SOLVING BURGER’S EQUANTION USING EXPLICIT FINITE...
Transcript of SOLVING BURGER’S EQUANTION USING EXPLICIT FINITE...
SOLVING BURGER’S EQUANTION USING EXPLICIT FINITE DIFFERENCE
METHOD AND METHOD OF LINE
CARRIE TAN SHIAU YING
UNIVERSITI TEKNOLOGI MALAYSIA
SOLVING BURGER’S EQUATION USING EXPLICIT FINITE DIFFERENCE
METHOD AND METHOD OF LINE
CARRIE TAN SHIAU YING
A dissertation submitted in partial fulfilment of the
requirements for the award of the degree of
Master of Science in
Engineering Mathematics
Faculty of Science
Universiti Teknologi Malaysia
JUNE 2017
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This dissertation is specially dedicated to:
My beloved fam ily especially to my parents, lecturers
and all my friends
They have always given me a warm encouragement and guided me throughout my
journey o f my education. Their moral support and contribution I will never forget.
I own them more than they will ever know
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ACKNOW LEDGEM ENT
First and foremost, I would like to express my gratefulness and appreciation to
my honorable supervisor of this research, Dr. Shazirawati Mohd Puzi and co
supervisor, Dr. Anati Ali for their consistence guidance and support that they have
given to me throughout the duration of this research.
Besides that, I would like to thank Dr. Yeak Su Hoe for his help in running the
programme simulations for this research. My sincere appreciation also extends to all
of my course mates and seniors who have provided assistance throughout the
completion of this research. I also feel grateful to PSZ for providing me information
and a comfortable place for my research findings.
Million thanks also to all my family members for endless care and support
during hardships and struggles. Last but not least, unrepaid debt to my precious friends
for all the encouragement and assistance through the ups and downs in this challenging
periods.
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ABSTRACT
Burgers’ equation is a quasilinear differential equation can be solve either
analytically or numerically. The analytical solutions use the Hopf-Cole transformation
and reduced to diffusion equation. The focus of this research was to solve Burgers’
equation numerically by using Finite Difference Method (FDM) and Method of Line
(MOL) by using Fourth Order Runge-Kutta (RK4). The accuracy of MOL obtained
solutions depends on the type of Ordinary Differential Equation (ODE) method used.
The results obtained from both numerical method were compared between Hopf-Cole
transformation analytical solutions. The simulations is coded by using MATLAB
software. From the comparison, both methods shown to be good numerical
approximation as the results obtained near to the exact solution. As the increase of
spatial step size, the solutions obtained with be more accurate followed by individual
methods’ restrictions. Different time and viscosity coefficient also tested to observe
the changes of Burgers’ equation solutions.
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ABSTRAK
Persamaan Burgers merupakan persamaan pembezaan quasilinear yang boleh
diselesaikan secara analitikal atau berangka. Bagi mendapatkan penyelesaian
analitikal, transformasi Hopf-Cole digunakan untuk menghasilkan persamaan resapan.
Fokus kajian ini adalah untuk menyelesaikan persamaan Burgers secara kaedah
berangka dengan menggunakan kaedah beza terhingga (FDM) dan kaedah garisan
(MOL) yang menggunakan kaedah Runge-Kutta peringkat keempat (RK4). Ketepatan
hasil pengiraan MOL bergantung kepada jenis kaedah yang digunakan untuk
menyelesaikan masalah seperti persamaan pembezaan biasa (ODE). Seterusnya,
perbandingan dibuat terhadap keputusan simulasi MATLAB untuk penyelesaian
menggunakan kaedah berangka dan penyelesaian analitikal yang melalui transformasi
Hopf-Cole. Berdasarkan perhatian didapati, penyelesaian berangka mampu memberi
penyelesaian yang baik. Pengurangan panjang saiz langkah akan memberikan
penyelesaian berangka yang lebih tepat. Masa dan pekali kelikatan yang berbeza juga
diuji untuk melihat kesannya terhadap penyelesaian persamaan Burgers.
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TABLE OF CONTENTS
CHAPTER TITLE PAGE
TITLE i
DECLARATION ii
DEDICATION iii
ACKNOW LEDGEM ENT iv
ABSTRACT v
ABSTRAK vi
TABLE OF CONTENTS vii
LIST OF TABLES x
LIST OF FIGURES xi
LIST OF ABBREVIATIONS xiv
LIST OF SYMBOLS xv
LIST OF APPENDICES xvii
1 INTRODUCTION 1
1.1 Background of the Study 1
1.2 Statements of the Problem 3
1.3 Objectives of the Study 4
1.4 Scope of the Study 4
1.5 Significance of the Study 5
1.6 Organization of the Research 5
2 LITERATURE REVIEW 6
2.1 Introduction 6
2.2 Burgers’ Equation 6
2.2.1 History Review 7
2.3 Finite Difference Method 8
2.3.1 Taylor Series 8
2.3.2 First Order Forward Difference 9
2.3.3 First Order Backward Difference 10
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2.3.4 First Order Central Difference 11
2.3.5 Second Order Central Difference 12
2.4 Method of Line 13
2.4.1 Formulate the Approximating System of ODE 13
2.5 Runge-Kutta Method 14
2.5.1 Derivation of Runge-Kutta Method 15
2.6 Summary 17
M ATHEM ATICAL FORM ULATION AND3 18
ALGORITM S
3.1 Introduction 18
3.2 Mathematical Examples 19
3.2.1 Finite Difference Approximations 19
3.2.2 Fourth Order Runge-Kutta 20
3.2.3 Example of Neumann Boundary Condition 20
Boundary Value Problem (BVP) solve using Finite
Difference Method
3.2.4 Example of Initial Value Problem (IVP) solved 24
using Fourth Order Runge-Kutta Method
3.3 Mathematical Problem 26
3.4 Mathematical Formulation of Burgers’ Equation 27
3.4.1 Hopf-Cole Transformation 27
3.4.2 Analytical Solution 28
3.5 Numerical Method 29
3.5.1 Explicit Finite Difference Method 29
3.5.2 Method of Line 31
3.5.3 Transformation of Diffusion equation solutions 33
into Burgers’ equation solutions
3.5.4 MATLAB 34
3.6 MATLAB Codes 34
3.6.1 MATLAB codes for Exact Solution of Burgers’ 34
Equation
3.6.2 MATLAB codes for FDM on transformed Burgers’ 36
Equation
3.6.3 MATLAB codes for RK4 on transformed Burgers’ 38
Equation
4 RESULTS AND DISCUSSION 41
4.1 Introduction 41
4.2 Burgers’ Equation Computation using Finite Difference 43
Method (FDM)
4.3 Burgers’ Equation Computation using Method of Line 47
(MOL)
4.4 Comparison of Transformed Burgers’ Equation for FDM 51
and MOL
4.5 Value of Viscosity coefficient in Burgers’ Equation 54
4.5.1 Time Varies for Small Value of Viscosity 55
Coefficient
5 CONCLUSION AND RECOM M ENDATIONS 60
5.1 Introduction 60
5.2 Conclusion 61
5.3 Recommendations 61
REFERENCES 63
APPENDIX A 66
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TABLE NO.
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
LIST OF TABLES
TITLE
The Calculation of Transformed Burgers’ equation
using explicit FDM with different step size, N
Error differences of Burgers’ Equation using
FDM with different step size, N
Computation of Burgers’ equation using explicit FDM
with different time, t
The Calculation of Transformed Burgers’ equation
using MOL with different step size, N
Error differences of Burgers’ Equation using
MOL with different step size, N
The Calculation of TransformedBurgers’ equation
using MOL with different time, t
Computation of Diffusion equation using FDM
and MOL with different step size, N
Absolute differences of Diffusion equation using
FDM and MOL with different step size, N
Computation of Burgers’ equation using MOL
with different viscosity coefficient, v
PAGE
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44
46
47
48
50
52
53
54
4.10
4.11
Computation of Burgers’ equation using MOL
at v=0.001 with different time, t 56
Computation of Burgers’ equation using MOL
at v=0.0001 with different time, t 58
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FIGURE NO.
3.1
3.2
4.1
4.2
4.3
4.4
4.5
4.6
LIST OF FIGURES
TITLE
Summary of nodes obtained from the given question
Summary of nodes from the given question
Burgers’ equation solution, u versus value of x with
different step size, N by using FDM
Relative difference of Burgers’ equation solution,
u versus value of x with different step size, N by
using explicit FDM
Burgers’ equation solution, u versus value of x
with different time, t by using explicit FDM
Burgers’ equation solution, u versus value of x
with different step size, N by using MOL
Error difference of Burgers’ equation solution,
u versus value of x with different step size, N
by using MOL
Burgers’ equation solution, u versus value of x
with different time, t by using MOL
PAGE
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24
44
45
46
48
49
50
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4.7 Burgers’ equation solution, u versus value of x with
different viscosity coefficient, v 55
4.8 Burgers’ equation solution, u versus value of x
at v=0.001 with different time, t 57
4.9 Burgers’ equation solution, u versus value of x at
v=0.0001 with different time, t 59
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LIST OF ABBREVIATIONS
FDM - Finite Difference Method
MOL - Method of Line
RK Runge-Kutta
RK2 ' Second Order Runge-Kutta
RK3 Third Order Runge-Kutta
RK4 Fourth Order Runge-Kutta
PDE - Pertial Differential Eqution
ODE - Ordinary Differential Equation
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LIST OF SYMBOLS
uux
vuxx
Ax
Ay
Unsteady term
Convective term
Viscous term
Step size in term of x
Step size in term of y
u
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APPENDIX
A
LIST OF APPENDICES
TITLE
MATLAB Codes for explicit FDM with Neumann
Boundary Condition
PAGE
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CHAPTER 1
INTRODUCTION
1.1 Background of the Study
Burgers’ equation is a quasilinear differential equation shows the nonlinear
convection and linear dissipation follows the evolution of time of the function u(x,t)
(Singh, 2016). The equation first discovered by Betemen (1915) used in the modelling
of the motion of viscous fluid. In 1948, Burgers (1948) tried to formulate a simplest
mathematical model that can related to turbulence. From that day onwards, the
equation earned its name as Burgers’ equation. In modern days, this equation are
widely formulated in modelling, turbulence, gas fluid dynamics, traffic flows and so
on. This equation also played as a model equation for the development in the
computation in nonlinear equation.
Burgers’ equation can be solved either analytically or numerically. The
analytical or also known as exact solution often related to the Hopf-Cole
transformation which is used to reduce the quasilinear equation into a diffusion
equation. In recent years, beside Hopf-Cole transformation, various works have been
produced on the findings of Burgers’ equation analytical solution such as G’/G
Expansion Method (Wang, Li and Zhang, 2008), Tanh Expansion Method (Malfliet
and Hereman, 1996) and Method using Transformation from Sine-Gordon equation
(Fu, Liu and Liu, 2002).
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Simultaneously, the good numerical approximations on Burgers’ equation also
grew as time goes. Examples like Nyuyen and Reynen (1984) presented a space-time
finite element approach to Burgers’ equation, Kakuda and Tosaka (1990) used
generalized boundary element method, Bar-Yoseph et al. (1995) used and discussed
the space time spectral element method on Burgers’s equation solution, Zhu et al.
(2009) applied a cubic B-spline quasi interpolation to Burgers equation, Siraj et
al.(2012) researched the numerical solution of Burgers’ equations using meshless
Method of Lines and many more.
In this research, the focus is to solve Burgers’ equation numerically by using
Finite Difference Method (FDM) and Method of Line (MOL) by using Fourth Order
Runge-Kutta (RK4). The results obtained from both numerical method will be
compared with Hopf-Cole transformation analytical solutions. The simulations is
coded using MATLAB software.
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In reality, most of the physical problems existed in form of nonlinear partial
differential equations. In this research, Burgers’ equation is chosen due to the
simplicity of one dimensional but contains the nonlinear properties.
Numerical method in other hand means the approximation of a solutions.
Number of methods have been introduces in decades. Among the well-knowns, FDM
and MOL are chosen to be used as the numerical approach to solved Burgers’ equation.
Although both come from different approach where FDM solved in partial differential
equation (PDE) form meanwhile MOL transform a PDE into a system of Ordinary
differential equation (ODE) and solved using various ODE solver methods. FDM is
one of the classic method to solve PDE. However, methods of lines said to be more
accurate and computational timewise compared to regular finite difference method
(Sadika and Obiozor,2000). However, the accuracy of the solutions also depends on
the methods used to solve the ODE after transformed by using MOL. Moreover, this
method can achieved the numerical stability and convergence efficiently due to the
separation of time and space discretization. There are several issues of concern
discussed in this research.
1.2 Statements of the Problem
1. How to numerically simulate Burgers’ equation using MATLAB software?
2. How accurate are the numerical methods used to solve Burgers’ equation?
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The following objectives is achieved from this research.
1. To review and understand the applications of numerical schemes on Burgers’
equation.
2. To simulate numerical computational of Burgers’ equation in MATLAB
software.
3. To determine the accuracy of result obtained using numerical approach in
Burgers’ equation.
4. To observe different numerical approach in solving Burgers’ equation.
1.3 Objectives of the Study
1.4 Scope of the Study
This research focused on one dimensional nonlinear Burgers’ equation that is
linearized by the used of Hopf-Cole transformation and solved by using explicit FDM
and MOL using RK4. The discretization will be carry out and code by using
MATLAB software.
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In recent years, the research of Burgers’ equation had contributes various
achievement especially in fluid dynamics field. Thus, the significant of this research
are:
1. This research determined the accuracy of result obtained using numerical
approach in Burgers’ equation.
2. This research also provides extra information on the comparison between finite
difference method and method of line in Burgers’ equation.
3. This research provides other alternative to solve Burgers’ equation by using
two different approach of numerical methods, the pde solver and the ode solver.
1.5 Significance of the Study
1.6 Organization of the Research
This research is organised into five chapters. Chapter 1 presents the
introduction, background of the study, statements of the problem, objectives, scope
and the significance of study. The theoretical information and literature review related
to the background of the study are discussed in Chapter 2. Literature review on the
background of Burgers’ equation and numerical methods that involved. On the other
hand, Chapter 3 is about the mathematical formulation and algorithms. Based on the
mathematical formulation, the equation is solved by using MATLAB software.
Chapter 4 discussed about the results and discussion of this research. The last chapter
of this research which is Chapter 5 concluded the whole thesis and some
recommendations for future research.
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