Numerical Analysis – Differential Equationcontents.kocw.net/KOCW/document/2014/hanyang/parkjongil/... · 2016-09-09 · Department of Computer Science and Engineering, Hanyang University

Numerical Analysis –Differential Equation

Hanyang University

Jong-Il Park

Department of Computer Science and Engineering, Hanyang University

Differential Equation

P13 FIGURE 1.2

P700 FIGURE PT17.2


Solving 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

Errors in Numerical Approach Discretization error

Stability error

y

t

Grid Points

Exact sol.

deD yye

ndS yye

sol. numerical:sol.ddiscretize:

sol.exact :

n

d

e

yyy


Errors Total error

SD eee truncation round-off

De Se0 increase

as t 0 as t 0

trade-off

t

eDe

Se


Local 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 in discretization

Consistency

Order

Convergence

00)( Deht

55

22

)(

)(

hehO

hehO

D

D

0h

eyy

t


Useful concepts(II) stability

stable

unstable

y

t

Consistentstable

Converge


Stability

Stability condition

0' )0(, yyAyy eg.

Exact sol.Euler method

Ateyy 0

01

1

)1(

)1(

yhA

yhAhAyyy

nn

nnn

For stability

AhhA 201|1|

Amplification factor


nnnn

nnn

yhhyhyyhfyy

)1(2.1)2.1(1

1

hhy

yhyyhfyy

n

nnn

nnn

12.1

)2.1( 11

11

Explicit :

Implicit :

h increase

h large

h small

현재 이 이미지를 표시할 수 없습니다 .

ye

y

t

y

texplicit implicit

“conditionally stable” “stable”

yyyyy 2.12.0)0(,2.1 ''eg.

= f

Implicit vs. Explicit Method


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: Concept

P702 FIGURE PT17.4P700 FIGURE PT17.3


Initial value problem Initial Value Problem

Simultaneous D.E.

High-order D.E.

00' )(,),( ytyytf

dtdyy

00,21

1001,2111

)(),,,,(

)(),,,,(

nnnnn

n

ytyyyytfdtdy

ytyyyytfdtdy

)1(00

)1('00

'0

)(

)1(''')(

)(,,)(,

),,,,,(0

nnt

nn

ytyytyyy

yyyytfy


Well-posed condition

problem value-initial Then the . allfor and b][a,in

for t continuous are , respect to with derivative partialfirst its , and f that Suppse

y

yf y

, with ,for , ay(a)btaf(t,y)y'

posed.- wellis problem theand ,for solution unique a has btay(t)


Taylor Series Method),(' ytfy

nnn

n Rhyhyhyyhty )(0!

12''0!2

1'000 )(

Truncation error)( 0 hty

)( 0ty

0t ht 0

hhtt

ynhR n

n

n

00

)1(1

,)()!1(

Taylor series method(I)


High order differentiation

Implementation

22

'''

''

2

)()(][

))](,([

fffffffff

fffy

fft

ffffffdtdy

fffdtdy

yf

tftytf

dtdy

yyytytytt

yytyttyt

yt

<Type 1>

Complicated computation

0t'0y''

0y'''

0y

hhh

y

t

Less computationaccuracy

<Type 2>

h h h

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

0y'''

0y

'1y''

1y'''

1y

'2y''

2y'''

2y....

y

t

More computationaccuracy

Requiring complicatedsource codes

Taylor series method(II)


Euler method(I) Euler Method

0t

....

y

t1t 2t 3t ....

0y 0y1y 2y

3y

h

00' )(),,( ytyytfy

Talyor series expansion at to

1002

0''

!21'

001 ,)( tthyhyyy

0

00 ),(f

ytf

P707 FIGURE 25.1


Euler method(II)

Error

Eg. y’ =-2x3+12x2-20x+8.5, y(0)=1

P702 FIGURE PT7.4

P708 FIGURE 25.2


Euler method(III)

12

2''!2

11

,)(

)(

nnnnn

nnnn

tthOhfy

hyhfyy

1

010112010 )()()()(

n

iiinnn yyyyyyyyyyy

)()()(21

)(21

)(

_''

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

httn

ttyy

n

n

n

iin

0

0

1

0

''1'' ,)()(

; 1st order


Eg. Euler method

.5.0)0(with ,20for ,12` yttyywith h=0.2, N=10, and ti=0.2i.

Suppose that Euler`s method is used to approximate the solution to the initial-valueproblem

have we0.5,y(0) wand 1t-yy)f(t, Since 02

2.0008.00088.02.1]104.0[2.0)1( 21

221 iiwiwwtwhww iiiiii


Modified Euler method: Heun’s method Modified Euler’s Method

Why a modification?

errormodify

nt nt 1nt

Cny 1

Pny 1

Pnf 1

nf2

1P

nn ff

1nt

Predictor

Average slope

Corrector

nnPn hfyy 1

2),(),(

211

'1

''

Pnnnnnn ytfytfyyy

)],(),([2 11

'1 nnnnnn

Cn ytfytfhyyhyy


Heun’s method with iteration

Iteration

significant improvement

P720 FIGURE 25.9


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 Euler EulerModified

Euler Method

improvement

.5.0)0(with ,20for ,12` yttyywith h=0.2, N=10, and ti=0.2i, and w0=0.5 in each case. The difference equations produced from the various formulas are

Suppose we apply the Runge-Kutta methods of order 1 to our usual example.


Runge-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

ytfkytfk

kakaka


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

),(),(

)(

112

1

22111

nn

nn

nn

ytfkytfk

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 ytfhahaaa


Modified Euler - revisited

)( 2121 kkyy hnn

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

set

1111 ,,21 hkha

21

2 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 methods Midpoint method

Ralston’s method

11121 2,

2,1,0 khhaa

))),(,(( 221 nnh

nh

nnn ytfytfhyy

)3( 2141 kkyy hnn

),(1 nn ytfk

),( 132

32

2 hkyhtfk nn

P725 FIGURE 25.12


Comparison: 2nd order R-K method

.5.0)0(with ,20for ,12` yttyyThe Runge-Kutta method of order 4 applied to the initial-value problem

with h=0.2, N=10, and ti=0.2i, and w0=0.5 in each case. The difference equationsproduced from the various formulas are

Midpoint method: Modified Euler method:

Heun`s method: 2173.0008.00088.022.1

216.0008.00088.022.1

218.0008.00088.022.1

211

211

211

iiww

iiww

iiww

ii

ii

ii

for each i=0,1...9.Table 5.5 on page 196 lists the results of these calculations.


Comparison: 2nd order R-K methodEg. y’ =-2x3+12x2-20x+8.5, y(0)=1

P731 FIGURE 25.14


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

),(),(),(

),(})(2{

34

2223

1222

1

432161

1

hkyhtfkkytfkkytfk

ytfkkkkkhyy

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 methods


Runge-Kutta method

P733 FIGURE 25.15


Eg. 4-th order R-K method

Significant improvement

.5.0)0(with ,20for ,12` yttyy

The Runge-Kutta method of order 4 applied to the initial-value problem

with h=0.2, N=10, and ti=0.2i, gives the results and errors listed in Table 5.6


Discussion

Better!

For the problem

.5.0)0(with ,20for ,12` yytyyEuler` method with h=0.025, the Modified Euler`s method with h=0.05, and the Runge-Kutta method of order 4 with h=0.1 are compared at the common mesh points of the three methods, 0.1,0.2,0.2,0.4 and 0.5. Each of these techniques requires 20 functional evaluations to approximate y(0.5). In this example, the fourth-order method is clearly superior, as it is in most situations.


Comparison

P736 FIGURE 25.16

Documents

Numerical Analysis – Differential Equationcontents.kocw.net/KOCW/document/2014/hanyang/parkjongil/... · 2016-09-09 · Department of Computer Science and Engineering, Hanyang University