CISE301_Topic8L4&5 KFUPM 1

CISE301: Numerical Methods

Topic 8 Ordinary Differential

Equations (ODEs)Lecture 28-36

KFUPM

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

CISE301_Topic8L4&5 KFUPM 2

Outline of Topic 8 Lesson 1: Introduction to ODEs Lesson 2: Taylor series methods Lesson 3: Midpoint and Heun’s method Lessons 4-5: Runge-Kutta methods Lesson 6: Solving systems of ODEs Lesson 7: Multiple step Methods Lesson 8-9: Boundary value Problems

CISE301_Topic8L4&5 KFUPM 3

Lecture 31Lesson 4: Runge-Kutta

Methods

CISE301_Topic8L4&5 KFUPM 4

Learning Objectives of Lesson 4 To understand the motivation for using

Runge-Kutta (RK) method and the basic idea used in deriving them.

To get familiar with Taylor series for functions of two variables.

To use RK method of order 2 to solve ODEs.

CISE301_Topic8L4&5 KFUPM 5

Motivation We seek accurate methods to solve ODEs

that do not require calculating high order derivatives.

The approach is to use a formula involving unknown coefficients then determine these coefficients to match as many terms of the Taylor series expansion as possible.

CISE301_Topic8L4&5 KFUPM 6

Runge-Kutta Method

1

2 1

1 1 2 2

1 2

( , )

( , )

( ) (

Second Order Rung

)

Problem:

, , ,

such that is as accurate as poss

e-Kutta (

ible

RK2

.

)

K h f t x

K h f t h x K

x t h x t w K w K

Find w w

x(t h)

CISE301_Topic8L4&5 KFUPM 7

Taylor Series in Two Variables

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

variable case.This is used to prove RK formula.

CISE301_Topic8L4&5 KFUPM 8

Taylor Series in One Variable

1( ) ( )

0

The Taylor Series expansion of ( )

( ) ( ) ( )! !

i nni n

i

f x

h hf x h f x f x

i n

where x is between x and x h

Approximation Error

CISE301_Topic8L4&5 KFUPM 9

Taylor Series in One Variable- Another Look -

hxxx

xfdx

dh

nxf

dx

dh

ihxf

f(x)

hxfdx

xfdhxf

dx

dh

nn

i

i

iii

ii

i

and between is

)(!

1)(

!

1)(

ofexpansion SeriesTaylor The

)()(

)(

Define

1

0

)(

CISE301_Topic8L4&5 KFUPM 10

Definitions

2

22

2

2

22

2

1

0

),(),(2

),(),(

),(),(),(

),(),(

),(

y

yxfk

yx

yxfkh

x

yxfhyxf

yk

xh

y

yxfk

x

yxfhyxf

yk

xh

yxfyxfy

kx

h

x

fhyxf

xh

Define

i

ii

i

CISE301_Topic8L4&5 KFUPM 11

Taylor Series Expansion

)0,0(

''

)0.0(

1

)0.0(

0

2

),(

4),(

(0,0)at evaluated sderivative Parial

)2)(1(,

yx fkfhyxfy

kx

h

yxfy

kx

h

yxxy)f(x

CISE301_Topic8L4&5 KFUPM 12

Taylor Series in Two Variables

),( and ),(between joining line on the is),(

),(!

1),(

!

1),(

, ofexpansion SeriesTaylor The

1

0

kyhxyxyx

errorionapproximat

yxfy

kx

hn

yxfy

kx

hi

kyhxf

y)f(xnn

i

i

x x+h

y

y+k

CISE301_Topic8L4&5 KFUPM 13

Runge-Kutta Method

1

2 1

1 1 2 2

1 2

( , )

( , )

( ) ( )

Problem:

RK

, , ,

such that is as accurate as possibl

2

e.

K h f t x

K h f t h x K

x t h x t w K w K

Find w w

x(t h)

CISE301_Topic8L4&5 KFUPM 14

Runge-Kutta Method

)( ),()()(

),(2

1 ),(

)),(,( ),()()(

...)('''6

)(''2

)(')()(

possible. as many terms asmatch to

,,, :Problem

3'22

'2221

2''

21

32

21

hOffhwfhwxtfhwwtxhtx

xtfx

ht

hffhfhffhxhtf

xtfhxhtfhwxtfhwtxhtx

txh

txh

thxtxhtx

wwFind

xt

xt

CISE301_Topic8L4&5 KFUPM 15

Runge-Kutta Method

)( ),()()(

...)('''6

)(''2

)(')()(

3'22

'2221

32

hOffhwfhwxtfhwwtxhtx

txh

txh

thxtxhtx

xt

1,1,5.0,5.0

solution possibleOne

5.0,5.0,1

21

2221

ww

wwww

CISE301_Topic8L4&5 KFUPM 16

Runge-Kutta Method

1

2 1

1 2

( , )

( , )

1( ) ( )

2

RK2

K h f t x

K h f t h x K

x t h x t K K

CISE301_Topic8L4&5 KFUPM 17

Runge-Kutta MethodAlternative Formula

1

2 1

1 2

( , )

( , )

( ) ( )2

RK2

F f t x

F f t h x hF

hx t h x t F F

CISE301_Topic8L4&5 KFUPM 18

Runge-Kutta MethodAlternative Formula

1

2 1

1 2

( , )

(

Alternative F

, )

( ) ( )2

orm

F f t x

F f t h x hF

hx t h x t F F

1

2 1

1 2

( , )

( , )

1( ) ( )

2

RK2

K h f t x

K h f t h x K

x t h x t K K

CISE301_Topic8L4&5 KFUPM 19

Runge-Kutta MethodAlternative Formulas

1

2 1

1 2

RK2 Formulas (select 0)

( , )

( , )

1 1( ) ( ) 1

2 2

K h f t x

K h f t h x K

x t h x t F F

2

1,

2

11,

number zero-nonany Pick

solutionanother

5.0,5.0,1

21

2221

ww

wwww

CISE301_Topic8L4&5 KFUPM 20

Runge-Kutta Method

1

2 1

3 2

4 3

1 2 3 4

( , )

1 1( , )

2 21 1

( , )2 2

( , )

Fourth Order Runge-Kutta (R

1( ) ( ) 2 2

6

K4)

K h f t x

K h f t h x K

K h f t h x K

K h f t h x K

x t h x t K K K K

CISE301_Topic8L4&5 KFUPM 21

Second order Runge-Kutta Method Example

2 3

Solve the following system to find (1.02) using RK2

( ) 1 ( ) , (1) 4, 0.01

x

x t x t t x h

CISE301_Topic8L4&5 KFUPM 22

Second order Runge-Kutta Method Example

2 3

2 31

2 32 1

1 2

Solve the following system to find (1.02) using RK2

( ) 1 ( ) , (1) 4, 0.01

STEP1:

( , ) 0.01(1 ) 0.18

( , ) 0.01(1 ( 0.18) ( .01) ) 0.1662

1 1(1 0.01) (1) 4 (0.18

2 2

x

x t x t t x h

K h f t x x t

K h f t h x K x t

x x K K

0.1662) 3.8269

CISE301_Topic8L4&5 KFUPM 23

Second order Runge-Kutta Method Example

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),(

2 STEP

21

3212

321

KKxx

txKxhtfhK

txxtfhK

CISE301_Topic8L4&5 KFUPM 24

RK2 Using

[1,2]for tSolution

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

xttxtx

CISE301_Topic8L4&5 KFUPM 25

Summary RK methods generate an accurate solution

without the need to calculate high order derivatives.

Second order RK have local truncation error of order O(h3).

Fourth order RK have local truncation error of order O(h5).

N function evaluations are needed in the Nth order RK method.

CISE301_Topic8L4&5 KFUPM 26

Lecture 32Lesson 5: Applications of Runge-Kutta Methods to Solve First Order ODEs

CISE301_Topic8L4&5 KFUPM 27

Learning Objectives of Lesson 5 Use Runge-Kutta methods of different

orders to solve first order ODEs.

CISE301_Topic8L4&5 KFUPM 28

Runge-Kutta Method

1

2 1

1 1 1 2 2

1 2

1

( , )

( , )

Problem:

, , ,

such that i

Second Order Runge Kutta (

s as accurate as possible.

RK2)

i i

i i

i i

i

K f x y

K f x h y K h

y y w K w K

Find w w

y

CISE301_Topic8L4&5 KFUPM 29

Runge-Kutta Methods

RK2

1

2 1

1 1 2

Second Order Runge-Kutta

( , )

( , )

R

2

( K2)

i i

i i

i i

K f x y

K f x h y K h

hy y K K

CISE301_Topic8L4&5 KFUPM 30

Runge-Kutta Methods

1

2 1

3 1 2

1 2 3

(

Third

, )

1 1( , )

2 2( , 2 )

1( ) ( ) 4

6

Order Runge Kutta (RK3)

i i

i i

i i

K f x y

K f x h y K h

K f x h y K h K h

y x h y x K K K

RK3

CISE301_Topic8L4&5 KFUPM 31

Runge-Kutta Methods

1

2 1

3 2

4 3

1 1 2 3 4

( , )

1 1( , )

2 21 1

( , )2

Fourth Order R

2(

unge Kutta (

, )

2 2

K4

6

R )

i i

i i

i i

i i

i i

K f x y

K f x h y K h

K f x h y K h

K f x h y K h

hy y K K K K

RK4

CISE301_Topic8L4&5 KFUPM 32

Runge-Kutta Methods

Higher order Runge-Kutta methods are available.

Higher order methods are more accurate butrequire more calculations.

Fourth order is a good choice. It offers good accuracy with a reasonable calculation effort.

CISE301_Topic8L4&5 KFUPM 33

Fifth Order Runge-Kutta Methods

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

CISE301_Topic8L4&5 KFUPM 34

Second Order Runge-Kutta Method

needed steps of# Determine

)(

),(

:Given

00

h

yxy

yxfdx

dy

CISE301_Topic8L4&5 KFUPM 35

Second Order Runge-Kutta Method

211

12

1

00

2

),(

),(

2

needed steps of# Determine

)(

),(

:Given

KKh

yy

hKyhxfK

yxfK

formulaRK

h

yxy

yxfdx

dy

ii

ii

ii

CISE301_Topic8L4&5 KFUPM 36

Second Order Runge-Kutta Method

2112

1112

111

01

2101

1002

001

2

),(

),(

:2 Step

2

),(

),(

:1 Step

KKh

yy

hKyhxfK

yxfK

hxx

KKh

yy

hKyhxfK

yxfK

211

12

1

00

2

),(

),(

2

needed steps of# Determine

)(

),(

:Given

KKh

yy

hKyhxfK

yxfK

formulaRK

h

yxy

yxfdx

dy

ii

ii

ii

CISE301_Topic8L4&5 KFUPM 37

Example 1Second Order Runge-Kutta Method

)02.1(),01.1(findto

4)1(,1

equation aldifferenti thesolve to

method Kutta Rungeorder second theUse

32

yy

yxydx

dy

CISE301_Topic8L4&5 KFUPM 38

Example 1Second Order Runge-Kutta Method

)02.1(),01.1(2

4)1(,1

:Problem

32

yyfindtoRKUse

yxydx

dy

4,1

1),(

0.01h

00

32

yx

xyyxf

CISE301_Topic8L4&5 KFUPM 39

Example 1Second Order Runge-Kutta Method

8269.3)62.1618(2

01.04

2

62.16))01.()18.0(1(),(

0.18)1(),(

:1 Step

2101

30

201002

30

20001

KKh

yy

xyhKyhxfK

xyyxfK

)02.1(),01.1(2

4)1(,1

:Problem

32

yyfindtoRKUse

yxydx

dy

4,1

1),(

0.01h

00

32

yx

xyyxf

CISE301_Topic8L4&5 KFUPM 40

Example 1Second Order Runge-Kutta Method

)02.1(),01.1(2

4)1(,1

:Problem

32

yyfindtoRKUse

yxydx

dy

8269.3,01.1

1),(

0.01h

11

32

yx

xyyxf

CISE301_Topic8L4&5 KFUPM 41

Example 1Second Order Runge-Kutta Method

6662.3)46.1568.16(2

01.08269.3

2

46.15))01.()1668.0(1(),(

68.16)1(),(

:2 Step

2112

31

211112

31

21111

KKh

yy

xyhKyhxfK

xyyxfK

)02.1(),01.1(2

4)1(,1

:Problem

32

yyfindtoRKUse

yxydx

dy

8269.3,01.1

1),(

0.01h

11

32

yx

xyyxf

CISE301_Topic8L4&5 KFUPM 42

Example 1Summary of the solution

6662.30212

8269.30111

0000.40010

.

.

.

yxi ii

)02.1(),01.1(2

4)1(,1

:Problem

32

yyfindtoRKUse

yxydx

dy

Summary of the solution

CISE301_Topic8L4&5 KFUPM 43

Solution after 100 steps

CISE301_Topic8L4&5 KFUPM 44

Example 24th-Order Runge-Kutta Method

)4.0()2.0(4

2.0

5.0)0(

1 2

yandycomputetoRKUse

h

y

xydx

dy

See RK4 Formula

CISE301_Topic8L4&5 KFUPM 45

Example 2Fourth Order Runge-Kutta Method

)4.0(),2.0(4

5.0)0(,1

:Problem

2

yyfindtoRKUse

yxydx

dy

CISE301_Topic8L4&5 KFUPM 46

Example 2Fourth Order Runge-Kutta Method

5.0,0

1),(

0.2h

00

2

yx

xyyxf

8293.0226

7908.12.016545.01),(

654.11.0164.01)2

1,

2

1(

64.11.015.01)2

1,

2

1(

5.11),(

:1Step

432101

2003004

2002003

2001002

200001

KKKKh

yy

xyhKyhxfK

xyhKyhxfK

xyhKyhxfK

xyyxfK

)4.0(),2.0(4

5.0)0(,1

:Problem

2

yyfindtoRKUse

yxydx

dy

See RK4 Formula

CISE301_Topic8L4&5 KFUPM 47

Runge-Kutta Methods

43211

34

23

12

1

226

),(

)2

1,

2

1(

)2

1,

2

1(

),(

(RK4) Kutta RungeOrder Fourth

KKKKh

yy

hKyhxfK

hKyhxfK

hKyhxfK

yxfK

ii

ii

ii

ii

ii

RK4

CISE301_Topic8L4&5 KFUPM 48

Example 2 Fourth Order Runge-Kutta Method

8293.0,2.0

1),(

0.2h

00

2

yx

xyyxf

2141.1226

2.0

0555.2),(

9311.1)2

1,

2

1(

9182.1)2

1,

2

1(

1.7893 ),(

:2Step

432112

3114

2113

1112

111

KKKKyy

hKyhxfK

hKyhxfK

hKyhxfK

yxfK

)4.0(),2.0(4

5.0)0(,1

:Problem

2

yyfindtoRKUse

yxydx

dy

CISE301_Topic8L4&5 KFUPM 49

Example 2Summary of the solution

2141.14.02

8293.02.01

5.00.00ii yxi

)4.0(),2.0(4

5.0)0(,1

:Problem

2

yyfindtoRKUse

yxydx

dy

Summary of the solution

CISE301_Topic8L4&5 KFUPM 50

Remaining Lessons in Topic 8Lesson 6:Solving Systems of high order ODE

Lesson 7:Multi-step methods

Lessons 8-9:Methods to solve Boundary Value Problems