ee3561_unit 8al-dhaifallah14351 ee 3561 : computational methods unit 8 solution of ordinary...

EE3561_Unit 8 Al-Dhaifallah1435 1

EE 3561 : Computational

MethodsUnit 8

Solution of Ordinary Differential Equations

Lesson 3: Midpoint and Heun’s Predictor corrector Methods


Lessons in Topic 8 Lesson 1: Introduction to ODE Lesson 2: Taylor series methods Lesson 3: Midpoint and Heun’s method Lessons 4-5: Runge-Kutta methods Lesson 6: Solving systems of ODE


Learning Objectives of Lesson 3 To be able to solve first order

differential equation using Midpoint Method

To be able to solve first order differential equation using Heun’s Predictor Corrector method


Outlines of Lesson 3Lesson 3: Midpoint and Heun’s Predictor-corrector methods

• Review Euler Method• Heun’s Method• Midpoint method


Euler MethodEuler Method

) Error Truncation Global

) Error Truncation Local

,...2,1

),()(

)(),()(

MethodEuler Problem

2

100

00

O(h

O(h

ifor

yxfhyyyxy

xyyyxfxy

iiii


We have seen Taylor series method• Euler method is simple but not accurate• Higher order Taylor series methods are accurate but require calculating higher order derivatives analytically

IntroductionIntroduction

00 )(),,()(

ODEorder first a is solved be toProblem

yxyyxfxy

hyy ii 1


The methods proposed in this lesson have the general form

For the case of Euler Different forms of will be used for

the midpoint and Heun’s methods

IntroductionIntroduction

hyy ii 1

),( ii yxf

00 )(),,()(

ODEorder first a is solved be toProblem

yxyyxfxy


Midpoint MethodMidpoint Method

) ErrorTruncation Global

) ErrorTruncation Local

),(

),(2

)(

)(),()(

ethodMidpoint MProblem

2

3

2

1

2

11

2

100

00

O(h

O(h

yxfhyy

yxfh

yyyxy

xyyyxfxy

iiii

iiii


Motivation The midpoint can be summarized as

Euler method is used to estimate the solution at the midpoint.

The value of the rate function f(x,y) at the mid point is calculated

This value is used to estimate yi+1.

Local Truncation error of order O(h3) Comparable to Second order Taylor series

method



1

2

10

ii

xxx

),( ii yx

),(,),(2 2

1

2

11

2

1

ii

iiiiii

yxfhyyyxfh

yy



1

2

10

ii

xxx

),( ii yxfslope

),( ii yx

),(,),(2 2

1

2

11

2

1

ii

iiiiii

yxfhyyyxfh

yy



1

2

10

ii

xxx

),( ii yxfslope

),( ii yx

),(,),(2 2

1

2

11

2

1

ii

iiiiii

yxfhyyyxfh

yy

),(2

1

2

1 iiyx



1

2

10

ii

xxx

),(2

1

2

1

iiyxfslope

),( ii yx

),(,),(2 2

1

2

11

2

1

ii

iiiiii

yxfhyyyxfh

yy

),(2

1

2

1 iiyx


Example 1

y(0.2) and y(0.1) Determine .1.0 Use

1)0(

1)(

ODE thesolve toMethodMidpoint theUse2

h

y

yxxy


Example 1

4446.1)3438.2(1.02103.1),(

3213.1)2103.101.01(05.2103.1),(2

:2Step

2103.1)1.10025.01(1.01),(

1.1)101(05.01),(2

:1Step

1.0,1)(,1),( :Problem

2

11

2

11

12

111

2

11

2

10

2

10

01

000

2

10

002

yxfhyy

yxfh

yy

yxfhyy

yxfh

yy

hxyyxyyxf


Summary The midpoint can be summarized as

Euler method is used to estimate the solution at the midpoint.

The value of the rate function f(x,y) at the mid point is calculated

This value is used to estimate yi+1.

Local Truncation error of order O(h3) Comparable to Second order Taylor series

method


Heun’s Predictor Corrector


Heun’s Predictor Corrector Heun’s Predictor Corrector MethodMethod

) ErrorTruncation Global

) ErrorTruncation Local

),(),(2

:Corrector

),(:Predictor)(

)(),()(

Method'HProblem

2

3

111

1

0100

00

O(h

O(h

yxfyxfh

yy

yxfhyyyxy

xyyyxfxy

seun

kiiiii

ki

iiii


Heun’s Predictor Corrector(Prediction)

1ii xx

),( ii yx

),(nPreidectio 01 iiii yxfhyy

),( 011 ii yx



1ii xx

),( ii yx


),( 011 ii yx

),( 011 ii yxfslope



1ii xx

),( ii yx


),( 011 ii yx

2

),(),( 011

iiii yxfyxfslope

),( 111 ii yx



10 ixx

),( 11 ii yxfslope

),( ii yx

),(),(2

,),( 011

111 iiiiiiiiii yxfyxf

hyyyxfhyy

),( 011 ii yx

),( 111 ii yx


Example 2

y(0.2) and y(0.1) Determine

only correction One .1.0 Use

1)0(

1)(

ODE thesolve toMethod sHeun' theUse2

h

y

yxxy


Example 2

4452.1),(),(2

:Corrector

4326.1),(:Predictor

:2Step

2105.1),(),(2

:Corrector

2.1)2(1.01),(:Predictor

:1Step

1.0,1)(,1),( :Problem

022111

12

11102

011000

11

00001

002

yxfyxfh

yy

yxfhyy

yxfyxfh

yy

yxfhyy

hxyyxyyxf


Summary Euler, Midpoint and Heun’s methods are

similar in the following sense:

Different methods use different estimates of the slope

Both Midpoint and Heun’s methods are comparable in accuracy to second order Taylor series method.

slopehyy ii 1


Comparison

),(

)()(),(2

Midpoint

),(),(2

:Corrector

)()(),(:Predictor

Method sHeun'

)()(),( MethodEuler

2

1

2

11

23

2

1

111

1

2301

21

iiii

iiii

kiiiii

ki

iiii

iiii

yxfhyy

hOhOyxfh

yy

yxfyxfh

yy

hOhOyxfhyy

hOhOyxfhyy

Method Local truncation error

Global truncation error


More in this Unit Lessons 4-5: Runge-Kutta Methods Lesson 6: Systems of High order ODE Lesson 7: Multi-step methods Lessons 8-9: Boundary Value Problems


EE 3561 : Computational Methods

Topic 8Solution of Ordinary

Differential Equations

Lesson 4: Runge-Kutta Methods


Lessons in Topic 8 Lesson 1: Introduction to ODE Lesson 2: Taylor series methods Lesson 3: Midpoint and Heun’s method Lessons 4-5: Runge-Kutta methods Lesson 6: Solving systems of ODE


Learning Objectives of Lesson 4 To understand the motivation for using

Runge Kutta method and basic idea used in deriving them.

To Familiarize with Taylor series for functions of two variables

Use Runge Kutta of order 2 to solve ODE


Motivation We seek accurate methods to solve ODE

that does not require calculating high order derivatives.

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


Runge-Kutta Method

possible. as accurate as is such that

,,,

:Problem

)()(

),(

),(

Kutta RungeOrder Second

21

2211

12

1

h)x(t

wwFind

KwKwtxhtx

KxhtfhK

xtfhK


Lecture 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


Taylor Series in One Taylor Series in One VariableVariable

hxandxbetweenisxwhere

xfn

hxf

i

hhxf n

ni

n

i

i

)(!

)(!

)(

f(x) ofexpansion SeriesTaylor The

)()(1

0

Approximation Error


Taylor Series in One Taylor Series in One VariableVariableanother lookanother 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

)(


DefinitionsDefinitions

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


Taylor Series Expansion 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


Taylor Series in Two Taylor Series in Two VariablesVariables

),( 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


Runge-Kutta Method

possible. as accurate as is such that

,,,

:Problem

)()(

),(

),(


21

2211

12

1

h)x(t

wwFind

KwKwtxhtx

KxhtfhK

xtfhK


Runge-Kutta Method

)(),()()(

),(2

1),(

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

...)('''6

)(''2

)(')()(

possible. as many terms asmatch to

,,, :Problem

322

2221

2

21

32

21

hOffhwfhwxtfhwwtxhtx

xtfx

ht

hfhfhffhxhtf

xtfhxhtfwxtfhwtxhtx

txh

txh

thxtxhtx

wwFind

tt

xt


Runge-Kutta Method

)(),()()(

...)('''6

)(''2

)(')()(

322

2221

32

hOffhwfhwxtfhwwtxhtx

txh

txh

thxtxhtx

tt

1,1,5.0,5.0

solution possibleOne

5.0,5.0,1

21

2221

ww

wwww


Runge-Kutta Method

21

12

1

2

1)()(

),(

),(


KKtxhtx

KxhtfhK

xtfhK


Runge-Kutta MethodAlternative Formula

21

12

1

2)()(

),(

),(


FFh

txhtx

hFxhtfF

xtfF


Runge-Kutta MethodAlternative Formula

21

12

1

2)()(

),(

),(

Form eAlternativ

FFh

txhtx

hFxhtfF

xtfF

21

12

1

2

1)()(

),(

),(


KKtxhtx

KxhtfhK

xtfhK


Runge-Kutta MethodAlternative Formulas

21

12

1

2

1

2

11)()(

),(

),(

)0(select Formulas Kutta RungeOrder Second

FFtxhtx

KxhtfhK

xtfhK

2

1,

2

11,

number zero-nonany Pick

solutionanother

5.0,5.0,1

21

2221

ww

wwww


Runge-Kutta Method

4321

34

23

12

1

226

1)()(

),(

)2

1,

2

1(

)2

1,

2

1(

),(

Kutta RungeOrder Fourth

KKKKtxhtx

KxhtfhK

KxhtfhK

KxhtfhK

xtfhK


Second order Runge-Kutta Method Example

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

RK2 using x(1.02)find tosystem following theSolve32 hxttxtx



8269.3)1662.018.0(2

14

2

1)1()01.01(

1662.0))01.()18.0(1(01.0),(

18.0)1(01.0),(

:1STEP

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

RK2using x(1.02) find tosystem following theSolve

21

3212

321

32

KKxx

txKxhtfhK

txxtfhK

hxttxtx



6648.3)1545.01666.0(2

18254.3

2

1)01.1()01.001.1(

1545.0))01.()1666.0(1(01.0),(

1666.0)1(01.0),(

2 STEP

21

3212

321

KKxx

txKxhtfhK

txxtfhK


RK2 Using

[1,2]for tSolution

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

xttxtx


Summary Runge Kutta methods generate 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 N th order RK method.


More in this unitLesson 5: Applications of Runge-Kutta

Methods To solve first order differential equations. Lessons 6: Solving Systems of high order

ODE.

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