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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� ���

��