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

i i i

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

iv

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

x

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

43

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

xi

xii

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

23

24

44

45

46

48

49

50

xiii

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

6 6

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).

2

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.

3

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?

4

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.

5

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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