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MODIFIED EXTENDED BLOCK BACKWARD
DIFFERENTIATION FORMULA FOR SOLVING
STIFF ODEs
ASMA IZZATI BINTI ASNOR
SCHOOL OF MATHEMATICAL SCIENCES
UNIVERSITI SAINS MALAYSIA
2015
MODIFIED EXTENDED BLOCK BACKWARD
DIFFERENTIATION FORMULA FOR SOLVING STIFF ODEs
by
ASMA IZZATI BINTI ASNOR
Dissertation submitted in partial fulfillment
of the requirements for the degree
of Master of Science in Mathematics
August 2015
ii
ACKNOWLEDGEMENT
In the Name of Allah, the Most Beneficient, the Most Merciful
Alhamdulillah, praise be to Allah. Peace and blessings be upon the Prophet of
Allah, Nabi Muhammad S.A.W. I am grateful to Allah that I have completed this
dissertation within the prescribed time. I would like to express my deepest gratitude to
my supervisor, Dr. Siti Ainor binti Mohd Yatim for her encouragement and guidance
throughout the learning process of this dissertation. Without her good supervision and
persistent help, this dissertation would never have been completed. I would like to thank
her for priceless advices, brilliant comments and valuable informations over this
semester. I am greatly appreciate it.
In addition, I also take this opportunity to offer my special thanks to the most
important people, my family members for their endless and continuous support. My
parents who have given their love and understanding towards me to get through this
period as a master student. Words cannot express how grateful I am for all the sacrifices
made by them. Alhamdulillah.
My special appreciation goes to all my friends who have supported me in
writing this project and assist me to achieve my goal and for those who always help me
during the completion of this dissertation. I will never forget your kindness. Thank you
so much.
Asma, 2015.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENT ii
TABLE OF CONTENTS iii
LIST OF TABLES vi
LIST OF FIGURES vii
LIST OF ABBREVIATIONS ix
ABSTRAK x
ABSTRACT xii
CHAPTER 1 : INTRODUCTION
1.1 Background 1
1.2 Objectives of the Study 6
1.3 Problem Statement 6
1.4 Scope of Study 7
1.5 Methodology 7
1.6 Outline of the Study 8
CHAPTER 2 : LITERATURE REVIEW
2.1 Introduction 10
2.2 Stiff initial value problem 10
2.3 Linear Multistep Method 12
2.4 Block method 13
ii
2.5 Review of Previous Work 15
CHAPTER 3 : FORMULATION OF THE METHOD
3.1 Introduction 18
3.2 Formulation of Predictor Method for VS-BBDF method 18
3.3 Formulation of Corrector Method for VS-BBDF method 21
3.4 Implementation of the Method 24
3.4.1 Newton’s Iteration 24
3.4.2 Choosing the step size 26
3.5 Stability of VS-BBDF Method 28
CHAPTER 4 : RESULTS AND DISCUSSIONS
4.1 Introduction 34
4.2 Test Problems 34
4.3 Numerical Results 38
4.4 Discussions 53
CHAPTER 5 : SUMMARY
5.1 Conclusions 55
5.2 Future Study 56
REFERENCES 57
APPENDIX
Appendix A : Algorithm for VS-BBDF method of order four
ii
LIST OF TABLES
Page
3.1 The formulae for predictor and corrector method for 23
two computed points
3.2 Local truncation error for the three step size ratios 27
3.3 Lists of stability polynomials and roots for the three step size ratios 30
iii
LIST OF FIGURES
Page
2.1 VS-BBDF method of order 4 (P4) 14
3.1 Stability region of for r = 1 31
3.2 Stability region of for r = 2 32
3.3 Stability region of for r = 5/9 32
3.4 Stability regions for r = 1, r = 2 and r = 5/9 33
4.1 Approximated solutions curves for variable step size ratios 39
for Test Problem 4.1
4.2 Graph of approximate solution and exact solution for 39
Test Problem 4.1
4.3 Total steps curves for Test Problem 4.1 40
4.4 Approximated solutions curves for variable step size ratios 41
for Test Problem 4.2
4.5 Graph of approximate solution and exact solution for 41
Test Problem 4.2
4.6 Total steps curves for Test Problem 4.2 42
4.7 Approximated solutions curves for variable step size ratios 43
for Test Problem 4.3
4.8 Graph of approximate solution and exact solution for 43
Test Problem 4.3
4.9 Total steps curves for Test Problem 4.3 44
4.10 Approximated solutions curves for variable step size ratios 45
for Test Problem 4.4
ii
4.11 Graph of approximate solution and exact solution for 45
Test Problem 4.4
4.12 Total steps curves for Test Problem 4.4 46
4.13 Approximated solutions curves for variable step size ratios 47
for Test Problem 4.5
4.14 Graph of approximate solution and exact solution for 47
Test Problem 4.5
4.15 Total steps curves for Test Problem 4.5 48
4.16 Approximated solutions curves for variable step size ratios 49
for Test Problem 4.6
4.17 Graph of approximate solution and exact solution for 49
Test Problem 4.6
4.18 Total steps curves for Test Problem 4.6 50
4.19 Graph of approximated solutions for variable step size ratios 51
for Test Problem 4.7
4.20 Graph of approximate solution and exact solution for 51
Test Problem 4.7
4.21 Total steps curves for Test Problem 4.7 52
ii
LIST OF ABBREVIATIONS
DE Differential Equation
ODE Ordinary Differential Equation
PDE Partial Differential Equation
IVP Initial Value Problem
BDF Backward Differentiation Formulae Method
BBDF Block Backward Differentiation Formulae Method
VS-BBDF Variable Step Block Backward Differentiation Formulae Method
LMM Linear Multistep Method
LTE Local Truncation Error
TOL Tolerance Limit
iii
PENGUBAHSUAIAN LANJUTAN FORMULA PEMBEZAAN
BLOK KEBELAKANG UNTUK MENYELESAIKAN PPB KAKU
ABSTRAK
Disertasi ini tertumpu kepada pengubahsuaian dan melanjutkan kaedah Formula
Pembezaan Blok Kebelakang (FPBK) yang sedia ada untuk menyelesaikan Persamaan
Pembezaan Biasa (PPB) kaku peringkat pertama. Kaedah baru ini akan diolah dengan
menggunakan saiz langkah 1.8. Kami memberi penekanan pada merumus kaedah blok
dua-titik berdasarkan kepada Langkah Berubah Formula Pembezaan Blok Kebelakang
(LB-FPBK) peringkat keempat dengan menggunakan pendekatan langkah berubah
untuk terbitan kaedah – kaedah tersebut. Strategi yang terlibat adalah pemilihan saiz
langkah. Jelasnya, matlamat kaedah blok dua-titik adalah untuk mengira dua nilai baru
dalam satu blok masa yang sama pada setiap langkah dan pada masa yang sama
menggunakan tiga nilai belakang blok sebelumnya. Beza Bahagi Newton digunakan
pada kaedah yang dicadangkan untuk menganggarkan masalah. Kemudian,
perbandingan prestasi untuk saiz langkah 1.8 dibuat dengan dua lagi saiz langkah, 1.6
dan 1.9 untuk menentukan kesan penyelesaian dengan menukar nilai nisbah saiz
langkah. Oleh yang demikian, rantau kestabilan kaedah LB-FPBK juga dibincangkan
dan illustrasi rantau kestabilan diplot di dalam graf. Kod program ditulis dalam
pengaturcaraan C. Keputusan berangka tersebut menunjukkan, sedikit perubahan pada
corak penyelesaian dengan menukar nilai nisbah saiz langkah. Kesimpulannya,
keputusan tersebut menunjukkan bahawa saiz langkah 1.8 memberi ketepatan yang
ii
paling baik untuk masalah terpilih yang diuji. Oleh itu, dengan meningkatkan nisbah
saiz langkah akan memberikan prestasi yang lebih baik dari segi ralat maksimum.
ii
ABSTRACT
This dissertation is concerned to modify and extend the existing block backward
differentiation formula (BBDF) method for solving first - order stiff IVP for ODE. The
new method will be modified by using step size 1.8. We emphasized on formulating
two-point block method based on fourth order variable step BBDF (VS-BBDF) method
by utilizing variable step approach for the derivation of the methods. The strategy for
choosing the step size are also involved. Apparently, the goal of two-point block
method is to compute two new values in a block simultaneously at every step and at the
same time using three back values of previous block. The Newton Divided Difference is
applied to the proposed method to approximate the problems. Then, the comparison for
performance step size 1.8 is made with the other two step sizes, 1.6 and 1.9 to determine
the effect of solutions by changing the value of step size ratios. Consequently, the
stability region of VS-BBDF method is also discussed and the illustration is plotted in a
graph. The source code is written in C language. The numerical results, shows a slight
change in the pattern of the solutions by changing the value of step size ratio. In
conclusions, the results show that step size 1.8 gives the best accuracy for the selected
tested problems. Thus, increasing the step size ratio will give better performance in
terms of maximum error.
1
CHAPTER 1
INTRODUCTION
1.1 Background
Numerical analysis is such a very wide area of mathematics and computer science
study that deal with most of the problems in real-world. It is the study that creates,
analyzes, and implements algorithms for solving problems numerically of continuous
mathematics with finding approximation solutions. Mathematicians interpret the real-
world situations into mathematical formulation. The development of sophisticated
computer at this century, the computer is beneficial for researchers or mathematicians to
solve more complicated mathematical models of the real world.
Several natural phenomena in most real-life situations are represented or modelled by
functions. The functions may depend on one or more independent variables and usually
time and space (location) variables are choosen as the independent variables. For instance,
in real-life situation, the position of the earth changes with time, the area of a circle
changes with the size of radius and many more. For that reasons, any equation involving
an unknown function with some or all of its derivatives, either ordinary derivatives or
partial derivatives is known as differential equation (DE).
DE is applied to model problems in science such as physics, biology and chemistry
as well as many other branches of studies (engineering and economic) involve the change
of some variable with respect to another. The problems in real-life situations are
2
complicated to be solved exactly. So, the easier way is to approximate the solutions.
Besides, it shows the relationship between physical quantities and the rates of change.
Physical quantities usually is defined as functions while the derivatives represent the rates
of change. There are two major kinds of DE, Ordinary differential equation (ODE) and
Partial differential equation (PDE).
Over the recent years, mathematical modelling problems has evolved in applied
science and engineering field of modern life. ODE is one of the problems arises in the
field. Many researchers and mathematicians have been solving ODE with various kind of
numerical methods from the earlier research with the intention to find the best
approximations for the solutions. Other than that, in 20th century the study on numerical
methods for the solution of initial value problem (IVP) ODE has become famous topic
and advanced in study as many researchers are interested in doing their researches. The
numerical methods used for solving ODE is to find the approximations to the solutions
of ODE where it provides an alternative way to some difficult problems. Some of the
numerical methods used in PDE convert the PDE to an ODE to solve it. The numerical
methods are classified as single-step methods or multi-step methods. In addition, the
methods can be divided into explicit methods or implicit methods. Since numerical
method has stability limitation on the step size, there are only few methods that can solve
stiff problems (Fatunla, 1990). Therefore, a commonly used method which is known as a
foundation method that still in use in the present for solving ODE is Backward
Differentiation Formula (BDF) (Suleiman et al., 2013) . It is also the most well-liked
implicit methods for solving stiff ODEs. This method has been proposed by Gear in 1970
who is also one of the well – known researcher in the study of stiff ODEs.
3
By defining IVP as a differential equation together with a specified initial condition
of the function at a given point in the domain of the solution. IVP of first-order ODE
considered in this project is of the form
where 0x is initial value in the given interval ],[ 0 nxxx . An ODE equation (1.1) contain
a function and its ordinary derivative of dependent variable with respect to single
independent variable. Order of ODE is the highest order of the derivative of the function
that appears in the equation. Besides, there are many numerical techniques for the
numerical solution of stiff IVP and the techniques depend on many factors such as
computational expense, data-storage requirements, speed of convergence, accuracy and
stability.
Stiff ODE has been large development of study by researchers since the last
century and until now they are still interested to do some researches on solving stiff ODE.
The problem will be having some difficulties when standard numerical techniques are
applied to approximate the solution of a differential equation when the exact solution has
terms of the form te , where is complex number with negative real part. This term will
decay to zero as t increases. Besides, this problem occurs in wide variety of application
including the study of spring and damping systems, problems in chemical kinetics and
many more. The system is stiff if its solution contains components with both slowly and
rapidly decaying rates because of large difference of time scales exhibited by the system
(Suleiman, 2013). The idea is that, it can lead to rapid variation in the solution. However,
it is important to determine either the ODE is stiff problem or not by the presence of very
),( yxfy , 00 )( yxy (1.1)
4
large negative eigenvalues of its Jacobian matrix, y
f
. However, the problem that is said
to be stiff requires very small step size to solve it. Otherwise, it will be unstable
(Mahayadin et al., 2014). For most stiff problems are difficult to solve since many
numerical methods have stability restriction on the step size unless the step size chosen
is very small to achieve the accuracy. Therefore, we must concern with choosing the
suitable numerical methods that solve stiff problem efficiently because if not the
numerical methods will become unbereably slow. Furthermore, stiff problem only works
with implicit method or in other words it can be said that explicit method cannot handle
the problem efficiently. Implicit methods on solving stiff ODEs are known to perform
better than explicit ones (Abasi, 2014). Theoretically, numerical methods that is suitable
for ODEs is usually implicit, which require repeated solutions of systems of linear
equations with coefficient matrix, JhI , here J is the Jacobian matrix (Ibrahim et al.,
2008).
BDF is linear multistep methods (LMM) that suitable for solving stiff initial value
problems. To improve the existing BDF method, Ibrahim et al. has proposed a new idea
in which the method generates block approximation knnn yyy ,...,, 21 (Ibrahim et al.,
2007). The reason is to produce better approximations in terms of computation time and
precision on solving first order stiff ODE. Hence, the most recent study on this new
method is known as Block Backward Differentiation Formula (BBDF). Over the years, a
block method has been discussed by a few researchers since the earlier research. This
block approximations has been used in different methods. For instance, block implicit
one step methods was proposed by Shampine and Watts (1969) is the earliest research on
block methods. Other studies done by Chu and Hamilton (1987) with multi-block
5
methods, Voss and Abbas (1997) with block predictor-corrector schemes and Ibrahim et
al. (2007) with block method based on BBDF method for first order stiff ODEs. The rapid
growth of the studies on the block methods for solving ODEs contribute to the
competition in developing and deriving an accurate method for solving many types of
ODEs by Zawawi et al. (2012). Block methods also have some advantages such as it
approximates the solution at more than one point and the number of point depends on the
structure of the block methods. So from the advantage, the execution time and the total
number of iterations can be reduced because it gives faster solutions. Therefore, this
method shows that it is more efficient than BDF method.
Basically, a block method is a method of a block of new values is obtained
simultaneously. It is called a block method because it compute previous k blocks and
calculate the current block where each block contain r points. In this dissertation, we
would only emphasize on two-points block method where two values are computed
simultaneously in a block by using the values of previous block with each block
containing two points. The solution of 1ny and 2ny are computed using three back
values, nnn yyy ,, 12 . There are five points for fourth order where two points will be
calculated and the rest three points are the previous points. Initially, this study is supposed
to formulate the BBDF method by using variable step size approach. The strategy
involved based on the value of Local Truncation Error (LTE) for choosing step size of
the method. The LTE is also depends on error tolerance limit where LTE will be less or
greater than error tolerance limit. The step size involve are constant step size, half of the
step size and increment of the step size. The doubling step size is not considered due to
zero instability. After that we can plot and investigate the stability region for BBDF
6
method. For a better performance, Newton iteration has been implemented to BBDF
method.
Nowadays, the algorithms are implemented in variety of programming languages for
numerical analysis. Some of the popular languages that are used to implement the
algorithms of numerical methods such as Fortran, C++ and Java. They are utilized to
make the complexity of the things for making more uncomplicated and efficient as well
as productive. In this dissertation, we only apply Microsoft Visual C++ to implement
BBDF method while Maple15 software is used to derive the formulae for predictor and
corrector of VS-BBDF methods.
1.2 Objectives
The main objective of the dissertation is to construct block BDF method by using variable
step size approach for solving first-order stiff ODE. The aim can be accomplished by :
• Derivation of the fourth order Variable Step Block Backward Differentiation
Formula (VS-BBDF) method.
• Solving numerically IVP of first order stiff ODE by applying Variable Step Block
Backward Differentiation Formula (VS-BBDF) method.
• Investigating the effect of the solutions by changing the value of step size ratio, r.
1.3 Problem Statement
An ordinary differential equation (ODE) that has a function and its ordinary derivative.
For this dissertation, we consider a system of first order stiff initial value problems (IVPs)
ODE. The problem to be considered in general form
)),(,()( xYxfxAYy iii )(aY ni ,...,2,1 (1.2)
7
where
x is the interval from a to b, ],[ bax ,
),,...,,,()( 211 nyyyyxY
),...,,( 21 n ,
A is mm matrices with large negative eigenvalues.
1.4 Scope of the Study
This dissertation is emphasized only on numerical solution of IVP of first - order stiff
ODE. We are interested to solve the problem by using two-point block method based on
BBDF method where two new values in a block will be computed simultaneously using
three back values. This study is focused on fourth order variable step BBDF (VS-BBDF)
method. We utilize variable step size approach to derive the formula for the predictor and
corrector block methods where the variable step size involve in this dissertation are
constant step size, half the step size and increment the step size by a factor of 1.8 at which
all of these satisfy the zero stability. The strategy for choosing the step size is based on
Local Truncation Error (LTE). In this dissertation, we only compare the numerical results
for step size of the proposed method within the selected test problems with the results of
the previous researches on step size 1.6 and 1.9. In this dissertation, we will use Microsoft
Visual C++ 6.0 as platform to implement the VS-BBDF method with Newton iteration
algorithm. The program will be written in C language.
1.5 Methodology
Next section is methodology. Basically, there are a few methods that are needed to
accomplish this dissertation in order to find the numerical results for the problems that
8
have been tested. Firstly, we formulate the predictor and corrector methods formulae for
VS-BBDF method using Maple 15 software. Then, we apply Newton’s iteration to the
corrector method. Microsoft Visual C++ is used as a compiler for the implementation of
the method. In this dissertation, we consider the variable step size approach where the
appropriate step size is chosen based on the Local Truncation Error (LTE). In addition,
the stability of the proposed method is investigated by applying the VS-BBDF method to
the test equation. We then check for the zero-stability and plot the stability region in the
complex plane.
1.6 Outline of the Project
This dissertation is organized as follows.
Chapter 1 is a brief introduction of this dissertation regarding the ODE and the
background of the method used for solving the problem. Besides, this chapter also
includes objectives of the study to achieve the goals for this dissertation. Then, provides
also problem statement, scope of the study and also methodology. Lastly, the outline of
the later chapters will be in the last part for this chapter.
In Chapter 2, we explain in detail for each of the basic concepts, theories and
definitions of the related study to support the completion of the current research. In the
same chapter, we provide some literature review where it discussed in detail about the
previous researches and overview from the early researches on background of existing
BBDF method. In this chapter, there are also review on other block methods that are used
to solve stiff ODE by previous researchers.
9
Chapter 3 provides the formulation of the method of this project especially further
explanation for the development of VS-BBDF using Lagrange polynomial as the
interpolating polynomial. The stability region is presented and the zero stability is
investigated. In addition, we develop the algorithm of VS-BBDF method that would solve
the stiff ODE problems by applying Newton iteration to the method. The source code is
written in C language and using Microsoft Visual C++ platform to implement the method.
The results of this dissertation are presented in Chaper 4. There are seven selected
tested problems of stiff IVP ODE that are tested. The numerical results for step size 1.8
is compared with the step size 1.9 and 1.6. Then, the comparisons are made between the
computed results and the existing results for step size 1.9 and 1.6. Last but not least, the
results are represented in figures.
Finally, the summary of this dissertation is presented in Chapter 5. We also made
some conclusions and a few recommendations for future research.
10
10
CHAPTER 2
LITERATURE REVIEW
2.1 Introduction
This chapter discusses the review of the previous researches on stiff ODE and block
methods. We discuss in detail for every main points in this chapter by providing some
definitions and theorems to strengthen the main points.
2.2 Stiff initial value problems
Stiff IVP were first encountered when varying stiffness occurs in the study of the
motion springs. The IVPs is considered as the following general linear systems with
constant coefficients,
),(xdAyy ,)( 0yay ,bxa (2.1)
where A is mm with real entries.
There are no universal definition of stiffness but the idea is more or less similar to one
another. However, a few experts have defined the definition of stiffness according to their
own way.
Definition 2.1 (Curtiss and Hirschfelder, 1952)
Stiff equations are equations where certain implicit methods, in particular BDF, perform
better, usually tremendously better, than explicit ones. The eigenvalues of the Jacobian
11
y
f
play certainly a role in this decision, but quantities such as the dimension of the
system, the smoothness of the solution or the integration interval are also important.
Definition 2.2 (Lambert, 1973;1991)
The system of IVP ODE is said to be stiff if
i. Re )( t < 0, t = 1, 2, …, m and
ii. |)Re(|min|)Re(|max tttt where t are the eigenvalues of the Jacobian
matrix, y
f
If the numerical method with a finite region of absolute stability, applied to a system
with any absolute stability, applied to a system with any initial conditions, is forced to
use in a certain interval of integration a step length which is excessively small in relation
to the smoothness of the exact solution in that interval, then the system is said to be stiff
in that interval.
There are other characteristics that exhibit by many examples of stiff problems, but for
each there are counter examples, so these characteristics do not make good definitions of
stiffness. Lambert refers to these as ‘statements’ rather than definitons. A few of these
are:
i. A linear constant coefficient system is stiff if all of its eigenvalues have negative
real part and the stiffness ratio is large.
ii. Stiffness occurs when stability requirements, rather than those of accurancy,
constrain the steplength.
iii. Stiffness occurs when some components of the solution decay much more rapidly
than others.
12
Definition 2.3 (Fatunla, 1987)
he stiffness ratio S of the system (2.1) is given as
,)ln(
)(||max
TOL
abS i
i
where )ln(TOL is the exponential logarithm of TOL.
Definition 2.4 (Dahlquist, 1974)
Systems containing very fast components as well as very slow components.
Definition 2.5 (Shampine, 1981)
A major difficulty is that stiffness is a complex of related phenomena, so that it is not
easy to say what stiffness is.
2.3 Linear Multistep Method
Basically, multistep method use the approximation values at more than one
previous value to approximate the subsequent value. Furthermore, multistep method is
more accurate than one-step method because it use more information about the known
portion of the solution than one-step. One of the category of multistep is Linear Multistep
Method (LMM). It can be written as linear combination of the value of solution and the
value of function at previous points.
The general LMM is
k
j
jnj
k
j
jnj fhy00
(2.2)
13
where j and j are constant, k is the number of steps used in multistep and h is the step
size. Coefficients are presumed to be real and satisfy the conditions 1k and
0|||| 00 . If 00 , so the method is explicit, otherwise it is an implicit method.
2.4 Block method
Apparently, block method has become one of the available solvers for stiff ODE.
Generally, block method is a method that obtain concurrently a block of new values by
computing k number of blocks. It uses values from the preceeding block to compute the
values for current block. Apart from that, block method will have r block size where it
will become r by r matrices of k blocks all together.
This method has some benefits in particular that can reduce the execution time
and total number of iterations where this block method generates new values at r points
concurrently at each step.
For instance, two-point block method introduced by Ibrahim et al. (2007), two
new values that are 1ny and 2ny are computed simultaneously by using the values from
previous block whereby each block will have maximum of two points.
In addition, the orders of the method are determined by the number of back values
contained in total blocks (Yatim et al., 2013). Step size ratio, r is defined as the ratio
distance between current step, nx and previous step, 1nx . For computed block, the step
size is 2h and 2rh is the step size for previous block.
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The figure below shows the illustration of the block method.
Previous block Current block
rh rh h h
2nx 1nx
nx 1nx
2nx
p4
Figure 2.1 VS-BBDF method of order 4 (p4) (Yatim et al., 2013)
2.5 Review of previous work
Shampine and Watts (1969) has discussed on block implicit one-step methods.
They studied a class of one-step methods for solving ODE at which they obtained block
of r new values at each step for solving linear and nonlinear ODE. In addition, they also
studied the stability and convergence for the particular method. The results showed that
Block one-step (BOS) method and Block one-step (BOSLS) method (as applied to linear
problems) were competitive with the other methods and BOS was more accurate than
BOSLS when small step size was used.
BBDF method was proposed by Ibrahim et al. (2007a) which was similar to
standard form of BDF method. But the difference was this method computed new values
in a block at the same time. The block method allowed us to store the coefficient of y
values. For that reason, it will avoid us to repeat the calculation of differentiation
coefficients at each step. In this research, they also presented the regions of absolute
15
stability for the method. They compared the efficiency of 2-point BBDF method with
conventional variable step variable order BDF (VSVOBDF) method. Hence, the results
showed that BBDF method gave more accurate results than VSVOBDF method with
lesser total steps and lesser computational time. In conclusion, this method was suitable
to solve stiff ODE.
According to Ibrahim et al. (2007b), they focused on implicit 2-point block
method based on BDF method. Therefore, two new values will be generated concurrently.
In this study, they also used variable step approach on variable step block backward
differentiation formula (VS-BBDF) method. Hence, the variable step size were constant
step size, half of the step size and increment the step size to 1.6. Then, they also have
plotted the absolute stability region for the method. The method applied to the selected
test problems were compared with variable step variable order non block BDF method
(NBDF). So, the results showed that 2-point BBDF method gave better performances than
non-block BDF (NBDF) with reduction of total step and lesser computational time.
In the same year, Ibrahim et al. (2007c) has released a new idea where they solved
ODE by using implicit r-point BBDF method. They derived a block of r new values at
each step and make a comparison between r-point block methods with the existing BDF
method. Hence, the results indicated that the r-point BBDF method was more efficient
than BDF method with reduction of the number of integration step and improve CPU
time.
Three years later Yatim et al. (2010) has extended the study by Ibrahim et. al in
2007. A few selected test problems have been tested with the increment of step size to
16
1.9. The main idea of this study was quite similar to Ibrahim et. al (2007b) with the
intention to optimize the performance of the method to produce better approximations
and good computation time. They derived the implicit block methods based on BDF
method for the solution stiff IVP while they also used variable step size approach.
Besides, the construction of variable step size block methods will store all the coefficients
of the method. Then they compared the results with the step size 1.6. In conclusion, this
study gave better results because the increment of step size can reduce the number of total
steps and also lesser computational time. Therefore, step size 1.9 was more efficient than
step size 1.6.
The research of Nasir et al. (2011) has presented a new method which called as
fifth order 2-point BBDF method. They derived the formula of the method where two
new values will be produced concurrently at each step using four back values. This new
method was used for solving first order ODE. Other than that, the study has shown that
this method was stiffly stable and satisfied the conditions to solve stiff problems. They
also compared the new method with classical BDF Method and ode15s in MATLAB.
Lastly, the study has concluded that the methods performed competitively and the
efficiency was over the BDF method and ode15s.
Implicit Continuous BBDF (CBBDF) method has been proposed by Akinfenwa
et al. (2013). This method was applied to solve ODE which they derived a block of p new
values at each step which simultaneously provided the approximate solutions for the stiff
ODEs. The performance of CBBDF method was compared with BDF method. So from
the numerical results, the proposed method was more efficient because it produced
accurate results and fewer number of function evaluations and computational steps.
17
Then, Yatim et. al. (2013) has discussed ‘A numerical algorithm for solving stiff
ordinary differential equations’. In this research, they applied variable step variable order
approach to the BBDF (VSVO-BBDF) method. Other than that, they also formulated
VSVO-BBDF method of order 3 to 5 and investigated the stability region of the particular
method. A comparison for the results was made between the proposed method and
MATLAB’s suite of ODEs solvers namely, ode15s and ode23s. Therefore, VSVO-BBDF
method outperformed ode15s and ode23s at which it managed to reduce the number of
total steps taken as well lesser computational time.
The recent study of Abasi et al. (2014) has introduced a formula for 2-point block
method with two off step of order 5 based on BDF method. The method was utilized for
finding the solution for stiff ODE. Furthermore, a strategy of the method was to produce
two new values ( 1ny and 2ny ) with step size h and two off step points 2
1n
y and 2
3n
y
with step size is halved simultaneously at each step. The formulae were computed using
two back values ny and 1ny with step size h in the previous block. This paper also
generated the stability region and convergence of the proposed methods. The method was
shown to be A-stable and convergent. In addition, they compared the proposed method
with the existing fifth oder BBDF method and the results showed that the performance of
methods were competitive in terms of accuracy and execution time.
18
CHAPTER 3
FORMULATION OF THE METHOD
3.1 Introduction
This chapter involve the derivation of predictor and corrector methods for VS-BBDF of
order four that are derived using Maple 15. We describe in detail one by one step that
include in the derivation of the methods. We only need three backvalues for predictor
methods whilst, five points are needed for corrector method as it is of order four. Then
for implementation of the method, we apply Newton’s iteration to the methods which we
derive later in this chapter. At the end of this chapter, we will present the further
implementation of the method and the strategy used for choosing the suitable step size.
We will use environment of Microsoft Visual C++ 6.0 to implement the algorithm in this
dissertation.
3.2 Formulation of predictor method for VS-BBDF method
The derivation of predictor method for VS-BBDF method presented here is conducted
using Maple 15 with ‘CurveFitting’ package. Therefore, there are three points, 12 , nn yy
and ny as the backvalues used to predict the values for points 1ny and 2ny . The
backvalues will be the interpolating points. Thus, we interpolates the points in Lagrange
form where Lagrange polynomial is the polynomial )(xPk degree k that passes through
the points ),(),,( 1122 nnnn yxyx and ),( nn yx .
19
The Lagrange basis polynomial is
k
jii injn
jn
jkxx
xxxL
0 11
1
,)(
)()( for each kj ,...,1,0 . (3.1)
So, the general form of Lagrange polynomial defined as follows
k
j
jkjnk xLxyxP0
,1 )()()( (3.2)
or expansion of the product (3.2) will be as below
))...((
))...((...
))...((
))...(()(
1
1
1
1
kknnkn
knkn
knnnn
knnnk
xxxx
xxxxy
xxxx
xxxxyxP
(3.3)
Start the derivation by using Lagrange’s interpolation formula to find the interpolating
polynomial through the points ),(),,( 1122 nnnn yxyx and ),( nn yx and obtained the
polynomial as follows
n
nnnn
nnn
nnnn
nnn
nnnn
nn yxxxx
xxxxy
xxxx
xxxxy
xxxx
xxxxxP
))((
))((
))((
))((
))((
))(()(
12
121
121
22
212
1
(3.4)
20
Then, define h
xxs n 1 and substitute 1)( nxhsx into (3.4) yield equation (3.5),
n
nnnn
nnnn
n
nnnn
nnnn
n
nnnn
nnnn
yxxxx
xxhsxxhs
yxxxx
xxhsxxhs
yxxxx
xxhsxxhsxP
))((
))((
))((
))((
))((
))(()(
12
1121
1
121
121
2
212
111
(3.5)
Next, replace ),( yxf in (1.1) by polynomial (3.5). Substitute 0s at point 1 nxx and
1s at point 2 nxx into P(x). So, for latest values of P(x), we replace step size ratio, r
by three distinct values that are 1, 2 and 5/9. Subsequently, the last part of derivation gives
the formulae for predictor method for fourth order VS-BBDF method. Step size ratio, r
is defined as the ratio distance between current step, nx and previous step, 1nx .
Let old
new
h
hr .
The predictor formulae for first and second points are
i. For r = 1 gives
nnnn
nnnn
yyyy
yyyy
683
33
122
121
(3.6)
ii. For r = 2 gives
nnnn
nnnn
yyyy
yyyy
33
8
15
4
5
8
3
122
121
(3.7)
21
iii. For r = 5/9 gives
nnnn
nnnn
yyyy
yyyy
25
322
25
504
25
207
25
133
25
171
25
63
122
121
(3.8)
3.3 Formulation of corrector method
The steps involve are quite similar to predictor method where the derivation of
corrector method are constructed using Maple 15 environment with “CurveFitting”
package as well as ‘PolynomialInterpolation’ command. The different from predictor is
corrector method need five points as interpolating points. Thus, we interpolates the points
in Lagrange form where Lagrange polynomial is the polynomial )(xPk degree k that
passes through the points ),(),,(),,(),,( 111122 nnnnnnnn yxyxyxyx and ),( 22 nn yx .
We begin our derivation by using Lagrange’s interpolation formula to find the
interpolating polynomial through the points ),(),,(),,(),,( 111122 nnnnnnnn yxyxyxyx
and ),( 22 nn yx and obtained the polynomial as follows
2
1221222
112
1
2111121
212
2112
2112
1
2111121
212
2
2212212
211
))()()((
))()()((
))()()((
))()()((
))()()((
))()()((
))()()((
))()()((
))()()((
))()()(()(
n
nnnnnnnn
nnnn
n
nnnnnnnn
nnnn
n
nnnnnnnn
nnnn
n
nnnnnnnn
nnnn
n
nnnnnnnn
nnnn
yxxxxxxxx
xxxxxxxx
yxxxxxxxx
xxxxxxxx
yxxxxxxxx
xxxxxxxx
yxxxxxxxx
xxxxxxxx
yxxxxxxxx
xxxxxxxxxP
(3.9)
22
Then, define h
xxs n 1 and substitute 1)( nxhsx into (3.9) yield equation (3.10)
2
1221222
1111121
1
2111121
2111121
2112
21111121
1
2111121
2111121
2
2212212
2111111
))()()((
)))(())(())(()((
))()()((
)))(())(())(()((
))()()((
)))(())(())(()((
))()()((
)))(())(())(()((
))()()((
)))(())(())(()(()(
n
nnnnnnnn
nnnnnnnn
n
nnnnnnnn
nnnnnnnn
n
nnnnnnnn
nnnnnnnn
n
nnnnnnnn
nnnnnnnn
n
nnnnnnnn
nnnnnnnn
yxxxxxxxx
xxhsxxhsxxhsxxhs
yxxxxxxxx
xxhsxxhsxxhsxxhs
yxxxxxxxx
xxhsxxhsxxhsxxhs
yxxxxxxxx
xxhsxxhsxxhsxxhs
yxxxxxxxx
xxhsxxhsxxhsxxhsxP
(3.10)
Next, replace ),( yxf in (1.1) by polynomial (3.10). Later, we differentiate (3.10) with
respect to s as well as substitute 0s at point 1 nxx and 1s at point 2 nxx into
P’(x). So, for latest values of P’(x), we replace step size ratio, r by three distinct values
that are 1, 2 and 5/9. Subsequently, the last part of derivation gives the formulae for
corrector methods for fourth order VS-BBDF method.
Let old
new
h
hr .
The formulae obtained are
i. Formula for r = 1 gives
nnnnnn
nnnnnn
yyyyhfy
yyyyhfy
25
36
25
16
25
3
25
48
25
12
5
9
5
3
10
1
10
3
5
6
12122
12211
(3.11)
23
ii. Formula for r = 2 gives
nnnnnn
nnnnnn
yyyyhfy
yyyyhfy
23
18
23
3
115
2
115
192
23
12
128
225
128
25
128
3
128
75
8
15
12122
12211
(3.12)
iii. Formula for r = 5/9 gives
nnnnnn
nnnnnn
yyyyhfy
yyyyhfy
35625
103684
11875
27216
225625
128547
27075
59248
1425
644
7425
17689
6325
9747
550
189
13662
2527
297
266
12122
12211
(3.13)
Table 3.1 below represents the formulae for predictor and corrector for the two computed
points.
Table 3.1 The formulae for predictor and corrector method for two computed points
Step
size
ratio
Points Coefficients of the points
r = 1
1ny
Predictor nnn yyy 33 12
Corrector nnnnn yyyyhf5
9
5
3
10
1
10
3
5
61221
2ny
Predictor nnn yyy 683 12
Corrector nnnnn yyyyhf25
36
25
16
25
3
25
48
25
121212
r = 2 1ny
Predictor nnn yyyy8
15
4
5
8
312
Corrector
nnnnn yyyyhf128
225
128
25
128
3
128
75
8
151221
24
2ny
Predictor nnn yyy 33 12
Corrector
nnnnn yyyyhf23
18
23
3
115
2
115
192
23
121212
r = 5/9
1ny
Predictor nnn yyy25
133
25
171
25
6312
Corrector
nn
nnn
yy
yyhf
7425
17689
6325
9747
550
189
13662
2527
297
266
1
221
2ny
Predictor nnn yyy25
322
25
504
25
20712
Corrector
nn
nnn
yy
yyhf
35625
103684
11875
27216
225625
128547
27075
59248
1425
644
1
212
3.4 Implementation of the method
There are many compiler that can be used to implement the algorithm of numerical
methods. In this dissertation, we choose to use Microsoft Visual C++ 6.0 as platform to
implement the VS-BBDF method with Newton iteration algorithm. The program will be
written in C language.
3.4.1 Newton’s iteration
Further explanation on implementation of the method will be discussed in this section.
Newton’s iteration is applied to fourth order VS-BBDF method for better performance in
finding the approximation solutions of 1ny and 2ny . From the derivation above, the
general form of corrector formulae (3.11-3.13) can be written in matrix form as
25
2
2
1
1
2
1
2
1
2
2
1
1
1
2
2
1
21
0
0
0
01
0
0
1
nnnnnnnnn yyyhfhfyyyy
(3.14)
where 2121212121 ,,,,,,,,, are the coefficients of the points. So, matrix form
in (3.14) can be simplified in this way with representation 21, as the backvalues
nnn yyy ,, 12 .
2
1
2
1
2
1
2
1
1
2 0
0
1
1
n
n
nnf
fhyy (3.15)
Thus, in simpler way we have (3.15) as
hBFYAI )(
(3.16)
where
2
1
2
1
2
1
2
1
2
1,,
0
0,,
0
0,
10
01
n
n
n
n
f
fFB
y
yYAI
Let the system equivalent to 0, we have
0)(ˆ hBFYAIF
(3.17)
Newton’s iteration is performed to the system (3.17) and we get a new equation in the
form of Newton’s iteration
)(
)()(
)2,1(
)(
)2,1()(
)2,1(
)1(
)2,1( i
nn
i
nni
nn
i
nnYF
YFYY
(3.18)
26
To approximate the solution, apply Newton’s iteration (3.18) to the system (3.17). The
new equation for approximating the solutions contain the Jacobian matrix of F with
respect to y shows in (3.19) as below
)(
)2,1(
)(
)2,1(
)(
)2,1()(
)2,1(
)1(
)2,1(
)(
)(
i
nn
i
nn
i
nni
nn
i
nn
Y
FhBAI
hBFYYAIYY
(3.19)
Hence, equation (3.19) is used to approximate the solutions.
3.4.2 Choosing the step size
Apart from that, there are three strategies that very significant in choosing
appropriate step size at each step of iteration. Two possibilities of adjustment the step size
for every successful step and only one possibilities for every failure step. Therefore, the
possibilities are remain the step size (r = 1), decrease the step size to half (r = 2) and
increase the step size to a factor 1.8 (r = 5/9). For every successful step, the chosen step
size is either maintained or increased the step size to a factor 1.8 whereby for failure step
the chosen step size is halved the current step size.
Furthermore, the values of 1ny and 2ny are accepted if the current step is successful and
otherwise the values of 1ny and 2ny are rejected. Consequently, cases for the step are
determined by checking the local truncation error (LTE) either it is less than or greater
than tolerance limit (TOL). The user will provide the TOL on any given step. The selected
test problems are solved with error tolerances limit of 10-2, 10-4, 10-6 and 10-8.
27
The formula of determining the LTE is given as
i
n
i
n yyLTE 2
1
2
(3.20)
where 1
2
i
ny is (i+1)-th order method and i
ny 2is the i-th order method.
Here are the lists of LTE for the proposed method. The lists are shown in the table below.
Table 3.2 Local truncation error for the three step size ratios
Step size
ratio Local Truncation Error, LTE
r = 1 nnnnnnn yyyyhfyy275
171
275
126
25
3
275
78
275
181212
3
2
4
2
r = 2 nnnnnnn yyyyhfyy161
34
483
40
115
2
2415
352
161
81212
3
2
4
2
r = 5/9
nn
nnnnn
yy
yyhfyy
344375
583487
2410625
4370598
225625
128547
1832075
826298
13775
1058
1
212
3
2
4
2
Basically, there are two possibilities in choosing the appropriate step size.
Case 1 : Successful step (LTE < TOL)
In this case, accept the values of 1ny and 2ny . But two possibilities of choosing the
suitable step size if we need to maintain the current step size (r = 1) or increase the step
size by a factor of 1.8 (r = 5/9). So, for this case it is significant to ensure the previous
28
step that it is successful or not. Let say the previous step is failed, then current step size
should be maintained. Or else the step size should be increase by a factor of 1.8. The step
size increment is given by
p
oldnewLTE
TOLhch
1
and if oldnew hh 8.1 then oldnew hh 8.1
where c is safety factor and we set it to 0.8, p is the order of the method, oldh is previous
step size and newh is current step size.
Case 2 : Failure step (LTE > TOL)
In this case, reject the values of 1ny and 2ny . Thus, reiterate the current step by halving
the current step size (r = 2).
3.5 Stability of VS-BBDF method
A method to be of practical importance it must have a region of absolute stability to ensure
that the method will be able to solve at least for the mildly stiff problems (Majid &
Suleiman, 2006). Absolute stability can indicate the stability of numerical methods. The
stability region is the region enclosed by the set of points determined by replacing
20 ,cossin iet i in stability polynomial. In order to determine the
absolute stability, we have to apply VS-BBDF method formula to the test equation. The
stability region is determined by finding the region for which |t| < 1.
Definition 3.1 (Lambert, 1991)
A method is said to be absolute stable in a region R for a given h , all the roots sr of the
stability polynomial 0)()(:);( rhrhr , satisfy 1|| sr where ks ,...,2,1 .
29
Definition 3.2 (Lambert, 1991)
A numerical method is said to be A stable if its region of absolute stability contains the
whole of the left-hand half-plane 0)Re( h .
Definition 3.3 (Lambert, 1991)
The LMM is said to be zero stable of no root of the first characteristic polynomial )(rp
has modulus greater than one, and if every root with unit modulus is simple.
Apply equation (1.2) to the the test equation, 𝑦′ = 𝑦 and then obtain
k
j
k
j
jnijjnij yhy0 0
22 (3.21)
or
k
j
jnijij yH0
2 0 (3.22)
where 𝐻 = ℎ and i = 1 and 2.
The equation (3.22) can be simplify in other form as
k
j
jjYA0
,0 (3.23)
where
],...,[ 0 rj AAA equivalent to
)12(,2)12(,2)12(,2
)12(,1)12(,12,1
jjj
jjj
j h
hA
,
30
rj YYY ,...,0 equivalent to
jn
jn
j y
yY
21
22 .
The stability polynomial of the method is
0det);(0
r
j
j
j tAHtR (3.24)
Then, substitute H = 0 of equation (3.24) to obtain the roots for stability polynomial. The
stability polynomial for the three step size ratios are listed in the table below.
Table 3.3 Lists of stability polynomials and roots for the three step size ratios
Step size
ratio, r
Stability polynomial, );( HtR Roots
r = 1
HtHtHt
Httttt
2324
4234
125
18
125
252
125
72
25
42
125
1
25
9
125
153
125
197
1,
0.0207917599,
-0.2441420137
r = 2
HtHtHt
Httttt
2324
4234
92
3
736
1155
46
45
184
441
2944
1
2944
289
93
173
46
91
1,
0.00325762197,
-0.05270817143
31
r = 5/9 HtHtHtHt
tttt
23244
234
130625
66654
556875
1839404
22275
9016
141075
190106
653125
59049
653125
755829
10580625
3575389
22275
31291
1,
0.0769489074
-0.8363962010
In order to determine the stability region, solve t in stability polynomial as stated in the
table by replacing the value of H. As we have mention earlier, the method is stable if the
absolute value of t that has been solved is less than 1. Therefore, the region of stability of
VS-BBDF method is plotted using MAPLE 15. Hence, figure 3.1- 3.4 show the region of
stability for VS-BBDF method.
Figure 3.1 Stability region for r = 1
32
Figure 3.2 Stability region for r = 2
Figure 3.3 Stability region for r = 5/9
33
Figure 3.4 Stability regions for r = 1, r = 2 and r = 5/9
From the figures, the stability region for VS-BBDF method lies outside the closed region.
Since all the roots for the step size ratios have modulus less than or equal to 1, thus the
proposed method are satisfied zero stability. The figures also show that the absolute
stability region for step size ratios 1 and 5/9 are almost A-stable while the stability region
for step size ratio 2 is A-stable since it contains the entire left half-plane of the complex
plane, 0)Re( h .
34
CHAPTER 4
RESULTS AND DISCUSSIONS
4.1 Introduction
The results of this dissertation will be presented in this chapter. Fourth order VS-BBDF
method with Newton’s iteration is used to solve first order stiff IVP ODEs problem. There
are seven selected test problems with different values of eigenvalues, . The results that
are obtained will be compared with the results of step size ratios 1.6 and 1.9 which have
been solved by the previous researchers. The results for each problem are illustrated in
three figures. The first figure is the approximated solutions curve for variable step size
ratio. The second figure is the graph of approximate solution and exact solution while the
third figure is total steps curve for every test problems considered in this dissertation.
4.2 Test Problems
There are seven test problems in this section. The following test problems are stiff IVP
ODEs. The tested problems of stiff ODEs are solved using the fourth order VS-BBDF
method with Newton’s iteration. The numerical results for step size 1.8 will be compared
with step size 1.6 and 1.9 in term of maximum errors.
35
The formula for maximum error is defined as
MAXE =
errorniTSi
maxmax11
,
where TS is the total steps, n is the number of equations and
))((
)(
i
ii
xyBA
xyyerror
with A = 1, B = 1 for mixed error test.
Problem 4.1
212
211
2019
1920
yyy
yyy
Interval : 200 x
Exact solution : xx
xx
eexy
eexy
39
2
39
1
)(
)(
Initial conditions : 0)0(
2)0(
2
1
y
y
Eigenvalues : 39
1
2
1
Source : Cheney and Kincaid (1999)
36
Problem 4.2
)1(
10001002
2212
2
211
yyyy
yyy
Interval : 200 x
Exact solution : x
x
exy
exy
)(
)(
2
2
1
Initial conditions : 1)0(
1)0(
2
1
y
y
Eigenvalue : 1002
Source : Kaps and Wanner (1981)
Problem 4.3
1)(20 xyy
Interval : 100 x
Exact solution : xexy x 20)(
Initial conditions : 1)0( y
Eigenvalues : 20
Source : Artificial problem
37
Problem 4.4
23 3)(100 xxyy
Interval : 100 x
Exact solution : 3)( xxy
Initial conditions : 0)0( y
Eigenvalues : 100
Source : Gear (1971)
Problem 4.5
212
211
1999999
1998998
yyy
yyy
Interval : 200 x
Exact solution : xx
xx
eexy
eexy
1000
2
1000
1
)(
2)(
Initial conditions : 0)0(
1)0(
2
1
y
y
Eigenvalues : 1000
1
2
1
Source : Gear (1971)
38
Problem 4.6
1)(100 xyy
Interval : 100 x
Exact solution : xexy x 100
1 )(
Initial conditions : 1)0( y
Eigenvalues : 100
Source : Gear (1971)
Problem 4.7
212
211
19971197
19951195
yyy
yyy
Interval : 200 x
Exact solution : xx
xx
eexy
eexy
8002
2
8002
1
86)(
810)(
Initial conditions : 2)0(
2)0(
2
1
y
y
Eigenvalues : 800
2
2
1
Source : Gerald and Wheatley (1989)
39
4.3 Numerical Results
In this section, we discussed the results for tested problem of first-order stiff IVP ODEs given
in the previous section. In this dissertation, we considered constant step size (r = 1), half the
step size (r = 2) and increment the step size to a factor 1.8 (r = 5/9). The results obtained are
then compared with the results of step size 1.6 and 1.9. Numerical results are presented in
the figures 4.1 until 4.21 where each problem have three different figure. Basically, the first
figure represented the approximated solutions for variable step size ratio, the second figure
is the approximate solutions and exact solutions for every tested problems and the third figure
is the total steps curve for each problem. Then, we make conclusions based on the results
obtained in those figures.
Figures below represent the results for every tested problems considered in this dissertation.
4.3.1 Results and Discussions for Test Problem 4.1
Figure 4.1 Approximated solutions curves for variable step size ratios
for Test Problem 4.1
-25
-20
-15
-10
-5
10e-2 10e-4 10e-6 10e-8
log|
Max
imu
m E
rro
r|
TOL
r = 1.6
r = 1.8
r = 1.9
40
Figure 4.2 Graph of approximate solution and exact solution
for Test Problem 4.1
Figure 4.3 Total steps curves for Test Problem 4.1
-16
-14
-12
-10
-8
-6
-4
-2
0
10e-2 10e-4 10e-6 10e-8
log|
Solu
tio
n|
TOL
Exact solution
Approximatesolution
0
50
100
150
200
250
300
350
400
10e-2 10e-4 10e-6 10e-8
T
o
t
a
l
s
t
e
p
s
TOL
r = 1.6
r = 1.8
r = 1.9
41
The three figures represent the results for the test problem 4.1. The first figure represents an
approximated solutions curves for variable step size ratios. As we can observe clearly from
the figure, the maximum errors of step size 1.8 were reducing as the value of tolerances were
decreasing. For the second figure, it can be seen that the approximate solution of step size
1.8 was lying almost in the same line with the exact solution. Besides, the third figure
represents as the total steps taken during the computation of the solution. Therefore, it was
obviously seen that the total steps taken were increasing as the value of tolerances were
decreasing.
4.3.2 Results and Discussions for Test Problem 4.2
Figure 4.4 Approximated solutions curves for variable step size ratios
for Test Problem 4.2
-20
-15
-10
-5
10e-2 10e-4 10e-6 10e-8
log|
Max
imu
m E
rro
r|
TOL
r = 1.6
r = 1.8
r = 1.9
42
Figure 4.5 Graph of approximate solution and exact solution
for Test Problem 4.2
Figure 4.6 Total steps curves for Test Problem 4.2
-7
-6
-5
-4
-3
-2
-1
0
10e-2 10e-4 10e-6 10e-8
log|
Solu
tio
n|
TOL
Exact solution
Approximatesolution
0
50
100
150
200
250
300
350
10e-2 10e-4 10e-6 10e-8
T
o
t
a
l
s
t
e
p
s
TOL
r = 1.6
r = 1.8
r = 1.9
43
The three figures represent the results for the test problem 4.2. The first figure shows an
approximated solutions curves for variable step size ratios. As we can observe clearly from
the figure, the maximum errors of step size 1.8 were also reducing as the value of tolerances
were decreasing. Next, for second figure, it can also be seen that the approximate solution of
step size 1.8 was lying almost in the same line with the exact solution. Then, it was obviously
seen that the total steps taken were also increasing as the value of tolerances were decreasing.
4.3.3 Results and Discussions for Test Problem 4.3
Figure 4.7 Approximated solutions curves for variable step size ratios
for Test Problem 4.3
-25
-20
-15
-10
-5
10e-2 10e-4 10e-6 10e-8
log|
Max
imu
m E
rro
r|
TOL
r = 1.6
r = 1.8
r = 1.9
44
Figure 4.8 Graph of approximate solution and exact solution
for Test Problem 4.3
Figure 4.9 Total steps curves for Test Problem 4.3
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
10e-2 10e-4 10e-6 10e-8
log|
Solu
tio
n|
TOL
Exact solution
Approximatesolution
0
50
100
150
200
250
10e-2 10e-4 10e-6 10e-8
T
o
t
a
l
s
t
e
p
s
TOL
r = 1.6
r = 1.8
r = 1.9
45
The three figures represent the results for the test problem 4.3. The first figure represents
an approximated solutions curves for variable step size ratios. Similarly to those
problems, the observation showed that the maximum errors of step size 1.8 were also
reducing when the tolerances value were decreasing. The approximate solution of step
size 1.8 was also located almost in the same line with the exact solution for the second
figure. Then, the third figure also shows that the total steps taken were increasing as the
value of tolerances were decreasing.
4.3.4 Results and Discussions for Test Problem 4.4
Figure 4.10 Approximated solutions curves for variable step size ratios
for Test Problem 4.4
-45
-40
-35
-30
-25
-20
10e-2 10e-4 10e-6 10e-8
log|
Max
imu
m E
rro
r|
TOL
r = 1.6
r = 1.8
r = 1.9
46
Figure 4.11 Graph of approximate solution and exact solution
for Test Problem 4.4
Figure 4.12 Total steps curves for Test Problem 4.4
-40
-35
-30
-25
-20
-15
-10
-5
0
10e-2 10e-4 10e-6 10e-8
log|
Solu
tio
n|
TOL
Exact solution
Approximatesolution
0
10
20
30
40
10e-2 10e-4 10e-6 10e-8
T
o
t
a
l
s
t
e
p
s
TOL
r = 1.6
r = 1.8
r = 1.9
47
The three figures represent the results for the test problem 4.4. So, the results was the
same as previous results where the maximum errors of step size 1.8 were reducing as the
value of tolerances were decreasing. The second figure also represents the approximate
solution of step size 1.8 and the exact solution. Then, the results also showed the
approximate solution of step size 1.8 was lying almost in the same line with the exact
solution. The third figure represents the total steps taken during the computation.
Consequently, the figure showed that the total steps taken were also increasing as the
value of tolerances were decreasing.
4.3.5 Results and Discussions for Test Problem 4.5
Figure 4.13 Approximated solutions curves for variable step size ratios
for Test Problem 4.5
-25
-20
-15
-10
-5
10e-2 10e-4 10e-6 10e-8
log|
Max
imu
m E
rro
r|
TOL
r = 1.6
r = 1.8
r = 1.9
48
Figure 4.14 Graph of approximate solution and exact solution
for Test Problem 4.5
Figure 4.15 Total steps curves for Test Problem 4.5
-2.5
-2
-1.5
-1
-0.5
0
10e-2 10e-4 10e-6 10e-8
log|
Solu
tio
n|
TOL
Exact solution
Approximatesolution
0
50
100
150
200
250
300
350
400
450
10e-2 10e-4 10e-6 10e-8
T
o
t
a
l
s
t
e
p
s
TOL
r = 1.6
r = 1.8
r = 1.9
49
The three figures represent the results for the test problem 4.5. We can observe the
maximum errors of step size 1.8 were also decreasing as the value of tolerances were
decreasing. Besides, we can also see on the second figure where the approximate solution
of step size 1.8 was also lying almost in the same line with the exact solution. Results for
the third figure was similar to the previous results at which the total steps taken during
the computation were also increasing as the value of tolerances were decreasing.
4.3.6 Results and Discussions for Test Problem 4.6
Figure 4.16 Approximated solutions curves for variable step size ratios
for Test Problem 4.6
-25
-20
-15
-10
-5
10e-2 10e-4 10e-6 10e-8
log|
Max
imu
m E
rro
r|
TOL
r = 1.6
r = 1.8
r = 1.9
50
Figure 4.17 Graph of approximate solution and exact solution
for Test Problem 4.6
Figure 4.18 Total steps curves for Test Problem 4.6
-12
-10
-8
-6
-4
-2
0
10e-2 10e-4 10e-6 10e-8
log|
Solu
tio
n|
TOL
Exact solution
Approximatesolution
0
50
100
150
200
250
10e-2 10e-4 10e-6 10e-8
T
o
t
a
l
s
t
e
p
s
TOL
r = 1.6
r = 1.8
r = 1.9
51
The three figures represent the results for the test problem 4.6. So, the results was similar
to the previous results where the maximum errors of step size 1.8 were reducing as the
value of tolerances were decreasing. The second figure was also similar to the previous
results whereby the approximate solution of step size 1.8 was lying almost in the same
line with the exact solution. Lastly, the third figure were also increasing as the value of
tolerances were decreasing.
4.3.7 Results and Discussions for Test Problem 4.7
Figure 4.19 Approximated solutions curves for variable step size ratios
for Test Problem 4.7
-25
-20
-15
-10
-5
10e-2 10e-4 10e-6 10e-8
log|
Max
imu
m E
rro
r|
TOL
r = 1.6
r = 1.8
r = 1.9
52
Figure 4.20 Graph of approximated solutions for variable step size ratios
for Test Problem 4.7
Figure 4.21 Total steps curves for Test Problem 4.7
-12
-10
-8
-6
-4
-2
0
2
410e-2 10e-4 10e-6 10e-8
log|
Solu
tio
n|
TOL
Exact solution
Approximatesolution
0
100
200
300
400
500
600
10e-2 10e-4 10e-6 10e-8
T
o
t
a
l
s
t
e
p
s
TOL
r = 1.6
r = 1.8
r = 1.9
53
The three figures represent the results for the test problem 4.7. The results for all the
figures were also similar to those results of previous tested problems where the results for
first figure was the maximum errors of step size 1.8 were also reducing as the value of
tolerances were decreasing. Then, the second figure also shows the approximate solution
of step size 1.8 was lying almost in the same line with the exact solution and last but not
least, the results for the third figure were also increasing as the value of tolerances were
decreasing.
4.4 Discussions
Generally, the first figure for each problem represented the approximate solutions for
every tested problems at three different values of step sizes (1.6, 1.8 and 1.9) in term of
maximum errors. From the figure, he maximum errors were also reduced when the value
of tolerances were reduced. Therefore, it can be seen that the performance was getting
better when step size ratio was increased. In addition, the second figure showed the results
for approximate solutions and exact solutions of all the selected tested problems. As we
can observed from the figures, it was clearly shown that the approximate solutions were
almost on the same line with the actual solutions in those figures. Subsequently, the
approximate solutions were converging to the exact solutions. Then, the third figures
represented all the total steps that were involved in computing the solutions. The total
steps were increasing as the value of tolerances were decreasing. As we can also see in
the same figure, mostly the step size 1.9 gave the least total steps for tolerances 10-2, 10-
4 and 10-6. However, mostly the step size 1.6 gave the least total steps for tolerance 10-8.
As a conclusion, we can conclude that the step size 1.9 gave the best results for all those
selected problems considered in this dissertation. However, the step size 1.8 was also
reliable to solve those selected problems of stiff ODEs based on the accuracy obtained
54
from the results. So, this showed that the proposed methods used were relevant for solving
stiff ODEs. Therefore, in conclusion, increasing the value of step size ratio will give more
accurate results.
57
CHAPTER 5
SUMMARY
5.1 Conclusions
The aim of this dissertation is to derive the fourth order Variable Step Block Backward
Differentiation Formula method. The aim is accomplished by applying the derived
method to the first order stiff ODE based on variable step size approach. In this
dissertation, we derived the method by using step size 1.8 where the proposed method
inspired from the fourth order Variable Step Block Backward Differentiation Formula of
step sizes 1.6 and 1.9. On the whole dissertation, the main objective is to investigate the
effect of solutions by changing the values of step size ratio, r where all the step size ratios
are satisfying zero stability.
There are seven selected tested problems that have been solved in this dissertation. Each
of the problems are tested at variable step sizes depending on the LTE. The numerical
results showed that the effectiveness of step size ratio 1.8 is relevant to solve the restricted
problems studied as the errors produced are within the tolerance given at each step of the
iteration. Hence, it also gave an accurate solutions. The conclusions made are restricted
only to the problems studied based on the results and discussions earlier. As the step size
ratio increased, the results produced better approximated solutions as the error goes to
zero and the solutions converged to the exact values. Therefore, by increasing the step
size ratio will
58
give better performance in terms of maximum error and reduction of total steps involved
at each iteration for all tested problems.
5.2 Future Study
A few recommendations and outlines stated for future study in this area as follows:
1. The first order stiff ODE has been solved in this dissertation. Therefore, it would
be interesting to solve higher order of stiff problems.
2. For next study, it would be exciting for solving first order stiff ODE by increasing
the number of points in the block for further improvement in performance of the
method by the aim of reducing the total number of steps.
3. For future research, it would be challenging to investigate the efficiency of the
method in terms of computation time.
56
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APPENDIX
Appendix A : Algorithm for VS-BBDF method of order four
Start
Input: Starting point a,
End point b,
Step size h,
Step size ratio r,
Initial value of X ]0[X ,
Point = 2,
Tolerance TOL,
IND2 = 1.
Step 1 : Compute backvalues points, ny , 1ny and 2ny (Euler method).
Step 2 : Compute predictor values for points 1ny and 2ny (Predictor method).
Step 3 : Compute Jacobian matrix
Step 4 : Compute Newton iteration matrix
Step 5 : Compute Error
Estimate the error (LTE)
Do Step 6 (Test for convergence) : Check
If LTE < =TOL & convergence
Accept values for 1ny and 2ny .
Step = Step + 1,
Compute maximum error, MAXE,
If IND2 = 0,
h = hold,
r = 1,
If IND2 = 1,
Calculate hnew for new step size
p
oldLTE
TOLhc
1
hacc
and if then
If oldh 8.1h acc then
oldhh 8.1
r = 5/9,
else
Reject values for 1ny and 2ny .
Failure step = Failure step + 1,
oldhh 2
1,
r = 2,
Step 7:
Output ),( 21 nn yy ,
Total steps,
Maximum error.
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