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Chapter �

One�Step Methods

��� Introduction

The explicit and implicit Euler methods for solving the scalar IVP

y� � f�t� y�� y��� � y�� �������

have an O�h� global error which is too low to make them of much practical value� With

such a low order of accuracy� they will be susceptible to round o� error accumulation�

Additionally� the region of absolute stability of the explicit Euler method is too small�

Thus� we seek higherorder methods which should provide greater accuracy than either

the explicit or implicit Euler methods for the same step size� Unfortunately� there is

generally a trade o� between accuracy and stability and we will typically obtain one at

the expense of the other�

Since we were successful in using Taylors series to derive a method� let us proceed

along the same lines� this time retaining terms of O�hk��

y�tn� � y�tn��� � hy��tn��� �h�

�y���tn��� � � � ��

hk

k y�k��tn��� �O�hk���� �������

Clearly� methods of this type will be explicit� Using �������

y��tn��� � f�tn��� y�tn����� ������a�

Di�erentiating �������

y���tn��� � �ft � fyy���tn���y�tn���� � �ft � fyf ��tn���y�tn����� ������b�

Continuing in the same manner

y����tn��� � �ftt � �ftyf � ftfy � fyyf� � f �

y f ��tn���y�tn����� ������c�

etc�

Speci�c methods are obtained by truncating the Taylors series at di�erent values of

k� For example� if k � � we get the method

yn � yn�� � hf�tn��� yn��� �h�

��ft � fyf ��tn�� �yn���� ������a�

From the Taylors series expansion �������� the local error of this method is

dn �h�

�y�����n�� ������b�

Thus� we succeeded in raising the order of the method� Unfortunately� methods of this

type are of little practical value because the partial derivatives are di�cult to evaluate

for realistic problems� Any software would also have to be problem dependent� By way

of suggesting an alternative� consider the special case of ������� when f is only a function

of t� i�e��

y� � f�t�� y��� � y��

This problem� which is of little interest� can be solved by quadrature to yield

y�t� � y� �

Z t

f���d��

We can easily construct highorder approximate methods for this problem by using nu

merical integration� Thus� for example� the simple leftrectangular rule would lead to

Eulers method� The midpoint rule with a step size of h would give us

y�h� � y� � hf�h��� �O�h���

Thus� by shifting the evaluation point to the center of the interval we obtained a higher

order approximation� Neglecting the local error term and generalizing the method to the

interval tn�� � t � tn yields

yn � yn�� � hf�tn�� � h����

Runge ���� sought to extend this idea to true di�erential equations having the form

of �������� Thus� we might consider

yn � yn�� � hf�tn � h��� yn�����

as an extension of the simple midpoint rule to �������� The question of how to de�ne

the numerical solution yn���� at the center of the interval remains unanswered� A simple

possibility that immediately comes to mind is to evaluate it by Eulers method� This

gives

yn���� � yn�� �h

�f�tn��� yn����

however� we must verify that this approximation provides an improved order of accuracy�

After all� Eulers method has an O�h�� local error and not an O�h�� error� Lets try to

verify that the combined scheme does indeed have an O�h�� local error by considering

the slightly more general scheme

yn � yn�� � h�b�k� � b�k��� ������a�

where

k� � f�tn��� yn���� ������b�

k� � f�tn�� � ch� yn�� � hak��� ������c�

Schemes of this form are an example of Runge�Kutta methods� We see that the proposed

midpoint scheme is recovered by selecting b� � �� b� � �� c � ���� and a � ���� We

also see that the method does not require any partial derivatives of f�t� y�� Instead� �the

potential� highorder accuracy is obtained by evaluating f�t� y� at an additional time�

The coe�cients a� b�� b�� and c will be determined so that a Taylors series expansion

of ������� using the exact ODE solution matches the Taylors series expansion �������

������ of the exact ODE solution to as high a power in h as possible� To this end� recall

the formula for the Taylors series of a function of two variables

F ��t� �� �y � �� � F ��t� �y� � ��Ft � �Fy���t��y� ��

��� �Ftt � ���Fty � ��Fyy���t��y� � � � �

�������

The expansion of ������� requires substitution of the exact solution y�t� into the formula

and the use of ������� to construct an expansion about �tn��� y�tn����� The only term

that requires any e�ort is k�� which� upon insertion of the exact ODE solution� has the

form

k� � f�tn�� � ch� y�tn��� � haf�tn��� y�tn������

To construct an expansion� we use ������� with F �t� y� � f�t� y�� �t � tn��� �y � y�tn����

� � ch� and � � haf�tn��� y�tn����� This yields

k� � f � chft � haffy ��

���ch��ftt � �ach�ffty � �ha��f �fyy� � O�h���

All arguments of f and its derivatives are �tn��� y�tn����� We have suppressed these to

simplify writing the expression�

Substituting the above expansion into ������a� while using ������b� with the exact

ODE solution replacing yn�� yields

y�tn� � y�tn��� � h�b�f � b��f � chft � haffy �O�h����� �������

Similarly� substituting ������� into �������� the Taylors series expansion of the exact

solution is

y�tn� � y�tn��� � hf �h�

��ft � ffy� �O�h��� �������

All that remains is a comparison of terms of the two expansions ������� and �������� The

constant terms agree� The O�h� terms will agree provided that

b� � b� � �� ������a�

The O�h�� terms of the two expansions will match if

cb� � ab� � ���� ������b�

A simple analysis would reveal that higher order terms in ������� and ������� cannot be

matched� Thus� we have three equations ������� to determine the four parameters a� b��

b�� and c� Hence� there is a one parameter family of methods and well examine two

speci�c choices�

�� Select b� � �� then a � c � ��� and b� � �� Using �������� this RungeKutta

formula is

yn � yn�� � hk� �������a�

with

k� � f�tn��� yn���� k� � f�tn�� � h��� yn�� � hk����� �������b�

Eliminating k� and k�� we can write �������� as

yn � yn�� � hf�tn�� � h��� yn�� � hf�tn��� yn������� �������a�

or

�yn���� � yn�� �h

�f�tn��� yn���� �������b�

yn � yn�� � hf�tn�� � h��� �yn������ �������c�

This is the midpoint rule integration formula that we discussed earlier� The � on

yn���� indicates that it is an intermediate rather than a �nal solution� As shown

in Figure ������ we can regard the twostage process �������b�c� as the result of two

explicit Euler steps� The intermediate solution �yn���� is computed at tn � h�� in

the �rst �predictor� step and this value is used to generate an approximate slope

f�tn � h��� �yn����� for use in the second �corrector� Euler step� According to Gear

����� this method has been called the EulerCauchy� improved polygon� Heun� or

modi�ed Euler method� Since there seems to be some disagreement about its name

and because of its similarity to midpoint rule integration� well call it the midpoint

rule predictorcorrector�

�� Select b� � ���� then a � c � � and b� � ���� According to �������� this Runge

Kutta formula is

yn � yn�� �h

��k� � k�� �������a�

with

k� � f�tn��� yn���� k� � f�tn�� � h� yn�� � hk��� �������b�

Again� eliminating k� and k��

yn � yn�� �h

��f�tn��� yn��� � f�tn� yn�� � hf�tn��� yn������ �������a�

This too� can be written as a twostage formula

�yn � yn�� � hf�tn��� yn���� �������b�

yn � yn�� �h

��f�tn��� yn��� � f�tn� �yn��� �������c�

The formula �������a� is reminiscent of trapezoidal rule integration� The combined

formula �������b�c� can� once again� be interpreted as a predictorcorrector method�

Thus� as shown in Figure ������ the explicit Euler method is used to predict a

solution at tn and the trapezoidal rule is used to correct it there� Well call ��������

������� the trapezoidal rule predictorcorrector� however� it is also known as the

improved tangent� improved polygon� modi�ed Euler� or EulerCauchy method

������ Chapter ���

Using De�nition ������ we see that the Taylors series method ������� and the Runge

Kutta methods �������� and �������� are consistent to order two since their local errors

are all O�h�� �hence� their local discretization errors are O�h����

Problems

�� Solve the IVP

y� � f�t� y� � y � t� � �� y��� � ���

on � � t � � using the explicit Euler method and the midpoint rule� Use several

step sizes and compare the error at t � � as a function of the number of evaluations

of f�t� y�� The midpoint rule has twice the number of function evaluations of the

Euler method but is higher order� Which method is preferred�

t t t

y

n-1 n

n

yn-1/2

h/2 h/2

yn-1

slopen-1 n-1/2

f(t + h/2,y )^

Figure ������ Midpoint rule predictorcorrector �������b�c� for one time step�

t t t

y

n-1 n

n

yn-1

h

yn

^

slopen n

f(t ,y )^

average slope

Figure ������ Trapezoidal rule predictorcorrector �������b�c� for one time step�

��� Explicit Runge�Kutta Methods

We would like to generalize the second order RungeKutta formulas considered in Section

��� to higher order� As usual� we will apply them to the scalar IVP �������� RungeKutta

methods belong to a class called one�step methods that only require information about

the solution at time tn�� to calculate it at tn� This being the case� its possible to write

them in the general form

yn � yn�� � h��tn��� yn��� h�� �������

This representation is too abstract and well typically consider an sstage RungeKutta

formula for the numerical solution of the IVP ������� in the form

yn � yn�� � hsX

i��

biki� ������a�

where

ki � f�tn�� � cih� yn�� � hsX

j��

aijkj�� i � �� �� � � � � s� ������b�

These formulas are conveniently expressed as a tableau or a �Butcher diagram�

c� a�� a�� � � � a�sc� a�� a�� � � � a�s���

������

� � ����

cs as� as� � � � assb� b� � � � bs

or more compactly as

c A

b

We can also write ������� in the form

yn � yn�� � hsX

i��

bif�tn�� � cih� Yi�� ������a�

where

Yi � yn�� � hsX

j��

aijf�tn�� � cjh� Yj�� i � �� �� � � � � s� ������b�

In this form� Yi� i � �� �� � � � � s� are approximations of the solution at t � tn � cih that

typically do not have as high an order of accuracy as the �nal solution yn�

An explicit RungeKutta formula results when aij � � for j � i� Historically� all

RungeKutta formulas were explicit� however� implicit formula are very useful for sti�

systems and problems where solutions oscillate rapidly� Well study explicit methods in

this section and take up implicit methods in the next�

RungeKutta formulas are derived in the same manner as the secondorder methods

of Section ���� Thus� we

�� expand the exact solution of the ODE in a Taylors series about� e�g�� tn���

�� substitute the exact solution of the ODE into the RungeKutta formula and ex

panded the result in a Taylors series about� e�g�� tn��� and

�� match the two Taylors series expansions to as high an order as possible� The

coe�cients are usually not uniquely determined by this process� thus� there are

families of methods having a given order�

A RungeKutta method that is consistent to order k �or simply of order k� will match

the terms of order hk in both series� Clearly the algebra involved in obtaining these for

mulas increases combinatorically with increasing order� A symbolic manipulation system�

such as MAPLE or MATHEMATICA� can be used to reduce complexity� Fortunately�

the derivation is adequately demonstrated by the secondorder methods presented in

Section ��� and� for the most part� we will not need to present detailed derivations of

higherorder methods�

There are three oneparameter families of threestage� thirdorder explicit Runge

Kutta methods ��� ���� However� the most popular explicit methods are of order four�

Their tableau has the general form

� � � � �c� a�� � � �c� a�� a�� � �c a� a� a� �

b� b� b� b

The Taylors series produce eleven equations for the thirteen nonzero parameters listed

above� The classical RungeKutta method has the following form�

yn � yn�� �h

��k� � �k� � �k� � k�� ������a�

where

k� � f�tn��� yn���� ������b�

k� � f�tn�� � h��� yn�� � hk����� ������c�

Order� k � � � � � � � � �Min� Fn� Evals� � � � � � � � �� ��

Table ������ Minimum number of function evaluations for explicit RungeKutta methodsof various orders�

k� � f�tn�� � h��� yn�� � hk����� ������d�

k � f�tn�� � h� yn�� � hk��� ������e�

Some observations about this method follow�

�� The local error of ������� is O�h�� In order to get an a priori estimate of the local

error we have to subtract the two Taylors series representations of the solution�

This is very tedious and typically does not yield a useful result� RungeKutta

methods do not yield simple a priori error estimates�

�� Four function evaluations are required per time step�

�� In the �unlikely� case when f is a function of t only� ������� reduces to

yn � yn�� �h

��f�tn��� � �f�tn����� � f�tn���

which is the same as Simpsons rule integration�

Our limited experience with RungeKutta methods would suggest that the number

of function evaluations increases linearly with the order of the method� Unfortunately�

Butcher ��� showed that this is not the case� Some key results are summarized in Table

������ The popularity of the fourstage� fourthorder RungeKutta methods are now clear�

From Table ������ we see that a �fth order RungeKutta method requires an additional

two function evaluations per step� Additionally� Butcher ��� showed that an explicit

sstage RungeKutta method will have an order of at least s� ��

Although RungeKutta formulas are tedious to derive� we can make a few general

observations� An order one formula must be exact when the solution of the ODE is a

linear polynomial� Were this not true� it wouldnt annihilate the constant and linear

��

terms in a Taylors series expansion of the exact ODE solution and� hence� could not

have the requisite O�h�� local error to be �rstorder accurate� Thus� the RungeKutta

method should produce exact solutions of the di�erential equations y� � � and y� � ��

The constantsolution condition is satis�ed identically by construction of the Runge

Kutta formulas� Using ������a�� the latter �linearsolution� condition with y�t� � t and

f�t� y� � � implies

tn � tn�� � hsX

i��

bi

or

sXi��

bi � �� ������a�

If we also require the intermediate solutions Yi to be �rst order� then the use of ������b�

with Yi � tn�� � cih gives

ci �sX

j��

aij� i � �� �� � � � � s� ������b�

This condition does not have to be satis�ed for loworder RungeKutta methods �����

however� its satisfaction simpli�es the task of obtaining order conditions for higherorder

methods� Methods that satisfy ������b� also treat autonomous and nonautonomous

systems in a symmetric manner �Problem ���

We can continue this process to higher orders� Thus� the RungeKutta method will

be of order p if it is exact when the di�erential equation and solution are

y� � �t� tn���l��� y�t� �

l�t� tn���

l� l � �� �� � � � � p�

�The use of t � tn�� as a variable simpli�es the algebraic manipulations�� Substituting

these solutions into ������a� implies that

hl

l� h

sXj��

bi�cih�l��

or

sXi��

bicl��i �

l� l � �� �� � � � � p� ������c�

��

Conditions ������c� are necessary for a method to be order p� but may not be su�cient�

Note that there is no dependence on the coe�cients aij i� j � �� �� � � � � s� in formulas

������a�c�� This is because our strategy of examining simple di�erential equations is

not matching all possible terms in a Taylors series expansion of the solution� This� as

noted� is a tedious operation� Butcher developed a method of simplifying the work by

constructing rooted trees that present the order conditions in a graphical way� They

are discussed in many texts �e�g�� ���� ����� however� they are still complex and we will

not pursue them here� Instead� well develop additional necessary order conditions by

considering the simple ODE

y� � y�

Replacing f�t� y� in ������� by y yields

yn � yn�� � hsX

i��

biYi�

Yi � yn�� � hsX

j��

aijYj� i � �� �� � � � � s�

Its simpler to use vector notation

yn � yn�� � hbTY� Y � yn��l� hAY

where

Y � �Y�� Y�� � � � � Ys�T � ������a�

l � ��� �� � � � � ��T � ������b�

A �

�����a�� a�� � � � a�sa�� a�� � � � a�s���

���� � �

���as� as� � � � ass

����� � ������c�

and

b � �b�� b�� � � � � bs�T � ������d�

��

Eliminating Y� we have

Y � yn���I� hA���l

and

yn � yn�� � hyn��bT �I� hA���l�

Assuming that yn�� is exact� the exact solution of this test equation is yn � ehyn���

Expanding this solution and �I� hA��� in series

� � h� � � ��hk

k � � � � � � � hbT �I� hA� � � �� hkAk � � � � �l�

Equating like powers of h yields the order condition

bTAk��l ��

k � k � �� �� � � � � p� �������

We recognize that this condition with k � � is identical to ������a�� Letting c �

�c�� c�� � � � � cs�T � we may write ������c� with l � � in the form bTc � ���� The

vector form of ������b� is Al � c� Thus� bTAl � ���� which is the same as ������� with

k � �� Beyond k � �� the order conditions ������c� and ������� are independent�

Although conditions ������� and ������� are only necessary for a method to be of order

p� they are su�cient in many cases� The actual number of conditions for a RungeKutta

method of order p are presented in Table ����� ����� These results assume that ������b�

has been satis�ed�

Order� p � � � � � � � � � ��No� of Conds� � � � � �� �� �� ��� ��� ����

Table ������ The number of conditions for a RungeKutta method of order p �����

Theorem ������ The necessary and su�cient conditions for a Runge�Kutta method

������� to be of second order are ������c�� l � �� �� and ������� k � �� If ������b� is

satised then �������� k � �� �� are necessary and su�cient for second�order accuracy�

Proof� We require numerous Taylors series expansions� To begin� we expand f�tn�� �

cih� Yi� using ������� to obtain

f�tn�� � cih� Yi� � f � ftcih� fy�Yi � y�tn���� ��

��ftt�cih�

� � �fty�cih��Yi � y�tn�����

��

fyy�Yi � y�tn������ �O�h���

All arguments of f and its derivatives are at �tn��� y�tn����� They have been suppressed

for simplicity�

Substituting the exact ODE solution and the above expression into ������a� yields

y�tn� � y�tn��� � hsX

i��

bi�f � ftcih� fy�Yi � y�tn���� �O�h����

The expansion of Yi�y�tn��� will� fortunately� only require the leading term� thus� using

������b�

Yi � y�tn��� � hsX

j��

aijf �O�h���

Hence� we have

y�tn� � y�tn��� � hsX

i��

bi�f � ftcih� hffy

sXj��

aij �O�h����

Equating terms of this series with the Taylors series ������� of the exact solution yields

������c� with l � �� ������c� with l � �� and ������� with k � �� We have demonstrated

the equivalence of these conditions when ������b� is satis�ed�

Remark �� The results of Theorem ����� and conditions ������� and ������� apply to

both explicit and implicit methods�

Let us conclude this section with a brief discussion of the absolute stability of explicit

methods� We will present a more detailed analysis in Section ���� however� the present

material will serve to motivate the need for implicit methods� Thus� consider an sstage

explicit RungeKutta method applied to the test equation

y� � �y� �������

Using ������� in ������� with the simpli�cation that aij � �� j � i� for explicit methods

yields

yn � yn�� � zsX

i��

biYi � yn�� � zbTY ������a�

��

where

Yi � yn�� � zi��Xj��

aijYj� i � �� �� � � � � s� ������b�

and

z � h�� ������c�

The vector form of ������� is

Y � yn��l� zAY� ������d�

Using this to eliminate Y in ������a�� we have

yn � yn���� � zbT �I� zA���l��

Expanding the inverse

yn � yn���� � zbT �I� zA� � � �� zkAk � � � � �l�

Using �������

yn � R�z�yn�� �������a�

where

R�z� � � � z �z�

�� � � ��

zp

p �

�Xj�p��

zjbTAj��l�

The matrix A is strictly lower triangular for an sstage explicit RungeKutta method�

thus� Aj�� � �� j s� Therefore�

R�z� � � � z �z�

�� � � ��

zp

p �

sXj�p��

zjbTAj��l� �������b�

In particular� for explicit sstage methods with p � s � �� we have

R�z� � � � z �z�

�� � � ��

zp

p � s � p � �� �������c�

The exact solution of the test equation ������� is

y�tn� � eh�y�tn����

��

thus� as expected� a pthorder RungaKutta formula approximates a Taylors series ex

pansion of the exact solution through terms of order p�

Using De�nition ����� and ��������� the region of absolute stability of an explicit

RungeKutta method is

jR�z�j ������� � z �

z�

�� � � ��

zp

p �

sXj�p��

zjbTAj��l

����� � �� �������a�

In particular�

jR�z�j ������ � z �

z�

�� � � ��

zp

p

���� � �� s � p � �� �������b�

Since no RungeKutta coe�cients appear in �������b�� we have the interesting result�

Lemma ������ All p�stage explicit Runge�Kutta methods of order p � � have the same

region of absolute stability�

Since jei�j � �� � � � ��� we can determine the boundary of the absolute stability

regions �������a�b� by solving the nonlinear equation

R�z� � ei�� ��������

Clearly� �������� implies that jyn�yn��j � �� For p � � �i�e�� for Eulers method�� the

boundary of the absolutestability region is determined as

� � z � ei��

which can easily be recognized as the familiar unit circle centered at z � �� � �i� For

real values of z the intervals of absolute stability for methods with p � s � � are shown

in Table ������ Absolute stability regions for complex values of z are illustrated for the

same methods in Figure ������ Methods are stable within the closed regions shown� The

regions of absolute stability grow with increasing p� When p � �� �� they also extend

slightly into the right half of the complex zplane�

Problems

�� Instead of solving the IVP �������� many software systems treat an autonomous

ODE y� � f�y�� Nonautonomous ODEs can be written as autonomous systems

��

Order� p Interval ofAbsolute Stability

� ������ ������ ��������� ��������

Table ������ Interval of absolute stability for pstage explicit RungeKutta methods oforder p � �� �� �� ��

−5 −4 −3 −2 −1 0 1−3

−2

−1

0

1

2

3

Re(z)

Im(z)

Figure ������ Region of absolute stability for pstage explicit RungeKutta methods oforder p � �� �� �� � �interiors of smaller closed curves to larger ones��

��

by letting t be a dependent variable satisfying the ODE t� � �� A RungeKutta

method for an autonomous ODE can be obtained from� e�g�� ������� by dropping

the time terms� i�e��

yn � yn�� � hsX

i��

bif�Yi��

with

Yi � yn�� � hsX

j��

aijf�Yj�� i � �� �� � � � � s�

The RungeKutta evaluation points ci� i � �� �� � � � � s� do not appear in this form�

Show that RungeKutta formulas ������� and the one above will handle autonomous

and nonautonomous systems in the same manner when ������b� is satis�ed�

��� Implicit Runge�Kutta Methods

Well begin this section with a negative result that will motivate the need for implicit

methods�

Lemma ������ No explicit Runge�Kutta method can have an unbounded region of abso�

lute stability�

Proof� Using ��������� the region of absolute stability of an explicit RungeKutta method

satis�es

jyn�yn��j � jR�z�j � �� z � h��

where R�z� is a polynomial of degree s� the number of stages of the method� Since R�z�

is a polynomial� jR�z�j � � as jzj � � and� thus� the stability region is bounded�

Hence� once again� we turn to implicit methods as a means of enlarging the region of

absolute stability�

Necessary order conditions for sstage implicit RungeKutta methods are given by

������c� ������ �with su�cient conditions given in Hairer et al� ����� Section II���� A

condition on the maximum possible order follows�

Theorem ������ The maximum order of an implicit s�stage Runge�Kutta method is �s�

��

Proof� cf� Butcher ����

The derivations of implicit RungeKutta methods follow those for explicit methods�

Well derive the simplest method and then give a few more examples�

Example ������ Consider the implicit �stage method obtained from ������� with s � �

as

yn � yn�� � hb�f�tn�� � c�h� Y��� ������a�

Y� � yn�� � ha��f�tn�� � c�h� Y��� ������b�

To determine the coe�cients c�� b�� and a��� we substitute the exact ODE solution into

������a�b� and expand ������a� in a Taylors series

y�tn� � y�tn��� � hb��f � c�hft � fy�Y� � y�tn���� �O�h����

where f �� f�tn��� y�tn����� etc� Expanding ������b� in a Taylors series and substituting

the result into the above expression yields

y�tn� � y�tn��� � hb��f � c�hft � ha��ffy �O�h����

Comparing the terms of the above series with the Taylors series

y�tn� � y�tn��� � hf �h�

��ft � ffy� �O�h��

of the exact solution yields

b� � �� a�� � c� ��

��

Substituting these coe�cients into �������� we �nd the method to be an implicit midpoint

rule

yn � yn�� � hf�tn�� � h��� Y��� ������a�

Y� � yn�� �h

�f�tn�� � h��� Y��� ������b�

The tableau for this method is

��

��

��

The formula has similarities to the midpoint rule predictorcorrector ��������� however�

there are important di�erences� Here� the backward Euler method �rather than the

forward Euler method� may be regarded as furnishing a predictor ������b� with the

midpoint rule providing the corrector ������a�� However� formulas ������a� and ������b�

are coupled and must be solved simultaneously rather than sequentially�

Example ������ The twostage method having maximal order four presented in the

following tableau was developed by Hammer and Hollingsworth �����

���

p��

��

p��

���

p��

��

p��

��

��

This method is derived in Gear ����� Section ����

Example ������ Let us examine the region of absolute stability of the implicit midpoint

rule �������� Thus� applying ������� to the test equation ������� we �nd

Y� � yn�� �h�

�Y�

and

yn � yn�� � h�Y��

Solving for Y�

Y� �yn��

�� h����

and eliminating it in order to explicitly determine yn as

yn �

�� �

h�

�� h���

�yn�� �

�� � h���

�� h���

�yn���

Thus� the region of absolute stability is interior to the curve

� � z��

�� z��� ei�� z � h��

Solving for z

z � ��� ei�

� � ei�� ��e

i��� � e�i���

ei��� � e�i���� ��i tan ���

��

λ)Re(h

Im(h λ)

Figure ������ Region of absolute stability for the implicit midpoint rule ��������

Since z is imaginary� the implicit midpoint rule is absolutely stable in the entire negative

half of the complex z plane �Figure �������

Let us generalize the absolute stability analysis presented in Example ����� before

considering additional methods� This analysis will be helpful since we will be interested

in developing methods with very large regions of absolute stability� Thus� we apply the

general method ������� to the test equation ������� to obtain

yn � yn�� � zbTY� ������a�

�I� zA�Y � yn��l� ������b�

where Y� l� A� and b and are de�ned by ������� and z � h��

Eliminating Y in ������a� by using ������b� we �nd

yn � R�z�yn��� ������a�

where

R�z� � � � zbT �I� zA���l� ������b�

The region of absolute stability is the set of all complex z where jR�z�j � �� While R�z�

is a polynomial for an explicit method� it is a rational function for an implicit method�

��

Hence� the region of absolute stability can be unbounded� As shown in Section ���� a

method of order p will satisfy

R�z� � ez �O�zp����

Rationalfunction approximations of the exponential are called Pad�e approximations�

De�nition ������ The �j� k� Pad�e approximation Rjk�z� is the maximum�order approx�

imation of ez having the form

Rjk�z� �Pk�z�

Qj�z��

p� � p�z � � � �� pkzk

q� � q�z � � � �� qjzj� ������a�

where Pk and Qj have no common factors�

Qj��� � q� � �� ������b�

and

Rjk�z� � ez �O�zk�j���� ������c�

With Rjk normalized by ������b�� there are k � j � � undetermined parameters in

������a� that can be determined by matching the �rst k � j � � terms in the Taylors

series expansion of ez� Thus� the error of the approximation should be O�zk�j���� Using

������c�� we have

k�jXi��

zi

i �

Pki�� piz

iPji�� qiz

i�O�zk�j��� �������

Equating the coe�cients of like powers of z determines the parameters pi� i � �� �� � � � � k

and qi� i � �� �� � � � � j�

Example ���� � Find the ����� Pad�e approximation of ez� Setting j � � and k � � in

������� gives

�� � z �z�

���� � q�z � q�z

�� � p��

Equating the coe�cients of zi� i � �� �� �� gives

p� � �� � � q� � ���

�� q� � q� � ��

��

Thus�

p� � �� q� � ��� q� � ����

Using �������� the ����� Pad�e approximation is

R���z� ��

�� z � z����

Additionally�

ez � R���z� �O�z���

Some other Pad�e approximations are presented in Table ������ We recognize that

the ����� approximation corresponds to Eulers method� the ����� method corresponds

to the backward Euler method� and the ����� approximation corresponds to the mid

point rule� �The ����� approximation also corresponds to the trapezoidal rule�� Methods

corresponding to the �s� s� diagonal Pad�e approximations are Butchers maximum order

implicit RungeKutta methods �Theorem �������

k � � � �j � � � � � z � � z � z���

� ���z

��z����z��

���z���z�����z��

� ���z�z���

��z�����z���z���

��z���z������z���z����

Table ������ Some Pad�e approximations of ez�

Theorem ������ There is one and only one �s�order s�stage implicit Runge�Kutta for�

mula and it corresponds to the �s� s� Pad�e approximation�

Proof� cf� Butcher ����

Well be able to construct several implicit RungeKutta methods having unbounded

absolutestability regions� Well want to characterize these methods according to their

behavior as jzj � � and this requires some additional notions of stability�

De�nition ������ A numerical method is A�stable if its region of absolute stability in�

cludes the entire left�half plane Re�h�� � ��

��

The relationship between Astability and the Pad�e approximations is established by

the following theorem�

Theorem ������ Methods that lead to a diagonal or one of the rst two sub�diagonals

of the Pad�e table for ez are A�stable�

Proof� The proof appears in Ehle ����� Without introducing additional properties of Pad�e

approximations� well make some observations using the results of Table ������

�� We have shown that the regions of absolute stability of the backward Euler method

and the midpoint rule include the entire lefthalf of the h� plane� hence� they are

Astable�

�� The coe�cients of the highestorder terms of Ps�z� and Qs�z� are the same for

diagonal Pad�e approximations Rss�z�� hence� jRss�z�j � � as jzj � � and these

methods are Astable �Table �������

�� For the subdiagonal ����� and ����� Pad�e approximations� jR�z�j � � as jzj � �and these methods will also be Astable�

It is quite di�cult to �nd highorder Astable methods� Implicit RungeKutta meth

ods provide the most viable approach� Examining Table ������ we see that we can intro

duce another stability notion�

De�nition ������ A numerical method is L�stable if it is A�stable and if jR�z�j � � as

jzj � ��

The backward Euler method and� more generally� methods corresponding to sub

diagonal Pad�e approximations in the �rst two bands are Lstable ������ Section IV����

Lstable methods are preferred for sti� problems where Re��� � � but methods where

jR�z�j � � are more suitable when Re��� � � but jIm���j � �� i�e�� when solutions

oscillate rapidly�

Explicit RungeKutta methods are easily solved� but implicit methods will require

an iterative solution� Since implicit methods will generally be used for sti� systems�

��

Newtons method will be preferred to functional iteration� To emphasize the di�culty�

well illustrate RungeKutta methods of the form ������� for vector IVPs

y� � f�t�y�� y��� � y�� �������

where y� etc� are mvectors� The application of ������� to vector systems just requires

the use of vector arithmetic� thus�

Yi � yn�� � hsX

j��

aijf�tn�� � cjh�Yj�� i � �� �� � � � � s� ������a�

yn � yn�� � hsX

i��

bif�tn�� � cih�Yi�� ������b�

Once again� yn etc� are mvectors�

To use Newtons method� we write the nonlinear system ������a� in the form

Fi�Y��Y�� � � � �Ys� � Yi � yn�� � hsX

j��

aijf�tn�� � hcj�Yj� � �� j � �� �� � � � � s�

������a�

and get

�����I� a��J

���� �ha��J���

� �ha�sJ���s

�ha��J���� I� ha��J

���� �ha�sJ���

s

������

� � ����

�has�J���� �has�J���

� I� hassJ���s

�����������Y

����

�Y�������

�Y���s

����� � �

�����F

����

F�������

F���s

����� � ������b�

Y�����i � Y

���i � �Y���

i � i � �� �� � � � � s� � �� �� � � � � ������c�

where

J���j � fy�tn�� � hcj�Y

���j �� F

���j � Fj�Y

���� �Y

���� � � � � �Y���

s �� j � �� �� � � � � s�

������d�

For an sstage RungeKutta method applied to an mdimensional system ��������

the Jacobian in ������b� has dimension sm sm� This will be expensive for highorder

methods and highdimensional ODEs and will only be competitive with� e�g�� implicit

��

multistep methods �Chapter �� under special conditions� Some simpli�cations are possi

ble and these can reduce the work� For example� we can approximate all of the Jacobians

as

J � fy�tn���yn���� �������a�

In this case� we can even shorten the notation by introducing the Kronecker or direct

product of two matrices as

A J �

�����a��J a��J a�sJa��J a��J a�sJ���

���� � �

���as�J as�J assJ

����� � �������b�

Then� ������b� can be written concisely as

�I� hA J��Y��� � �F��� �������c�

where A was given by ������c� and

�Y��� �

������Y

����

�Y�������

�Y���s

����� � F��� �

�����F

����

F�������

F���s

����� � �������d�

The approximation of the Jacobian does not change the accuracy of the computed solu

tion� only the convergence rate of the iteration� As long as convergence remains good�

the same Jacobian can be used for several time step and only be reevaluated when

convergence of the Newton iteration slows�

Even with this simpli�cation� with m ranging into the thousands� the solution of

�������� is clearly expensive and other ways of reducing the computational cost are nec

essary� Diagonally implicit Runge�Kutta �DIRK� methods o�er one possibility� A DIRK

method is one where aij � �� i � j and at least one aii �� �� i� j � �� �� � � � � s� If� in

addition� a�� � a�� � � � � � ass � a� the technique is known as a singly diagonally implicit

Runge�Kutta �SDIRK� method� Thus� the coe�cient matrix of an SDIRK method has

��

the form

A �

�����

aa�� a���

���� � �

as� as� a

����� � ��������

Thus� with the approximation ��������� the system Jacobian in �������c� is

�I� hA J� �

�����I� haJ ��ha��J I� haJ �

������

� � ����

�has�J �has�J I� haJ

����� �

The Newton system �������� is lower block triangular and can be solved by forward

substitution� Thus� the �rst block of �������c� is solved for �Y���� � Knowing Y� the

second equation is solved for �Y���� � etc� The Jacobian J is the same for all stages� thus�

the diagonal blocks need only be factored once by Gaussian elimination and forward and

backward substitution may be used for each solution�

The implicit midpoint rule ������� is a onestage� secondorder DIRK method� Well

examine a twostage DIRK method momentarily� but �rst we note that the maximum

order of an sstage DIRK method is s� � ����

Example ������ A twostage DIRK formula has the tableau

c� a�� �c� a�� a��

b� b�

and it could be of third order� According to Theorem ������ the conditions for second

order accuracy are ������c� with l � �� � when ������b� is satis�ed� i�e��

b� � b� � �� c� � a��� c� � a�� � a��� b�c� � b�c� � ����

�As noted earlier� satisfaction of ������b� is not necessary� but it simpli�es the algebraic

manipulations�� We might guess that the remaining conditions necessary for third order

accuracy are ������c� with l � � and ������� with k � �� i�e��

b�c�� � b�c

�� � ���

and

bTA�l � bAc � b�a��c� � b��a��c� � a��c�� � ����

��

where ������b� was used to simplify the last expression� After some e�ort� this system of

six equations in seven unknowns can be solved to yield

c� ��� �c��� �c�

� b� �c� � ���

c� � c�� b� �

���� c�c� � c�

a�� � c�� a�� ����� b�c

��

b��c� � c��� a�� � c� � a���

As written� the solution is parameterized by c�� Choosing c� � ��� gives

� � � � �� � � � �

� � � �

Using �������� the method is

Y� � yn�� �h

�f�tn�� �

h

�� Y��� Y� � yn�� �

h

��f�tn�� �

h

�� Y�� � f�tn� Y����

yn � yn�� �h

���f�tn�� �

h

�� Y�� � f�tn� Y����

We can check by constructing a Taylors series that this method is indeed third order�

Hairer et al� ����� Section II��� additionally show that our necessary conditions for third

order accuracy are also su�cient in this case�

The computation of Y� can be recognized as the backward Euler method for onethird

of the time step h� The computation of Y� and yn are not recognizable in terms of simple

quadrature rules� Since the method is thirdorder� its local error is O�h��

We can also construct an SDIRK method by insisting that a�� � a��� Enforcing this

condition and using the previous relations gives two methods having the tableau

� � ��� � �� �� �

��� ���

where

� ��

���� �p

���

The method with � � �� � ��p���� is Astable while the other method has a bounded

stability region� Thus� this would be the method of choice�

��

Let us conclude this Section by noting a relationship between implicit RungeKutta

and collocationmethods� With u�t� a polynomial of degree s in t for t � tn��� a collocation

method for the IVP

y� � f�t� y�� y�tn��� � yn�� �������a�

consists of solving

u�tn��� � yn�� �������b�

u��tn�� � cih� � f�tn�� � cih� u�tn�� � cih��� i � �� �� � � � � s� �������c�

where ci� i � �� �� � � � � s� are nonnegative parameters� Thus� the collocation method

consists of satisfying the ODE exactly at s points� The solution u�tn���h� may be used

as the initial condition yn for the next time step�

Usually� the collocation points tn�� � cih are such that ci ��� ��� i � �� �� � � � � s� but

this need not be the case ��� ��� ����

Generally� the ci� i � �� �� � � � � s� are distinct and we shall assume that this is the case

here� �The coe�cients need not be distinct when the approximation u�t� interpolates

some solution derivatives� e�g�� as with Hermite interpolation�� Approximating u��t��

t � tn��� by a Lagrange interpolating polynomial of degree s� �� we have

u��t� �sX

j��

kjLj�t� tn��

h� �������a�

where

Lj��� �sY

i���i��j

� � cicj � ci

� �������b�

� �t� tn��

h� �������c�

The polynomials Lj���� j � �� �� � � � � s� are a product of s � � linear factors and are�

hence� of degree s� �� They satisfy

Lj�ci� � �ji� j� i � �� �� � � � � s� �������d�

��

where �ji is the Kronecker delta� Using �������a�� we see that u��t� satis�es the interpo

lation conditions

u��tn�� � cih� � ki� i � �� �� � � � � s� �������e�

Transforming variables in �������a� using �������c�

u�tn�� � �h� � yn�� � h

Z �

u��tn�� � �h�d�� ��������

By construction� �������� satis�es �������b�� Substituting �������e� and �������� into

�������c�� we have

ki � f�tn�� � cih� yn�� � h

sXj��

kj

Z ci

Lj���d���

This formula is identical to the typical RungeKutta formula ������b� provided that

aij �

Z ci

Lj���d�� �������a�

Similarly� using �������a� in �������� and evaluating the result at � � � yields

u�tn�� � h� � yn � yn�� � hsX

j��

kj

Z �

Lj���d��

This formula is identical to ������a� provided that

bj �

Z �

Lj���d�� �������b�

This view of a RungeKutta method as a collocation method is useful in many situations�

Let us illustrate one result�

Theorem ������ A Runge�Kutta method with distinct ci� i � �� �� � � � � s� and of order

at least s is a collocation method satisfying ��������� �������� if and only if it satises

the order conditions

sXj��

aijcq��j �

cqiq� i� q � �� �� � � � � s� ��������

Remark �� The order conditions �������� are related to the previous conditions ������c�

������ �cf� ����� Section II����

��

Proof� We use the Lagrange interpolating polynomial �������� to represent any polyno

mial P ��� of degree s� � as

P ��� �sX

j��

P �cj�Lj����

Regarding P ��� as u��tn � �h�� integrate to obtain

u�tn�� � cih�� yn�� �Z ci

P ���d� �sX

j��

P �cj�

Z ci

Lj���d�� i � �� �� � � � � s�

Assuming that �������a� is satis�ed� we haveZ ci

P ���d� �sX

j��

aijP �cj�� i � �� �� � � � � s�

Now choose P ��� � � q��� q � �� �� � � � � s� to obtain ��������� The proof of the converse

follows the same arguments �cf� ����� Section II����

Now� we might ask if there is an optimal way of selecting the collocation points� Ap

propriate strategies would select them so that accuracy and or stability are maximized�

Lets handle accuracy �rst� The following theorems discuss relevant accuracy issues�

Theorem ������ �Alekseev and Gr�obner� Let x� y� and z satisfy

x��t� �� z���� � f�t� x�t� �� z������ x��� �� z���� � z���� �������a�

y��t� � f�t� y�t��� y��� � y�� �������b�

z��t� � f�t� z�t�� � g�t� z�t��� z��� � y�� �������c�

with fy�t� y� C�� t �� Then�

z�t�� y�t� �

Z t

�x�t� �� z����

�zg��� z����d�� �������d�

Remark �� Formula �������d� is often called the nonlinear variation of parameters�

Remark �� The parameter � identi�es the time that the initial conditions are applied

in �������a�� A prime� as usual� denotes t di�erentiation�

Remark � Observe that y�t� � x�t� �� y���

��

Proof� cf� Hairer et al� ����� Section I���� and Problem ��

Theorem ����� makes it easy for us to associate the collocation error with a quadrature

error as indicated below�

Theorem ������ Consider the quadrature ruleZ tn

tn��

F �t�dt � h

Z �

F �tn�� � �h�d� � hsX

i��

biF �tn�� � cih� � Ep �������a�

where

Ep � Chp��F �p���n�� �n �tn��� tn�� �������b�

F Cp�tn��� tn�� and C is a constant� Then the collocation method �������� has order p�

Proof� Consider the identity

u� � f�t� u� � �u� � f�t� u��

and use Theorem ����� on �tn��� tn� with z�t� � u�t� and g�t� u� � u� � f�t� u� to obtain

u�tn�� y�tn� �

Z tn

tn��

xu�tn� �� u�����u����� f��� u�����d��

Replace this integral by the quadrature rule �������� to obtain

u�tn�� y�tn� � hsX

i��

bixu�tn� tn�� � cih� u�tn�� � cih���u��tn�� � cih��

f�tn�� � cih� u�tn�� � cih��� � Ep�

All terms in the summation vanish upon use of the collocation equations ��������� thus�

ju�tn�� y�tn�j � jEpj � jCjhp�� max���tn�� �tn

j �p

��pxu�tn� �� u�����u

����� f��� u��j�

It remains to show that the derivatives in the above expression are bounded as h � ��

Well omit this detail which is proven in Hairer et al� ����� Section II��� Thus�

jy�tn�� u�tn�j � �Chp�� ��������

and the collocation method �������� is of order p�

��

At last� our task is clear� We should select the collocation points ci� i � �� �� � � � � s�

to maximize the order p of the quadrature rule ��������� Well review some of the details

describing the derivation of ��������� Additional material appears in most elementary

numerical analysis texts ���� Let �F ��� � F �tn�� � �h� and approximate it by a Lagrange

interpolating polynomial of degree s� � to obtain

�F ��� �sX

j��

�F �cj�Lj��� �Ms���

s �F �s����� � ��� ��� �������a�

where

Ms��� �sY

i��

�� � ci�� �������b�

�Di�erentiation in �������a� is with respect to � � not t��

Integrate �������a� and use �������b� to obtain

Z �

�F ���d� �sX

j��

bj �F �cj� � �Es �������a�

where

�Es ��

s

Z �

Ms��� �F�s�������d� �

s

Z �

sYi��

�� � ci� �F�s�������d�� �������b�

In NewtonCotes quadrature rules� such as the trapezoidal and Simpsons rules� the

evaluation points ci� i � �� �� � � � � s� are speci�ed a priori� With Gaussian quadrature�

however� the points are selected to maximize the order of the rule� This can be done by

expanding �F �s������� in a Taylors series and selecting the ci� i � �� �� � � � � s� to annihilate

as many terms as possible� Alternatively� and equivalently� the quadrature rule can be

designed to integrate polynomials exactly to as high a degree as possible� The actual

series expansion is complicated by the fact that �F �s� is evaluated at ���� in �������b��

Isaacson and Keller ���� provide additional details on this matter� however� well sidestep

the subtleties by assuming that all derivatives of ���� are bounded so that �F �s� has an

expansion in powers of � of the form

�F �s���� � �� � ��� � � � �� �r���r�� �O�� r��

��

d Pd�x�

� �

� x

� x� � ��

� x� � �x

� x � �x�

�� �

� x � ��x�

�� x

��

Table ������ Legendre polynomials Pd�x� of degree d ��� �� on �� � x � ��

The �rst r terms of this series will be annihilated by �������b� if Ms��� is orthogonal

to polynomials of degree r � �� i�e�� ifZ �

M���� q��d� � �� q � �� �� � � � � r� ��������

Under these conditions� were we to transform the integrals in �������� and �������� back

to t dependence using �������c�� we would obtain the error of �������b� with p � s � r�

With the s coe�cients ci� i � �� �� � � � � s� we would expect the maximum value of r to

be s� According to Theorem ������ this choice would lead to a collocation method of

order �s� i�e�� a method having p � r � s � �s and an O�h�s��� local error� These are

Butchers maximal order formulas �Theorem ������ corresponding to the diagonal Pad�e

approximations�

The maximumorder coe�cients identi�ed above are the roots of the s thdegree

Legendre polynomial scaled to the interval ��� ��� The �rst six Legendre polynomials are

listed in Table ������ Additional polynomials and their roots appear in Abromowitz and

Stegun ���� Chapter ���

Example ������ According to Table ������ the roots of P��x� are x��� � ���p� on

���� ��� Mapping these to ��� �� by the linear transformation � � �� � x���� we obtain

the collocation points for the maximalorder twostage method as

c� ��

���� �p

��� c� �

��� �

�p���

��

Since this is our �rst experience with these techniques� let us verify our results by a direct

evaluation of �������� using �������b�� thus�Z �

�� � c���� � c��d� � ��

Z �

�� � c���� � c���d� � ��

Integrating�

�� c� � c�

�� c�c� � ��

�� c� � c�

��c�c��

� ��

These may easily be solved to con�rm the collocation points obtained by using the roots

of P��x�� In this case� we recognize c� and c� as the evaluation points of the Hammer

Hollingsworth formula of Example ������

With the collocation points ci� i � �� �� � � � � s� determined� the coe�cients aij and bj�

i� j � �� �� � � � � s� may be determined from �������a�b�� These maximal order collocation

formulas are Astable since they correspond to diagonal Pad�e approximations �Theorem

�������

We may not want to impose the maximal order conditions to obtain� e�g�� better

stability and computational properties� With Radau quadrature� we �x one of the coef

�cients at an endpoint� thus� we set either c� � � or cs � �� The choice c� � � leads to

methods with bounded regions of absolute stability� Thus� the methods of choice have

cs � �� They correspond to the subdiagonal Pad�e approximations and are� hence� A

and Lstable �Theorem ������� They have orders of p � �s � � ����� Section IV��� Such

excellent stability and accuracy properties makes these methods very popular for solving

sti� systems�

The Radau polynomial of degree s on �� � x � � is

Rs�x� � Ps�x�� s

�s� �Ps���x��

The roots of Rs transformed to ��� �� �using � � ��� x���� are the ci� i� �� �� � � � � s� All

values of ci� i � �� �� � � � � s� are on ��� �� with� as designed� cs � �� The onestage Radau

method is the backward Euler method� The tableau of the twostage Radau method is

�Problem ��

��

��

���

� �

��

Well conclude this Section with a discussion of singly implicit RungeKutta �SIRK�

methods� These methods are of order s� which is less than the Legendre ��s�� Radau

��s � ��� and DIRK �s � �� techniques� They still have excellent A and Lstability

properties and� perhaps� o�er a computational advantage�

A SIRK method is one where the coe�cient matrix A has a single sfold real eigen

value� These collocation methods were Originally developed by Butcher ��� and have

been subsequently extended ��� ��� �� ���� Collocating� as described� leads to the system

��������������� The intermediate solutions Yi� i � �� �� � � � � s� have the vector form spec

i�ed by ������d� with the elements of A given by �������a�� Multiplying ������d� by a

nonsingular matrix T��� we obtain

T��Y � ynT��l� hT��ATT��f

where Y� l� A� and f are� respectively� given by ������ac� and

f �

�����f�tn�� � c�h�f�tn�� � c�h�

���f�tn�� � csh�

����� � ��������

Let

�Y � T��Y� �l � T��l� �A � T��AT� �f � T��f � ��������

Butcher ��� chose the collocation points ci � ��i� i � �� �� � � � � s� where �i is the i th

root of the s thdegree Laguerre polynomial Ls�t� and � is chosen so that the numerical

method has favorable stability properties� butcher also selected T to have elements

Tij � Li����j��

Then

�A �

�������

�� �

�� � �

������� � ��������

��

Thus� �A is lower bidiagonal with the single eigenvalue �� The linearized system �������

is easily solved in the transformed variables� �A similar transformation also works with

Radau methods ������ Butcher ��� and Burrage ��� show that it is possible to �nd A

stable SIRK methods for s � �� These methods are also Lstable with the exception of

the sevenstage method�

Problems

�� Verify that �������d� is correct when f�t� y� � ay with a a constant�

�� Consider the method

yn � yn�� � h���� �f�tn��� yn��� � f�tn� yn��

with ��� ��� The method corresponds to the Euler method with � �� the

trapezoidal rule with � ���� and the backward Euler method and when � ��

���� Write the RungeKutta tableau for this method�

���� For what values of is the method Astable� Justify your answer�

�� Radau or Lobatto quadrature rules have evaluation points at one or both endpoints

of the interval of integration� respectively� Consider the two twostage RungeKutta

methods based on collocation at Radau points� In one� the collocation point c� � �

and in the other the collocation point c� � �� In each case� the other collocation

point �c� for the �rst method and c� for the second method� is to be determined so

that the resulting method has as high an order of accuracy as possible�

���� Determine the parameters aij� bj� and ci� i� j � �� � for the two collocation

methods and identify their orders of accuracy�

���� To which elements of the Pad�e table do these methods correspond�

���� Determine the regions of absolute stability for these methods� Are the meth

ods A and or Lstable�

��

��� Convergence� Stability� Error Estimation

The concepts of convergence� stability� and a priori error estimation introduced in Chap

ter � readily extend to a general class of �explicit or implicit� onestep methods having

the form

yn � yn�� � h��tn��� yn��� h�� ������a�

Again� consider the scalar IVP

y� � f�t� y�� y��� � y�� ������b�

and� to begin� well show that onestep methods are stable when � satis�es a Lipschitz

condition on y�

Theorem ������ If ��t� y� h� satises a Lipschitz condition on y then the one�step method

��� ��a� is stable�

Proof� The analysis follows the lines of Theorem ������ Let yn and zn satisfy method

������� and

zn � zn�� � h��tn��� zn��� h�� z� � y� � ��� �������

respectively� Subtracting ������� from �������

yn � zn � yn�� � zn�� � h���tn��� yn��� h�� ��tn��� zn��� h���

Using the Lipschitz condition

jyn � znj � �� � hL�jyn�� � zn��j�

Iterating the above inequality leads to

jyn � znj � �� � hL�njy� � z�j�

Using ��������

jyn � znj � enhLj��j � eLT � � k��

since nh � T and j��j � ��

��

Example �� ��� The function � satis�es a Lipschitz condition whenever f does� Con

sider� for example� the explicit midpoint rule which has the form of ������a� with

��t� y� h� � f�t� h��� y � hf�t� y�����

Then�

j��t� y� h�� ��t� z� h�j � jf�t� h��� y � hf�t� y����� f�t� h��� z � hf�t� z����j

Using the Lipschitz condition on f

j��t� y� h�� ��t� z� h�j � Ljy � hf�t� y���� z � hf�t� z���j

or

j��t� y� h�� ��t� z� h�j � L�jy � zj � �h���jf�t� y�� f�t� z�j�

or

j��t� y� h�� ��t� z� h�j � L�� � hL���jy � zj�

Thus� we can take the Lipschitz constant for � to be L�� � �hL��� for h ��� �h��

In addition to a Lipschitz condition� convergence of the onestep method ������a�

requires consistency� Recall �De�nition ������� that consistency implies that the local

discretization error limh�� �n � �� Consistency is particularly simple for a onestep

method�

Lemma ������ The one�step method ��� ��a� is consistent with the ODE y� � f�t� y� if

��t� y� �� � f�t� y�� �������

Proof� The local discretization error of ������a� satis�es

�n �y�tn�� y�tn���

h� ��tn��� y�tn���� h��

Letting h tend to zero

limh��

�n � y��tn���� ��tn��� y�tn���� ���

Using the ODE to replace y� yields the result�

��

Theorem ������ Let ��t� y� h� be a continuous function of t� y� and h on � � t � T �

�� � y � �� and � � h � �h� respectively� and satisfy a Lipschitz condition on y�

Then the one�step method ��� ��a� converges to the solution of ��� ��b� if and only if it

is consistent�

Proof� Let z�t� satisfy the IVP

z� � ��t� z� ��� z��� � y�� �������

and let zn� n � �� satisfy

zn � zn�� � h��tn��� zn��� h�� n � �� z� � y�� �������

Using the mean value theorem and �������

z�tn�� z�tn��� � hz��tn�� � hn� � h��tn�� � hn� z�tn�� � hn�� ��� �������

where n ��� ��� Let

en � z�tn�� zn �������

and subtract ������� from ������� to obtain

en � en�� � h���tn�� � hn� z�tn�� � hn�� ��� ��tn��� zn��� h���

Adding and subtracting similar terms

en � en�� � h���tn�� � hn� z�tn�� � hn�� ��� ��tn��� z�tn���� ��

� ��tn��� z�tn���� h�� ��tn��� zn��� h�

� ��tn��� z�tn���� ��� ��tn��� z�tn���� h��� ������a�

Using the Lipschitz condition

j��tn��� z�tn���� h�� ��tn��� zn��� h�j � Ljenj� ������b�

Since ��t� y� h� C�� it is uniformly continuous on the compact set t ��� T �� y � z�t��

h ��� �h�� thus�

��h� � maxt����T

j��tn��� z�tn���� ��� ��tn��� z�tn���� h�j � O�h�� ������c�

��

Similarly�

��h� � maxt����T

j��tn�� � hn� z�tn�� � hn�� ��� ��tn��� z�tn���� ��j � O�h�� ������d�

Substituting ������b�c�d� into ������a�

jenj � jen��j� h�Ljen��j� ��h� � ��h��� �������

Equation ������� is a �rst order di�erence inequality with constant �independent of n�

coe�cients having the general form

jenj � Ajen��j�B �������a�

where� in this case�

A � � � hL� �������b�

B � h���h� � ��h��� �������c�

The solution of �������a� is

jenj � Anje�j��An � �

A� �

�B� n � ��

Since e� � �� we have

jenj ���� � hL�n � �

hL

�h���h� � ��h���

or� using ��������

jenj ��eLT � �

L

����h� � ��h���

Both ��h� and ��h� approach zero as h� �� therefore�

limh��� n�� Nh�T

zn � z�tn��

Thus� zn converges to z�tn�� where z�t� is the solution of �������� If the onestep method

satis�es the consistency condition �������� then z�t� � y�t�� Thus� yn converges to y�tn��

n � �� This establishes su�ciency of the consistency condition for convergence�

��

In order to show that consistency is necessary for convergence� assume that the one

step method ������a� converges to the solution of the IVP ������b�� Then� yn � y�tn�

for all t ��� T � as h � � and N � �� Now� zn� de�ned by �������� is identical to yn�

so zn must also converge to y�tn�� Additionally� we have proven that zn converges to

the solution z�t� of the IVP �������� Uniqueness of the solutions of ������� and ������b�

imply that z�t� � y�t�� This is impossible unless the consistency condition ������� is

satis�ed�

Global error bounds for general onestep methods ������� have the same form that we

saw in Chapter � for Eulers method� Thus� a method of order p will converge globally

as O�hp��

Theorem ������ Let � satisfy the conditions of Theorem �� �� and let the one�step

method be of order p� Then� the global error en � y�tn�� yn is bounded by

jenj � Chp

L�eLT � ��� ��������

Proof� Since the onestep method is of order p� there exists a positive constant C such

that the local error dn satis�es

jdnj � Chp���

The remainder of the proof follows the lines of Theorem ������

Problems

�� Prove Theorem ������

��� Implementation� Error and Step Size Control

We would like to design software that automatically adjusts the step size so that some

measure of the error� ideally the global error� is less than a prescribed tolerance� While

automatic variation of the step size is easy with onestep methods� it is very di�cult to

compute global error measures� A priori bounds� such as ��������� tend to be too conser

vative and� hence� use very small step sizes �cf� ����� Section II���� Other more accurate

procedures �cf� ����� pp� ����� tend to be computationally expensive� Controlling a

��

measure of the local �or local discretization� error� on the other hand� is fairly straight

forward and this is the approach that we shall study in this section�

A pseudocode segment illustrating the structure of a onestep method

yn � yn�� � h��tn���yn��� h� ������a�

that performs a single integration step of the vector IVP

y� � f�t�y�� y��� � y�� ������b�

is shown in Figure ������ On input� y contains an approximation of the solution at time

t� On output� t is replaced by t � h and y contains the computed approximate solution

at t � h� The step size must be de�ned on input� but may be modi�ed each time the

computed error measure fails to satisfy the prescribed tolerance ��

procedure onestep �f vector function� � real� var t� h real� var y vector��

begin

repeat

Integrate ������b� from t to t � h using ������a��Compute errormeasure at t � h�if errormeasure � then Calculate a new step size h�

until errormeasure � ��t � t� h�Suggest a step size h for the next step

end�

Figure ������ Pseudocode segment of a onestep numerical method with error controland automatic step size adjustment�

In addition to supplying a onestep method� the procedure presented in Figure �����

will require routines to compute an error measure and to vary the step size� Well

concentrate on the error measure �rst�

Example ������ Let us calculate an estimate of the local discretization error of the

midpoint rule predictorcorrector� We do this by subtracting the Taylor Taylors series

expansion of the exact solution ������� ������ from the expansion of the RungeKutta

formula ������� with a � c � ���� b� � �� and b� � �� The result is

dn �h�

����ftt � �ffty � f �fyy�� �ftfy � ff �

y ���tn�� �y�tn��� �O�h��

��

Clearly this is too complicated to be used as a practical error estimation scheme�

Two practical approaches to estimating the local and local discretization errors of

RungeKutta methods are �i� Richardsons extrapolation �or step doubling� and �ii� em

bedding� Well study Richardsons extrapolation �rst�

For simplicity� consider a scalar onestep method of order p having the following form

and local error

yn � yn�� � h��tn��� yn��� h�� ������a�

dn � Cnhp�� �O�hp���� ������b�

The coe�cient Cn may depend on tn�� and y�tn��� but is independent of h� Typically�

Cn is proportional to y�p����tn���� Of course� the ODE solution must have derivatives of

order p� � for this formula to exist�

Let yhn be the solution obtained from ������a� using a step size h� Calculate a second

solution yh��n at t � tn using two steps with a step size h�� and an �initial condition�

of yn�� at tn��� �Well refer to the solution computed at tn���� � tn�� � h�� as yh��n�����

Assuming that the error after two steps of size h�� is twice that after one step �i�e��

Cn���� � Cn�� the local errors of both solutions are

yhn � y�tn� � Cnhp�� �O�hp���

and

yh��n � y�tn� � �Cn�h���p�� �O�hp���

Subtracting the two solutions to eliminate the exact solution gives

yhn � yhn�� � Cnhp����� ��p� �O�hp����

Neglecting the O�hp��� term� we estimate the local error in the solution of ������a� as

jdnj � jCnjhp�� �jyhn � yh��n j�� ��p

� ������a�

Computation of the error estimate requires �s additional function evaluations �to

compute yn���� and yn� for an sstage RungeKutta method� If s � p then approximately

��

�p extra function evaluations �for scalar systems�� This cost for mdimensional vector

problems is approximately �pm function evaluations per step� Richardsons extrapolation

is particularly expensive when used with implicit methods because the change of step

size requires another Jacobian evaluation and �possible� factorization� It may� however�

be useful with DIRK methods because of their lower triangular coe�cient matrices�

Its possible to estimate the error of the solution yh��n as

jdh��n j � jCnjhp��

�p�jyhn � y

h��n j

�p � �� ������b�

Proceeding in this manner seems better than accepting yhn as the solution� however� it

is a bit risky since we do not have an estimate of the error of the intermediate solution

yh��n�����

Finally� the local error estimate ������a� or ������b� may be added to yhn or yh��n �

respectively� to obtain a higherorder method� For example� using ������b��

y�tn� � yh��n �yhn � yh��n

�p � ��O�hp����

Thus� we could accept

�yh��n � yh��n �yhn � yh��n

�p � �

as an O�hp��� approximation of y�tn�� This technique� called local extrapolation� is also

a bit risky since we do not have an error estimate of �yh��n � Well return to this topic in

Chapter ��

Embedding� the second popular means of estimating local �or local discretization�

errors� involves using two onestep methods having di�erent orders� Thus� consider cal

culating two solutions using the p th and p� � storder methods

ypn � yn�� � h�p�tn��� yn��� h�� dpn � Cpnh

p�� ������a�

and

yp��n � yn�� � h�p���tn��� yn��� h�� dp��

n � Cp��n hp��� ������b�

�The superscripts on yn and dn are added to distinguish solutions of di�erent order�� The

local error of the porder solution is

jdpnj � jypn � y�tn�j � jypn � yp��n � yp��

n � y�tn�j�

��

Using the triangular inequality

jdpnj � jypn � yp��n j� jyp��

n � y�tn�j�

The last term on the right is the local error of the order p � � method ������b� and is

O�hp���� thus�

jdpnj � jypn � yp��n j� jdp��

n j�

The higherorder error term on the right may be neglected to get an error estimate of

the form

jdpnj � jypn � yp��n j� �������

Embedding� like Richardsons extrapolation� is also an expensive way of estimating errors�

If the number of RungeKutta stages s � p� then embedding requires approximately

m�p� �� additional function evaluations per step for a system of m ODEs�

The number of function evaluations can be substantially reduced by embedding the

p thorder method within an �s���stage method of order p��� For explicit RungeKutta

methods� the tableau of the �s � ��stage method would have the form

�c� a��c� a�� a�����

������

� � �

cs�� as���� as���� � � � as���s

�b� �b� � � � �bs �bs��

�Zeros on an above the diagonal in A are not shown�� Assuming that the p thorder

RungeKutta method has s stages� it would be required to have the form

�c� a��c� a�� a�����

������

� � �

cs as� as� � � � as�s��b� b� � � � bs�� bs

With this form� only one additional function evaluation is needed to estimate the

error in the �lower� p thorder method� However� the derivation of such formula pairs is

��

not simple since the order conditions are nonlinear� Additionally� it may be impossible

to obtain a p��order method by adding a single stage to an sstage method� Formulas�

nevertheless� exist�

Example ������ The forward Euler method is embedded in the trapezoidal rule

predictorcorrector method� The tableaux for these methods are

� � �� � �

� � � �

� ��

The two methods are

k� � f�tn��� yn���� k� � f�tn�� � h� yn�� � hk��

y�n � yn�� � hk�

y�n � yn�� �h

��k� � k���

Example ������ There is a threestage� secondorder method embedded in the classical

fourthorder RungeKutta method� Their tableaux are

�� � � �� � � � �� � � �

� � � � � � � �

�� � � �� � � � �

� � �

These formulas are

k� � f�tn��� yn���� k� � f�tn�� � h��� yn�� � hk�����

k� � f�tn�� � h��� yn�� � hk����� k � f�tn�� � h� yn�� � hk���

y�n � yn�� � hk��

��

yn � yn�� �h

��k� � �k� � �k� � k��

Example ���� � Fehlberg ���� constructed pairs of explicit RungeKutta formulas for

nonsti� problems� His fourth and �fthorder formula pair is

��

���

���

����

��������

���������

��������

� �����

� ������

� ���

��� �

��� ��

�������

����

����

� �����

������

��

� ����

� �������

�������

� ��

The � denotes the coe�cients in the higher �fthorder formula� Thus� after determining

ki� i � �� �� � � � � �� the solutions are calculated as

yn � yn�� � h���

���k� �

����

����k� �

����

����k � �

�k�

and

yn � yn�� � h���

���k� �

����

�����k� �

�����

�����k � �

��k �

��k���

Hairer et al� ����� Section II�� give several Fehlberg formulas� Their fourth and �fthorder

pair is slightly di�erent than the one presented here�

Example ������ Dormand and Prince ���� develop another fourth and �fthorder pair

that has been designed to minimize the error coe�cient of the higherorder method so

that it may be used with local extrapolation� Its tableau follows�

��

���

��

��

���

���

��

��������

���������

�����

�������

� ��������

����

������

����

� �������

� ���

� ������

�����

��������

���

���

� ������

�����

��������

���

� �������

� �������

�����

� �����������

�������

��

Having procedures for estimating local �or local discretization� errors� we need to

develop practical methods of using them to control step sizes� This will involve the

selection of an appropriate �i� error measure� �ii� error test� and �iii� re�nement strategy�

As indicated in Figure ������ we will concentrate on step changing algorithms without

changing the order of the method� Techniques that automatically vary the order of the

method with the step size are more di�cult and are not generally used with RungeKutta

methods �cf�� however� Moore and Flaherty ������

For vector IVPs ������b�� we will measure the �size� of the solution or error estimate

by using a vector norm� Many such metrics are possible� Some that suit our needs are

�� the maximum norm

ky�t�k� � max��i�m

jyi�t�j� ������a�

�� the L� or sum norm

ky�t�k� �mXi��

jyi�t�j� ������b�

�� and the L� or Euclidean norm

ky�t�k� �

mXi��

jyi�t�j����

� ������c�

��

The two most common error tests are control of the absolute and relative errors� An

absolute error test would specify that the chosen measure of the local error be less than

a prescribed tolerance� thus�

k!dnk � �A�

where the ! signi�es the local error estimate rather than the actual error� Using a relative

error test� we would control the error measure relative to the magnitude of the solution�

e�g��

k!dnk � �Rkynk�

It is also common to base an error test on a combination of an absolute and a relative

tolerance� i�e��

k!dnk � �Rkynk� �A�

When some components of the solution are more important than others it may be

appropriate to use a weighted norm with yi�t� in ������� replaced by yi�t��wi� where

w � �w�� w�� � � � � wm�T ������a�

is a vector of positive weights� As an example� consider the weighted maximum norm of

the local error estimate

k!dnkw�� � max��i�m

�����!dn�iwi

����� � ������b�

where !dn�i denotes the local error estimate of the i th component of dn�

Use of a weighted test such as

k!dnkw � � ������c�

adds "exibility to the software� Users may assign weights prior to the integration in

proportion to the importance of a variable� The weighted norm may also be used to

simulate a variety of standard tests� Thus� for example� an absolute error test would be

obtained by setting wi � �� i � �� �� � � � � m� and � � �A� A mixed error test where the

integration step is accepted if the local error estimate of the i th ODE does not exceed

�Rjyn�ij� �A

��

may be speci�ed by using the maximum norm and selecting

� � max��A� �R�

and

wi � ��Rjyn�ij� �A����

Present RungeKutta software controls�

�� the local error

k!dnkw � �� ������a�

�� the local error per unit step

k!dnkw � h�� ������b�

�� or the indirect �extrapolated� local error per unit step

k!dnkw � Ch�� ������c�

where C is a constant depending on the method�

The latter two formulas are attempts to control a measure of the global error�

Let us describe a step size selection process for controlling the local error per unit

step in a p th order RungeKutta method� Suppose that we have just completed an

integration from tn�� to tn� We have computed an estimate of the local error !dn using

either Richardsons extrapolation or order embedding� We compare k!dnkw with the

prescribed tolerance and

�� if k!dnkw � we reject the step and repeat the integration with a smaller step size�

�� otherwise we accept the step and suggest a step size for the subsequent step�

In either case�k!dnkw

h� Cnh

p�

��

Ideally� we would like to compute a step size hOPT so that

� � CnhpOPT �

Eliminating the coe�cient Cn between the two equations

k!dnkw� hpOPT

hp���

or

hOPTh

��

h�

k!dnkw

���p

� ������a�

The error estimates are based upon an asymptotic analysis and are� thus� not com

pletely reliable� Therefore� it is best to include safety factors such as

hOPT � hmin

��MAX �max

�MIN � �s

��

k!dnkw

���p�

� ������b�

The factors �MAX and �MIN limit the maximum step size increase and decrease� respec

tively� while �s tends to make step size changes more conservative� Possible choices of the

parameters are �MAX � �� �MIN � ���� and �s � ���� Step size control based on either

������a� or ������c� works similarly� In general� the user must also provide a maximum

step size hMAX so that the code does not miss interesting features in the solution�

Selection of the initial step size is typically left to the user� This can be somewhat

problematical and several automatic initial step size procedures are under investigation�

One automatic procedure that seems to be reasonably robust is to select the initial step

size as

h �

��

��T p� � kf���y����kp����p�

where T is the �nal time and p� � p� � for local error control and p� � p for local error

per unit step control�

Example ������ ������ Section II���� We report results when several explicit fourth

order explicit RungeKutta codes were applied to

y�� � �ty� log�max�y�� ������� y���� � ��

y�� � ��ty� log�max�y�� ������� y���� � e�

��

��

��

��

��

��

���

��

� ���

�� �

��

����

��

��� �

������

����

���

���

� � ����

����

���

������

����

����

�����

� ���

���

�����

Table ������ Butchers sevenstage sixthorder explicit RungeKutta method�

The exact solution of this problem is

y��t� � esin t�

� y��t� � ecos t�

Hairer et al� ���� solved the problem on � � t � � using tolerances ranging from ����

to ����� The results presented in Figure ����� compare the base �� logarithms of the

maximum global error and the number of function evaluations�

The several methods that are not identi�ed in Figure ����� are the more traditional

formulas� including the classical RungeKutta method �solid line�� All of these are listed

in Hairer et al� ����� Section II��� �Fehlbergs method� is the fourth and �fthorder pair

given in Example ������ The �DormandPrince� method is the fourth and �fthorder

pair of Example ������ �Butchers method� is the sixthorder sevenstage formula shown

in Table ������ It is the only formula that is beyond fourth or �fth order� Results in the

lower graph of Figure ����� use local extrapolation� thus� the higherorder solution of the

pair is kept� even though it has no local error estimate�

Of all the methods shown in Figure ������ the DormandPrince and Fehlberg methods

appear to have the greatest accuracy for a given cost� The higherorder Butcher formula

gains appeal as accuracy increases� The DormandPrince method has a distinct advantage

relative to the Fehlberg method when local extrapolation is used� As noted� the Dormand

Prince method was designed for this purpose� For this problem� which has a smooth

��

Figure ������ Accuracy vs� e�ort for several explicit RungeKutta methods �����

��

solution� the DormandPrince method is nearly a factor of �� more accurate with local

extrapolation than without it�

Numerous other implementation details have not been addressed� These include �i�

Newton iteration for implicit systems and �ii� portability� We will return to these topics

when discussing multistep methods in Chapter �� While we have not discussed compet

ing methods� well nevertheless conclude that explicit RungeKutta formulas should be

considered when

�� the functions de�ning the di�erential system are simple or

�� the solution has discontinuities�

The latter case is interesting� Discontinuities can be ignored and treated with the au

tomatic stepsize selection procedures� This is easier to do with RungeKutta methods

than with competing approaches such as multistep methods� Explicit treatment of dis

continuities by locating the discontinuity and restarting the solution there is also simpler

with RungeKutta methods than with competing methods�

Implicit RungeKutta methods are useful when high accuracy and A or Lstability

are needed simultaneously� This occurs with problems where fy�t�y� has eigenvalues with

large �neagative� real or imaginary parts� We will postpone a comparison of methods

for these problems until examining multistep methods in Chapter �� At this time� well

note that software based on �fth� seventh and ninthorder Radau methods ���� has

done extremely well when solving sti� IVPs� The STRIDE software ��� based on SIRK

methods has been successful� but less so than the Radau methods�

Problems

�� The aim of this problem is to write a subroutine or procedure for performing one

step of a fourthorder variable step RungeKutta method applied to vector IVPs of

the form

y� � f�t�y�� y�t�� � y��

���� Write a subroutine or procedure to perform one step of a fourthorder explicit

RungeKutta method� You may use the classical� Fehlberg� Dormand and

Prince� or other formula as long as it is fourth order�

��

���� Test your procedure using �xed step integration with step sizes h � � �� � ��

� �� � � � using the test IVPs

y� � y� y��� � �� � �� �

andd�y

dt�� �k

dy

dt� n�y � �� � � t � ��

y��� � �� dyoverdt��� � ��

For the second example use n � ��� k � � and n � ��� k � �� In each

case� present results �tables and or graphs� of the global error and number of

function evaluations as functions of h� Estimate the rate of convergence of the

method and compare it with the theoretical rate�

���� Replace the �xed step size strategy above with a variable step size technique of

your choice� Base step size selection on control of the local error� which may be

estimated using either step doubling or embedding� Compare the performance

of your code on the above problems with the �xed step performance�

��

Bibliography

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York� �����

��� R� Alexander� Diagonally implicit rungekutta methods for sti� o�d�e�s� SIAM J�

Numer� Anal�� �������#����� �����

��� U�M� Ascher and L�R� Petzold� Computer Methods for Ordinary Di�erential Equa�

tions and Di�erential�Algebraic Equations� SIAM� Philadelphia� �����

��� R�L� Burden and J�D� Faires� Numerical Analysis� PWSKent� Boston� �fth edition�

�����

��� K� Burrage� A special family of rungekutta methods for solving sti� di�erential

equations� BIT� ����� # ��� �����

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rungekutta methods� BIT� ������ # ���� �����

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# ���� �����

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������ # ���� �����

��

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Wiley and Sons� New York� �����

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Di�erential Algebraic Problems� SpringerVerlag� Berlin� �����

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��