Numerical Analysis Differential Equations

8/22/2019 Numerical Analysis Differential Equations

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Numerical Analysis –

Differential Equation

Hanyang University

Jong-Il Park



Department of Computer Science and Engineering, Hanyang University

Differential Equation




Solving Differential Equation

Differential EquationOrdinary D.E.

Partial D.E.

Ordinary D.E.

Linear eg.

Nonlinear eg.

Initial value problem

Boundary value problem

)('' t f y y

)(''' t f y y y

Usually no closed-form solution

linearization

numerical solution

0)0()0(,0. ''' y y y yeg

3)1(,0)0(,1054. ''' y y y y yeg




Discretization in solving D.E.

Discretization

Errors in Numerical

Approach

Discretization error

Stability error

y

t

Grid Points

Exact sol.

d e D y ye nd S y ye

sol.numerical:

sol.ddiscretize:

sol.exact:

n

d

e

y

y

y




Errors

Total error

S D eee truncation round-off

De S e0 increase

as t 0 as t 0

trade-off

t

e

DeS e




Local error & global error

Local error

The error at the given step if it is assumed that all theprevious results are all exact

Global error

The true, or accumulated, error




Useful concepts(I)

Useful concepts in discretization

Consistency

Order

Convergence

00)( Deht

55

22

)()(

hehOhehO

D

D

0h

e y

y

t




Useful concepts(II)

stability

stable

unstable

y

t

Consistent

stableConverge




Stability

Stability condition

0

' )0(, y y Ay y eg.

Exact sol.

Euler method

At e y y 0

0

1

1

)1(

)1(

yhA

yhA

hAy y y

nn

nnn

For stability

AhhA 201|1|

Amplification factor




nnnn

nnn

yhh yh y y

hf y y

)1(2.1)2.1(1

1

h

h y

yh y y

hf y y

n

nnn

nnn

1

2.1

)2.1( 11

11

Explicit :

Implicit :

h increase

h large

h small

ye

y

t

y

txplicit implicit

“conditionally stable” “stable”

y y y y y 2.12.0)0(,2.1 ''eg.

= f

Implicit vs. Explicit Method




Modification to solve D.E.

Modified Differential Eq.

)(. ' t g Ay yeg

Diff.

eq.Modified

D.E.

Discretization

Discretization by Euler method)(1 nnnn Ay g h y y

<Consistency check>

''

!21'

''2

!2

1'

''2

!21'

1

|

)(||

||

hy g Ay y

Ay g h y yhhy y

yhhy y y

nnn

nnnnnn

nnnn

0Let h nnn g Ay y |'consistent;

<Order>

)(|'

hO g Ay y nnn




Initial Value Problem: Concept




Initial value problem

Initial Value Problem

Simultaneous D.E.

High-order D.E.

00' )(,),( yt y yt f

dt dy y

00,21

1001,2111

)(),,,,(

)(),,,,(

nnnnn

n

yt y y y yt f dt

dy

yt y y y yt f dt

dy

)1(

00

)1('

00

'

0

)(

)1(''')(

)(,,)(,

),,,,,(0

nnt

nn

yt y yt y y y

y y y yt f y




Well-posed condition




Taylor Series Method

),(

'

yt f y

n

nn

nRh yh yh y yht y )(

0!12''

0!21'

000 )(

Truncation error)( 0 ht y

)( 0t y

0t ht 0

h

ht t

yn

h R n

n

n

00

)1(1

,)()!1(

Taylor series method(I)




High order differentiation

Implementation

22

'''

''

2

)()(][

))](,([

f f f f f f f f f

f f f y

f f t

f f f f f f dt

d y

f f f dt

dy

y

f

t

f t yt f

dt

d y

yy yt ytytt

y ytytt yt

yt

<Type 1>

Complicated computation

0t '

0 y''

0 y'''

0

y

hhh

y

t

Less computation

accuracy

<Type 2>

h h h

0t 1t 2t 3t '

0 y''

0 y'''

0 y

'

1 y''

1 y'''

1 y

'

2 y''

2 y'''

2 y

....

y

t

More computation

accuracy

Requiring complicated

source codes

Taylor series method(II)




Euler method(I)

Euler Method

0t

....

y

t1t 2t 3t ....

0 y0 y

1 y2 y

3 y

h

00

' )(),,( yt y yt f y

Talyor series expansion at to

100

2

0

''

!21'

001 ,)( t t h yh y y y

0

00 ),(

f

yt f




Euler method(II)

Error

Eg. y’ =-2x3+12x2-20x+8.5, y(0)=1




Euler method(III)

1

2

2''

!21

1

,)(

)(

nnnnn

nnnn

t t hOh f y

h yh f y y

1

0

10112010 )()()()(n

i

iinnn y y y y y y y y y y y

)()()(

2

1

)(2

1

)(

_ ''

0

2 _

''0

2''1

02

1

hOh yt t

h yh

t t

h ye

n

n

i

n

it

Generalizing the relationship

Euler’s approx. truncation error

Error Analysis

Accumulated truncation error

h

t t n

t t y y

n

n

n

iin

0

0

1

0

''1''

,)()(

; 1st order




Eg. Euler method




Modified Euler method: Heun’s method

Modified Euler’s Method

Why a modification?

errormodify

nt nt 1n

t

C

n y 1

P

n y 1 P

n f 1

n f 2

1

P

nn f f

1nt

Predictor

Average slope

Corrector

nn

P

n hf y y 1

2),(),(

2

11

'

1

''

P

nnnnnn yt f yt f y y y

)],(),([2

11

'

1 nnnnnn

C

n yt f yt f h

y yh y y




Heun’s method with iteration

Iteration

significant

improvement




Error analysis

Error Analysis

Taylor series

Total error

)(][

)()(

)(

3'

1

'

2

'''3

!31

''

12

21'

'''3

!31''2

21'

1

hO y y y

yhhOh

y yhhy y

yh yhhy y y

nnh

n

nnnn

nnnn

truncation

3rd order

)( 2hO ; 2nd order method

※ Significant improvement over Euler’s method!




Eg. Euler vs. Modified Euler

Euler Modified

Euler Method

improvement




Runge-Kutta method

Runge-Kutta Method

Simple computation

very accurate

The idea

codesourceEasy.,,no ''' y y

),,(1 h yt h y y nnnn

where

),(

),(

),(

11

112

1

2211

nnnnn

nn

nn

nn

yt f k

yt f k

yt f k

k ak ak a




Second-order Runge-Kutta method

Second-order Runge-Kutta method

),(

),(

)(

112

1

22111

nn

nn

nn

yt f k

yt f k

k ak ah y y ①

Taylor series expansion

nn y

f

nt

f

n

hn yt hnnn

R f k

y f f f f h y y

112

'''!3!21

)(

)()()(32

nn yt nnn hR f f ha f aah y y

)())(( 112211

②

③

④ ③→①

Equating ② and ④

),(2

,2

,1 121221 nn yt f h

ah

aaa




Modified Euler - revisited

)( 2121 k k y y hnn

),(1 nn yt f k

),( 12 hk yht f k nn Modified Euler method

set

1111 ,,2

1hk ha

2

12 a

Modified Euler method is a kind

of 2nd-order Runge-Kutta method.

P1

P2

),(1 nn yt f k

),( 12 hk yht f k nn

1n y




Other 2nd order Runge-Kutta methods

Midpoint method

Ralston’s method

111212

,2

,1,0 k hhaa

))),(,((221 nnh

nh

nnn yt f yt f h y y

)3( 2141 k k y y hnn

),(1 nn yt f k

),( 132

32

2 hk yht f k nn




Comparison: 2nd order R-K method




Comparison: 2nd 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

hk yht f k

k yt f k

k yt f k

yt f k

k k k k h y y

nn

hn

hn

hn

hn

nn

nn

)( 1 P f

nt 2h

nt ht n

P1

P2

P3

P4

)( 3 P f

)( 2 P f

)( 1 P f

※ significant improvement over

modified Euler’s method

4-th order Runge-Kutta methods




Runge-Kutta method




Eg. 4-th order R-K method

Significant

improvement




Discussion

Better!



Comparison

(5th order)

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Numerical Analysis Differential Equations

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