Numerical Analysis – Differential Equation

Numerical Analysis – Numerical Analysis – Differential EquationDifferential Equation

Hanyang University

Jong-Il Park

Department of Computer Science and Engineering, Hanyang University

Differential EquationDifferential Equation


Solving Differential EquationSolving Differential Equation

Differential EquationOrdinary D.E.

Partial D.E. Ordinary D.E.

Linear eg. Nonlinear eg.

Initial value problem

Boundary value problem

)('' tfyy )(''' tfyyy

Usually no closed-form solution

linearizationnumerical solution

0)0()0(,0. ''' yyyyeg

3)1(,0)0(,1054. ''' yyyyyeg


Discretization in solving D.E.Discretization in solving D.E.

Discretization

Errors in Numerical Approach Discretization error

Stability error

y

tGrid Points

Exact sol.

deD yye

ndS yye

sol. numerical:

sol.ddiscretize:

sol.exact :

n

d

e

y

y

y


ErrorsErrors Total error

SD eee truncation round-off

De Se0 increaseas t 0 as t 0

trade-off

t

eDeSe


Local error & global errorLocal error & global error

Local error The error at the given step if it is assumed that all the

previous results are all exact Global error

The true, or accumulated, error


Useful concepts(I)Useful concepts(I)

Useful concepts in discretization Consistency

Order

Convergence

00)( Deht

55

22

)(

)(

hehO

hehO

D

D

0h

eyy

t


Useful concepts(II)Useful concepts(II) stability

stable

unstable

y

t

Consistentstable

Converge


StabilityStability

Stability condition

0' )0(, yyAyy eg.

Exact sol.Euler method

Ateyy 0

01

1

)1(

)1(

yhA

yhA

hAyyy

n

n

nnn

For stability

AhhA 201|1|

Amplification factor


nnnn

nnn

yhhyhyy

hfyy

)1(2.1)2.1(1

1

h

hy

yhyy

hfyy

n

nnn

nnn

1

2.1

)2.1( 11

11

Explicit :

Implicit :

h increase

h large

h small

ye

y

t

y

texplicit implicit

“conditionally stable” “stable”

yyyyy 2.12.0)0(,2.1 ''eg.= f

Implicit vs. Explicit MethodImplicit vs. Explicit Method


Modification to solve D.E.Modification to solve D.E.

Modified Differential Eq.

)(. ' tgAyyeg

Diff.eq.

ModifiedD.E.

Discretization

Discretization by Euler method)(1 nnnn Ayghyy

<Consistency check>

''!2

1'

''2!2

1'

''2!2

1'1

|

)(||

||

hygAyy

Ayghyyhhyy

yhhyyy

nnn

nnnnnn

nnnn

0Let h nnn gAyy |' consistent;

<Order>)(|' hOgAyy nnn


Initial Value Problem: ConceptInitial Value Problem: Concept


Initial value problemInitial value problem

Initial Value Problem

Simultaneous D.E.

High-order D.E.

00' )(,),( ytyytfdt

dyy

00,21

1001,2111

)(),,,,(

)(),,,,(

nnnnn

n

ytyyyytfdt

dy

ytyyyytfdt

dy

)1(00

)1('00

'0

)(

)1(''')(

)(,,)(,

),,,,,(0

nnt

nn

ytyytyyy

yyyytfy


Well-posed conditionWell-posed condition


Taylor Series Method

),(' ytfy

nnn

n Rhyhyhyyhty )(0!

12''0!2

1'000 )(

Truncation error)( 0 hty

)( 0ty

0t ht 0

h

htt

yn

hR n

n

n

00

)1(1

,)()!1(

Taylor series method(I)Taylor series method(I)


High order differentiation

Implementation

22

'''

''

2

)()(][

))](,([

fffffffff

fffy

fft

ffffffdt

dy

fffdt

dy

y

f

t

ftytf

dt

dy

yyytytytt

yytyttyt

yt

<Type 1>

Complicated computation

0t'0y''

0y'''

0y

hhh

y

tLess computation accuracy

<Type 2>

h h h

0t 1t 2t 3t'0y''

0y'''

0y

'1y''

1y'''

1y

'2y''

2y'''

2y

....

y

tMore computation accuracy

Requiring complicated source codes

Taylor series method(II)Taylor series method(II)


Euler method(I)Euler method(I)

Euler Method

0t

....y

t1t 2t 3t ....

0y0y

1y2y

3y

h

00' )(),,( ytyytfy

Talyor series expansion at to

1002

0''

!21'

001 ,)( tthyhyyy

0

00 ),(

f

ytf


Euler method(II)Euler method(II)

Error

Eg. y’ =-2x3+12x2-20x+8.5, y(0)=1


Euler method(III)Euler method(III)

12

2''!2

11

,)(

)(

nnnnn

nnnn

tthOhfy

hyhfyy

1

010112010 )()()()(

n

iiinnn yyyyyyyyyyy

)()()(2

1

)(2

1

)(

_''

0

2_

''0

2''1

021

hOhytt

hyh

tt

hye

n

n

i

n

it

Generalizing the relationship

Euler’s approx.truncation error

Error Analysis

Accumulated truncation error

h

ttn

ttyy

n

n

n

iin

0

0

1

0

''1'' ,)()(

; 1st order


Eg. Euler methodEg. Euler method


Modified Euler method: Heun’s methodModified Euler method: Heun’s method

Modified Euler’s Method Why a modification?

errormodify

nt nt 1nt

Cny 1

Pny 1

Pnf 1

nf2

1Pnn ff

1nt

Predictor

Average slope

Corrector

nnPn hfyy 1

2

),(),(

211

'1

''

Pnnnnnn ytfytfyy

y

)],(),([2 11

'1 nnnnnn

Cn ytfytf

hyyhyy


Heun’s method with iterationHeun’s method with iteration

Iteration

significant improvement


Error analysis Error analysis Error Analysis

Taylor series

Total error

)(][

)()(

)(

3'1

'2

'''3!3

1''

1221'

'''3!3

1''221'

1

hOyyy

yhhOh

yyhhyy

yhyhhyyy

nnh

n

nnnn

nnnn

truncation3rd order

)( 2hO ; 2nd order method

※ Significant improvement over Euler’s method!


Eg. Euler vs. Modified EulerEg. Euler vs. Modified Euler EulerModified

Euler Method

improvement


Runge-Kutta methodRunge-Kutta method

Runge-Kutta Method Simple computation

very accurate

The idea

code sourceEasy .,,no ''' yy

),,(1 hythyy nnnn

where

),(

),(

),(

11

112

1

2211

nnnnn

nn

nn

nn

ytfk

ytfk

ytfk

kakaka


Second-order Runge-Kutta methodSecond-order Runge-Kutta method

Second-order Runge-Kutta method

),(

),(

)(

112

1

22111

nn

nn

nn

ytfk

ytfk

kakahyy ①

Taylor series expansion

nnyf

ntf

n

hnyt

hnnn

Rfk

yffffhyy

112

'''!3!21

)(

)()()(32

nnytnnn hRffhafaahyy )())(( 112211

②

③

④③→①

Equating ② and ④

),(2

,2

,1 121221 nn ytfh

ah

aaa


Modified Euler - revisitedModified Euler - revisited

)( 2121 kkyy hnn

),(1 nn ytfk ),( 12 hkyhtfk nn Modified Euler method

set

1111 ,,2

1hkha

2

12 a

Modified Euler method is a kind of 2nd-order Runge-Kutta method.

P1

P2

),(1 nn ytfk

),( 12 hkyhtfk nn

1ny


Other 2nd order Runge-Kutta methodsOther 2nd order Runge-Kutta methods

Midpoint method

Ralston’s method

11121 2,

2,1,0 k

hhaa

))),(,(( 221 nnh

nh

nnn ytfytfhyy

)3( 2141 kkyy hnn

),(1 nn ytfk

),( 132

32

2 hkyhtfk nn


Comparison: 2Comparison: 2ndnd order R-K method order R-K method


Comparison: 2Comparison: 2ndnd order R-K method order R-K method

Eg. y’ =-2x3+12x2-20x+8.5, y(0)=1


Fourth-order Runge-Kutta Taylor series expansion to 4-th order accurate short, straight, easy to use

),(

),(

),(

),(

})(2{

34

2223

1222

1

432161

1

hkyhtfk

kytfk

kytfk

ytfk

kkkkhyy

nn

hn

hn

hn

hn

nn

nn

)( 1Pf

nt 2h

nt htn

P1

P2

P3

P4

)( 3Pf

)( 2Pf

)( 1Pf

※ significant improvement over modified Euler’s method

4-th order Runge-Kutta methods4-th order Runge-Kutta methods


Runge-Kutta methodRunge-Kutta method


Eg. 4-th order R-K methodEg. 4-th order R-K method

Significant improvement


DiscussionDiscussion

Better!


ComparisonComparison

(5th order)

Documents

Numerical Analysis – Differential Equation