A STUDY OF HIGHER ORDER IMPLICIT RUNGE-KUTTA METHODS IN SOLVING STIFF NONLINEAR PROBLEMS - By Amira...

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A STUDY OF HIGHER ORDER IMPLICITRUNGE-KUTTA METHODS IN SOLVING STIFF

NONLINEAR PROBLEMS

Amira binti Ismail

November 26, 2013

Amira binti Ismail A STUDY OF HIGHER ORDER IMPLICIT RUNGE-KUTTA M
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Chapter 1:INTRODUCTION

1 Problem Statements.

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2 Importance of Research.

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3 Research Objectives.

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3 Research Objectives.4 Scope of Study

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3 Research Objectives.4 Scope of Study

5 Thesis Outline

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Problem Statements

1 Solving an explicit method for stiff problems is not advisabledue to the stability of the explicit methods which are notA-stable (Butcher,1975)

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Problem Statements


2 Implicit methods especially higher order methods are the

solution for solving stiff problems since implicit methods havegood stability region (Dalquist,1963 )


S
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Problem Statements


2 Implicit methods especially higher order methods are the

solution for solving stiff problems since implicit methods havegood stability region (Dalquist,1963 )

3 It is shown that the higher the order if the methods the moreaccuracy the methods will be although the implementation for

implicit methods are difficult compared to explicit methods(S. Gonzalez-Pinto, J.I Montijano and L.Randez, 2001)


I f R h
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Importance of Research

1 Most real life problems ( chemical reaction, biomedical, solarsystems etc) using numerical methods are stiff and nonlinear.


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2 Numerical analysts are still finding the best methods to usewhen solving this real life problems.


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3

Implementation nonlinear problems with implicit methods aremore advisable since they are more stable and efficientespecially the well known Radau II A methods (Hairer andWanner,1993)


I t f R h
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3

Implementation nonlinear problems with implicit methods aremore advisable since they are more stable and efficientespecially the well known Radau II A methods (Hairer andWanner,1993)

4 Implicit methods are shown to be efficient in solving partial

differential equations problems using semi-implicit approach(Scheffel.J (2006))


R s h Obj ti s


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Research Objectives

1 To extend the study of symmetric Runge-Kutta methodespecially the higher order Gauss methods in solving stiffnon-linear problems.


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Research Objectives


2 To study the efficiency of the symmetric methods numerically

for nonlinear problems.


Research Objectives


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Research Objectives



for nonlinear problems.3 To investigate the order behavior of symmetric Runge-Kutta

methods for stiff problem theoretically.


Research Objectives


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Research Objectives



for nonlinear problems.3 To investigate the order behavior of symmetric Runge-Kutta

methods for stiff problem theoretically.

4 To apply extrapolation technique and compare the efficiency

with other well-known method such as the Radaus methodnumerically.


Scope of Study
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Scope of Study

This research is focusses on the higher order implicit Runge-Kuttamethods such as the 2-stage and 3-stage Gauss methods. These

methods are tested numerically particularly for stiff nonlinearproblems.


Thesis Outline
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Thesis Outline

1 Chapter 1 is about introduction of the the research including

some knowledge about ordinary differential equations,numerical methods, objective and scope of study.


Thesis Outline
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Thesis Outline



2 Chapter 2 is the literature review. In this chapter, we willdiscuss the theoritical concept of Runge-Kutta methods and

some history. This part is important to know the theory andorder condition of Runge-Kutta methods.


Thesis Outline
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Thesis Outline



2 Chapter 2 is the literature review. In this chapter, we willdiscuss the theoritical concept of Runge-Kutta methods andsome history. This part is important to know the theory andorder condition of Runge-Kutta methods.

3 In Chapter 3, we will discuss the construction of higher orderimplicit Gauss Runge-Kutta methods.


Thesis Outline


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Thesis Outline





4 Chapter 4 is about the analysis of this research. This chapter

will discuss about solving stiff problem using Matlab software.This analysis based on numerical and theoretical approach.


Thesis Outline
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Thesis Outline





4 Chapter 4 is about the analysis of this research. This chapter

will discuss about solving stiff problem using Matlab software.This analysis based on numerical and theoretical approach.

5 Discussion about this research is given in Chapter 5.


Chapter 2:Literature Review
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p

1 Ordinary Differential Equations (ODEs).




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p


2 Runge-Kutta Methods.


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p



3 Explicit Methods.


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p



3 Explicit Methods.4 Implicit Methods.


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3 Explicit Methods.4 Implicit Methods.

5 Explicit vs Implicit.


Ordinary Differential Equations (ODEs)
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( )

1 All derivatives of unknown solution in the differentialequations are respect to a single independent variable.


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2 Consider an ODE in this form

y =f(x, y), y(x0) =y0, f : [x0, xn] RN RN. (1)


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3 If the value ofx0 and y0 is given, then equation (1) is knownas initial values.


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3 If the value ofx0 and y0 is given, then equation (1) is knownas initial values.

4

It can be solve analytically for linear equations and numericallyfor nonlinear equations especially higher order equations.


Runge-Kutta methods
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1 One step methods due to Runge (1895), Heun (1990) andKutta (1991).


Runge-Kutta methods
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2 Generalization Euler method that computes f only once in

each step to evaluate f two or more times with differentargument.


Runge-Kutta methods


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2 Generalization Euler method that computes f only once in

each step to evaluate f two or more times with differentargument.

3 Give greater accuracy and better stability compared Eulermethod.


Runge-Kutta Methods
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Define as

Yi=yn1+hs

j=1

aijf (xn1+cjh, Yj) , (2a)

yn =yn1+hs

j=1

bjf(xn1+cjh, Yj). (2b)

Yi represent the internal stage value,yn represent the update ofyat the n

th step.

y0 yn

x0 xn|| |

Y

< >h


Runge-Kutta methods
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We can display (2a) and (2b) by a Butcher tableau of the form

c A

bT,

cis the vector of abscissae, Matrix A are the coefficients, Vector brepresents the quadrature of weight indication.Where

ci=s

j=1

aij, i= 1, 2, . . . , s. (3)


Runge-Kutta methods
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1

Divided into two main types according to the style of thematrix A


Runge-Kutta methods
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1


Explicit methods - Where the matrix A is strictly lowertriangular.


Runge-Kutta methods
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1



Implicit methods:-


Runge-Kutta methods
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1



Implicit methods:-

Fully implicit - Matrix A is not lower triangular,


Runge-Kutta methods
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1



Implicit methods:-

Fully implicit - Matrix A is not lower triangular, Semi-implicit - Matrix A is lower triangular with at least one

non-zero diagonal element,


Runge-Kutta methods
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1



Implicit methods:-


non-zero diagonal element, Diagonal implicit - Matrix A is a lower triangular with all

elements are equal and non-zero,


Runge-Kutta methods
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1



Implicit methods:-


non-zero diagonal element, Diagonal implicit - Matrix A is a lower triangular with all

elements are equal and non-zero,

Singly implicit - Matrix Ais a non-singular matrix with singleeigenvalue.


Explicit Methods
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1 Example of explicit methods


Explicit Methods


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1 Example of explicit methods Euler method

0 01

,


Explicit Methods
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0 01

,

Midpoint method

0 0 012

1

2 0

0 1

,


Explicit Methods
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0 01

,

Midpoint method

0 0 012

1

2 0

0 1

,

Trapezoidal rule

0 0 0

1 1 01

2

1

2

.


Implicit Methods
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1 3 Family of implicit methods


Implicit Methods
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Radau methods - Implicit Euler


Implicit Methods
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Gauss methods - Implicit midpoint


Implicit Methods
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Lobatto IIIA - Implicit Trapezoidal


Implicit Methods


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Lobatto IIIA - Implicit Trapezoidal

2 Example of implicit methods

Table: Family of Implicit Method

1 1

1

1

2

1

2

1

1 12

1

21

2

1

2

Implicit Euler Implicit Midpoint Rule Implicit Trapezoidal Rule


Explicit vs Implicit
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Table: Explicit vs Implicit

Subject Explicit Implicit

Implementation Very easy Expensive and difficult

Stiff problem Cannot solve Can solve

Stability Poor Better Stability


Explicit vs Implicit:Implementation


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Example: Lets y0 = 1

Explicit Euler:y1 =y0+hf(y0)

Implicit Midpoint rule:Y =y0+

h2

f(x0+ 1

2h, Y)

yn =y0+hf(Y)

*Hard to solve nonlinear equations for implicit midpoint rule.


Explicit vs Implicit:Stiffness
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By applying Dahlquist test problem to any methods,

y =qY, y(0) = 1, y(x) =y0eqx

.

where q is stiff constant.A methods are stiff if there are stiff constant with large negative

value.Example of problem with large negative value is Holsapple, Iyerand Domans problem :

y = 1000y+sin(x)

By applying implicit Runge-Kutta methods, the problem can besolved. Explicit methods also can solve the problem, but we needto use small h which will have lot of iterations.


Example of implementation of Holsapple, Iyer andDomans problem using explicit method with h=0.01
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5 4.5 4 3.5 3 2.5 214

12

10

8

6

4

2

0

Log of stepsize

Log

ofError

Holsapple, Iyer & Doman problem for explicit method


Example of implementation of Holsapple, Iyer andDomans problem using Implicit method with h=0.01
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3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 211

10.5

10

9.5

9

8.5

8

7.5

Log of stepsize

Log

ofError

Implicit RK method for Hosaple


Explicit vs Implicit:Stability

Applying Dahlquist test equation on an s-stage Runge-Kutta
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pp y g q q g gmethods yields,

Y =ey0+hAF(Y),=ey0+hqAY,

= (I zA)1ey0,

yn =y0+hbTF(Y),

=y0+hbT(qY),

=y0+zbT(I zA)1ey0,

= (1 zbT(I zA)1)ey0,

yn =R(z)y0,

where e is a vector of unit 1. The stability function is thereforedefined by

R(z) = 1 zbT(I zA)1e.


Explicit vs Implicit:Stability
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For explicit Euler methods, stability function is

R(z) = 1 zbT(I zA)1e= 1 +z

It is show polynomial function and does not satisfy A-stablecondition where R(z) 1.

For implicit methods, stability function is

R(z) = 1 zbT(I zA)1e= 1

1 z

Implicit Euler method show rational function and will satisfyA-stable condition.

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Extrapolation


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1 Consider numerical integration of an initial value problem overthe time interval from t=t0 to t=tn. Ifh is the constantstep size used, then the number of steps taken is given byn= tn

h.

2

Lety

h(tn) denote the numerical approximation of a givenmethod at time tn using stepsize h and yh

2

(tn) the numerical

solution at time tn using stepsize h

2.


Extrapolation
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1 Consider numerical integration of an initial value problem overthe time interval from t=t0 to t=tn. Ifh is the constantstep size used, then the number of steps taken is given byn= tn

h.

2

Lety

h(tn) denote the numerical approximation of a givenmethod at time tn using stepsize h and yh

2

(tn) the numerical

solution at time tn using stepsize h

2.

3 Extrapolation means apply the linear combination ofyh(tn)and yh

2

(tn) to eliminate error term.


Extrapolation
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Extrapolation formula is given as

2pyh2

(tn) yh(tn)

2p 1 (4)

as extrapolation formula for first level.By applying extrapolation, 2nd methods resulted order 3, if themethods is unsymmetrical methods and order 4 if it is symmetry.


Chapter 3:Methodology
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Chapter 4: Numerical Analysis
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1 Some studies about explicit RK methods: Accuracy andefficiency graph.


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2 Implementation and extrapolation of implicit method for

Prothero Robinsons problem:y = (q(y sin(x)) +cos(x)), q= 10.


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2 Implementation and extrapolation of implicit method for

Prothero Robinsons problem:y = (q(y sin(x)) +cos(x)), q= 10.

3 Efficiency graph of Prothero Robinsons problem with andwithot extrapolation.


Some studies about explicit RK methods: Accuracy graph

The Accuracy Diagram


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2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.211

10

9

8

7

6

5

4

3

2

Log of stepsize

Logo

fError

y g

Midpoint

Euler

Trapezoidal

3/8 RK

1/6 2/6 1/6 RK

Classical

1/6 2/6 1/6 RK2


Some studies about explicit RK methods: Efficiency graph

The Efficiency Diagram
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1 0.5 0 0.5 1 1.5 211

10

9

8

7

6

5

4

3

2

Log of Cputime

Logo

fError

MidpointEuler

Trapezoidal

3/8 RK

1/6 2/6 1/6 RK

Classical

1/6 2/6 1/6 RK2


Implementation and extrapolation of implicit method forProthero Robinsons problem
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3.5 3 2.5 2 1.5 1 0.5 015

10

5

0Prothero Robinson Problem with and without Extrapolation

Log of stepsize

Log

ofError

IMR without extrapolation

IMR with extrapolation


Efficiency graph of Prothero Robinsons problem with andwithot extrapolation
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0.5 0 0.5 1 1.5 2 2.5 315

10

5

0

Log of Cputime

Log

ofError

The Efficiency Diagram

With extrapolation

without extrapolation


Refference1 J. C. Butcher, A Stability Property of Implicit Runge-Kutta

th d BIT 15 358 361 1769
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methods, BIT., 15, 358-361, 1769.



methods BIT 15 358 361 1769
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methods, BIT., 15, 358-361, 1769.2 J.C. Butcher and G.Wanner, (1996). Runge-Kutta methods:

some historical notes. Applied Numerical Mathematics 22(113-151)

3 G. Dahlquist, A special stability problem for linear multistepmethods, BIT.,3,27-43, 1963.



methods BIT 15 358 361 1769
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4 E. Hairee, G. Wanner, Solving ordinary differential equations,II. Stiff and differential-algebraic problems, Springer Series inComputational Mathematics, 14. Springer-Verlag, Berlin,1993.

5 S.Gonzalez-Pinto, J.I. Montijano and L.Randez,

Implementation of Higher Order Implicit Runge-KuttaMethods, Computer and Maths. with applications, 41,1009-1024,2001.



methods BIT 15 358 361 1769
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4 E. Hairee, G. Wanner, Solving ordinary differential equations,II. Stiff and differential-algebraic problems, Springer Series inComputational Mathematics, 14. Springer-Verlag, Berlin,1993.

5 S.Gonzalez-Pinto, J.I. Montijano and L.Randez,

Implementation of Higher Order Implicit Runge-KuttaMethods, Computer and Maths. with applications, 41,1009-1024,2001.

6 J. Scheffel, Solution of Systems of Nonlinear Equations- ASemi-implicit Approach, KTH ElectricalEngineering,Royal

Institute of Technolo Sweden 2006Amira binti Ismail A STUDY OF HIGHER ORDER IMPLICIT RUNGE-KUTTA M
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A STUDY OF HIGHER ORDER IMPLICIT RUNGE-KUTTA METHODS IN SOLVING STIFF NONLINEAR PROBLEMS - By Amira...

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