p� J bounded derivatives� Thus� expand w�x� in a Taylor�s series
of the form
w�x� � Pp�J���x� �O�hp�Ji �
where Pp�J���x� is a polynomial of degree p� J � ��
If the one�step method is accurate to order p then
Z xj
xj��
Pp�J���x�JY
j��
�x� �ij�dx � ��
SinceJY
j��
�x� �ij� � O�hJi �
we have Z xj
xj��
G��� x�Le�x�dx � O�hP�Ji �O�hJi �hi� � �� �xj��� xj�� �������a�
The result �������a� is obtained by summing the above relation over the subintervals�
When � � �xj��� xj� then we are only able to show thatZ xj
xj��
G��� x�Le�x�dx � O�hJ��i �� � � �xj��� xj�� �������b�
Summing �������a�b� yields �������b��
Superconvergence occurs whenever p � J � ��
��
Bibliography
�� U�M� Ascher� R� Mattheij� and R� Russell� Numerical Solution of Boundary Value
Problems for Ordinary Di erential Equations� SIAM� Philadelphia� second edition�
�����
�� C� de Boor� A Practical Guide to Splines� Springer�Verlag� New York� �����
�� C� de Boor and B� Swartz� Collocation at gaussian points� SIAM J� Numer� Anal��
���������� �����
�� J�E� Flaherty and W� Mathon� Collocation with polynomial and tension splines for
singularly perturbed boundary value problems� SIAM J� Sci� Stat� Comput� �����
���� �����
�� E� Isaacson and H�B� Keller� Analysis of Numerical Methods� John Wiley and Sons�
New York� ���
��