CISE301-Topic8L4&5.ppt

7/29/2019 CISE301-Topic8L4&5.ppt

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CISE301_Topic8L4&5 1

SE301: Numerical Methods

Topic 8Ordinary Differential Equations (ODEs)

Lecture 28-36

KFUPM(Term 101)

Section 04

Read 25.1-25.4, 26-2, 27-1


2/36

CISE301_Topic8L4&52

Outline of Topic 8

Lesson 1: Introduction to ODEs

Lesson 2: Taylor series methods

Lesson 3: Midpoint and Heuns method

Lessons 4-5: Runge-Kutta methods

Lesson 6: Solving systems of ODEs

Lesson 7: Multiple step Methods

Lesson 8-9: Boundary value Problems


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CISE301_Topic8L4&53

Lecture 31Lesson 4: Runge-Kutta Methods


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CISE301_Topic8L4&54

Learning Objectives of Lesson 4

To understand the motivation for usingRunge Kutta method and the basic ideaused in deriving them.

To Familiarize with Taylor series forfunctions of two variables.

Use Runge Kutta of order 2 to solveODEs.


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CISE301_Topic8L4&55

Motivation

We seek accurate methods to solve ODEsthat do not require calculating high orderderivatives.

The approach is to use a formulainvolving unknown coefficients thendetermine these coefficients to match asmany terms of the Taylor series

expansion.


6/36

CISE301_Topic8L4&56

Second Order Runge-Kutta Method

possible.asaccurateasisthatsuch,,,

:Problem

),(

),(

1

21

22111

12

1

i

ii

ii

ii

ywwFind

KwKwyy

KyhxfhK

yxfhK


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CISE301_Topic8L4&57

Taylor Series in TwoVariables

The Taylor Series discussed in Chapter 4is extended to the 2-independent

variable case.

This is used to prove RK formula.


8/36

CISE301_Topic8L4&58

Taylor Series in One Variable

hxandxbetweenisxwhere

xfn

h

xfi

h

hxf

f(x)n

nn

i

n

i

i

)()!1()(!)(

ofexpansionSeriesTaylororderThe

)1(1

)(

0

th

Approximation Error


9/36

CISE301_Topic8L4&59

Derivation of 2nd Order

Runge-Kutta Methods1 of 5

)(),('2

),(

:aswritteniswhich

)(2

),(:ODEsolvetoUsed

ExpansionSeriesTaylorOrderSecond

32

1

3

2

22

1

hOyxfh

yxfhyy

hOdx

ydh

dx

dyhyy

yxfdx

dy

iiiiii

ii


10/36



CISE301_Topic8L4&510

)(2),(),(

:

),(

),(),(

),('

ationdifferentirule-chainbyobtainedis),('where

32

1 hOh

yxfy

f

x

fhyxfyy

ngSubstituti

yxfy

f

x

f

dx

dy

y

yxf

x

yxf

yxf

yxf

iiiiii


11/36


Taylor Series in Two Variables

),(and),(betweenjoininglinetheonis),(

),()!1(

1),(

!

1

...

2

!2

1

),(),(

1

0

2

2

22

2

22

kyhxyxyx

errorionapproximat

yxfy

kx

hn

yxfy

kx

hi

yx

fhk

y

fk

x

fh

y

fk

x

fhyxfkyhxf

nn

i

i


12/36




),(),(

:ngSubstituti

),(

),(

thatsuch,,,:Problem

1211

22111

12

1

21

Kyhxfhwyxfhwyy

KwKwyyKyhxfhK

yxfhK

wwFind

iiiiii

ii

ii

ii


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...),(),()(

...),()(

...),(),(

:

...),(),(

22

22211

12211

1211

11

iiiiii

iiii

iiiiii

iiii

yxfy

fhw

x

fhwyxfhwwyy

y

fK

x

fhhwyxfhwwyy

y

fK

x

fhyxfhwyxfhwyy

ngSubstituti

y

fK

x

fhyxfKyhxf


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2

1,1:solutionpossibleOne

solutionsinfiniteunknowns4withequations32

1and,

2

1,1

:equationsthreefollowingtheobtainweterms,Matching

)(2

),(),(

...),(),()(

:forexpansionstwoderivedWe

21

2221

3

2

1

22

22211

1

ww

wwww

hOhyxfyf

xfhyxfyy

yxfy

fhw

x

fhwyxfhwwyy

y

iiiiii

iiiiii

i


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2nd Order Runge-Kutta Methods


2

1and,

2

1,1

:thatsuch,,,Choose

),(

),(

2221

21

22111

12

1

wwww

ww

KwKwyy

KyhxfhK

yxfhK

ii

ii

ii


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Alternative Form

22111

12

1

),(

),(

FormeAlternativ

kwkwhyy

khyhxfk

yxfk

ii

ii

ii

22111

12

1

),(

),(

KuttaeOrder RungSecond

KwKwyy

KyhxfhK

yxfhK

ii

ii

ii


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Choosing , , w1and w2

CorrectorSingleawith'isThis

),(),(22

1

),(

),(

:becomesmethodKutta-eOrder RungSecond

2

1,1then,1choosingexample,For

0

11211

12

1

21

s MethodHeun

yxfyxf

h

yKKyy

KyhxfhK

yxfhK

ww

iiiiiii

ii

ii


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Choosing , , w1and w2

MethodMidpointtheisThis

)2

,2

(

)2

,2

(

),(

:becomesmethodKutta-eOrder RungSecond

1,0,2

1then

2

1Choosing

121

12

1

21

KyhxfhyKyy

Ky

hxfhK

yxfhK

ww

iiiii

ii

ii


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2nd Order Runge-Kutta MethodsAlternative Formulas

211

1i2

1

2

1

2

11

),(

),(

)0(selectmulasKutta ForeOrder RungSecond

KKyy

KyhxfhK

yxfhK

ii

i

ii

2

11,

2

1,:numbernonzeroanyPick

1,2

1,

2

1

12

2122

ww

wwww


20/36


Second order Runge-Kutta MethodExample

8269.32/)1662.018.0(4

2/)1()01.01(1662.0))01.()18.0(1(01.0

),(

18.0)1(01.0)4,1(

:1STEP

1,01.0,4)1(,1)(

RK2using(1.02)findtosystemfollowingtheSolve

21

3020

1002

30

20001

32

KKxxtx

KxhtfhK

txxtfhK

hxtxtx

x


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Second order Runge-Kutta MethodExample

6662.3)1546.01668.0(2

18269.3

2

1)01.1()01.001.1(

1546.0))01.()1668.0(1(01.0

),(

1668.0)1(01.0)8269.3,01.1(

2STEP

21

31

21

1112

31

21111

KKxx

tx

KxhtfhK

txxtfhK


22/36


1RK2,Using

[1,2]for tSolution

,4)1(,)(1)( 32

xttxtx


23/36


Lecture 32Lesson 5: Applications of Runge-Kutta

Methods to Solve First Order ODEs

Using Runge-Kutta methods of different

orders to solve first order ODEs


24/36


2nd Order Runge-Kutta RK2

)(iserrorglobaland)(iserrorLocal

2

),(

),(

correctorsingleawithmethodsHeun'toEquivalent

RK2asKnow1,ofvalueTypical

23

211

12

1

hOhO

kkh

yy

hkyhxfk

yxfk

ii

ii

ii


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Higher-Order Runge-Kutta

Higher order Runge-Kutta methods are available.

Derived similar to second-order Runge-Kutta.

Higher order methods are more accurate butrequire more calculations.


26/36


3rd Order Runge-Kutta RK3

)(iserrorGlobaland)(iserrorLocal

46

)2,(

)2

1,2(

),(

RK3asKnow

34

3211

213

12

1

hOhO

kkkh

yy

hkhkyhxfk

hkyh

xfk

yxfk

ii

ii

ii

ii


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4th Order Runge-Kutta RK4


226

),(

)2

1,

2(

)2

1,

2(

),(

45

43211

3i4

2i3

12

1

hOhO

kkkkh

yy

hkyhxfk

hkyh

xfk

hkyh

xfk

yxfk

ii

i

i

ii

ii


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Higher-Order Runge-Kutta

654311

543216

415

324

213

12

1

7321232790

)7

8

7

12

7

12

7

2

7

3,(

)16

9

16

3,

4

3(

)2

1,

2

1(

)

8

1

8

1,

4

1(

)4

1,

4

1(

),(

kkkkkh

yy

hkhkhkhkhkyhxfk

hkhkyhxfk

hkhkyhxfk

hkhkyhxfk

hkyhxfk

yxfk

ii

ii

ii

ii

ii

ii

ii


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Example4th-Order Runge-Kutta Method

)4.0()2.0(4

2.0

5.0)0(

1 2

yandycomputetoRKUse

h

y

xydx

dy

RK4


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Example: RK4

)4.0(),2.0(4

5.0)0(,1

:Problem2

yyfindtoRKUse

yxydx

dy


31/36


4th Order Runge-Kutta RK4


226

),(

)2

1,

2(

)2

1,

2(

),(

45

43211

3i4

2i3

12

1

hOhO

kkkkhyy

hkyhxfk

hkyh

xfk

hkyh

xfk

yxfk

ii

i

i

ii

ii


32/36


Example: RK4

5.0,0

1),(

0.2

00

2

yx

xyyxf

h

8293.0226

7908.12.016545.01),(

654.11.0164.01)

2

1,

2

1(

64.11.015.01)2

1,

2

1(

5.1)1(),(

432101

2003004

2002003

2001002

200001

kkkkh

yy

xyhkyhxfk

xyhkyhxfk

xyhkyhxfk

xyyxfk

)4.0(),2.0(4

5.0)0(,1

:Problem2

yyfindtoRKUse

yxydx

dy

See RK4 Formula

Step1


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Example: RK4

8293.0,2.0

1),(

0.2

11

2

yx

xyyxf

h

2141.1226

2.0

0555.2),(

9311.1)

2

1,

2

1(

9182.1)2

1,

2

1(

1.7893),(

432112

3114

2113

1112

111

kkkkyy

hkyhxfk

hkyhxfk

hkyhxfk

yxfk

)4.0(),2.0(4

5.0)0(,1

:Problem2

yyfindtoRKUse

yxydx

dy

Step2


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Example: RK4

)4.0(),2.0(4

5.0)0(,1

:Problem2

yyfindtoRKUse

yxydx

dy

xi yi

0.0 0.5

0.2 0.8293

0.4 1.2141

Summary of the solution


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Summary

Runge Kutta methods generate anaccurate solution without the need tocalculate high order derivatives.

Second order RK have local truncationerror of order O(h3) and global truncationerror of order O(h2).

Higher order RK have better local and

global truncation errors.

N function evaluations are needed in theNth order RK method.


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Remaining Lessons in Topic 8

Lesson 6:Solving Systems of high order ODE

Lesson 7:Multi-step methods

Lessons 8-9:

Methods to solve Boundary Value Problems

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