Komputasi Numerik: Integrasi dan Differensiasi Numerik Agus Naba Physics Dept., FMIPA-UB.

Komputasi Numerik:Integrasi dan

Differensiasi Numerik

Agus Naba

Physics Dept., FMIPA-UB

AGUS NABA-Computational Physics-Physics. Dept., FMIPA-UB 2

Ordinary Differential Equation

Ordinary Differential Equation (ODE) is a differential equation in which all dependent variables are functions

of a single independent variable.


ODE’s Problem

00 ')0(';)0(

)),(,()(')(

yyyy

tytftydt

tdy

First-order Ordinary Differential Equation (ODE):

?)t(y


Euler’s Method

t

tyty

dt

dy

ntnttt

nn

t

n

n

)()(

,..2,1,0 ,

1

0

tytfytdt

dyyy nnn

tnn

n

),(1

)tt(yy);t(yy nnnn 1


tn tn+1

tdt

dyyy

ntnn 1

It enables us to calculate all of yn = y (tn), given y(t0).

y(t)

tdt

dy

nt

yn

ntdt

dySlope:


Numerical Errors

Truncation Errors, depending on numerical methods

Round-off Errors, depending on capability of computer in storing floating-point number


Truncation ErrorsThe curve y(t) is not generally a straight-line between the neighbouring grid-times tn and tn+1 as assumed.

According to Taylor Series:

...t

dt

ydt

dt

dyyy

nn ttnn

2

2

2

2

1

O(t2)Truncation Error

Each step incurs truncation error ~ t2 Net truncation errors of Euler’s Method ~ t


Round-off Errors

For every type of computer, there is a charasteristic number, , which defined as the smallest number which when added to a number of order unity gives rise to a new number. For example: = 2.2 x 10-16 (for double precision number in IBM-PC ) = 1.19 x 10-7 (for single precision number in IBM-PC )

The net round-off errors of Euler’s Method /t.


Net Numerical Errors of Euler’s Method

tt

~

At large t, the error is dominated by the truncation errors, whereas the round-off errors dominates at small t .

Minimum net numerical errors are achieved when

2121 // ~;~t


t

t~1/2

~1/2


1)0(0)()(

y,,tydt

tdy

t

y(0)=1

tety )(

Numerical Instalibilities


Solusi Numerik,...,,n,tnttn 210 0

nn y)t(y 11

/t 2 /t 2

Numerical Instabilitiesy(0) y(0)

t t


Defect of Euler’s Method

Not generally used in scientific computing: Truncation errors is far larger than other,

more advanced, methods. Too prone to numerical instabilities


Main reason of large truncation errors:

Euler’s method only evaluates derivative at the beginning of the interval [tn,tn+1], i.e., at tn.

（ Very asymetric with respect to the beginning and the end of the interval)


tn tn+1tn+ h/2

h

y(t)

Runge-Kutta (RK) Methods

221

2

ky,

htff nn

nn y,tff 1

11 hfk

k1 k2

yn

f1

f2

y(t)

22 hfk

k1 /2

11 kyy nn

21 kyy nn

The 2nd order RK Method

Euler’s Method


Modified Euler’s Method

12 ky,htff nn

nn y,tff 1

11 hfk

tn tn + h

h

y(t)

f1

yn

k1

f2

k2

(k1+k2)/2

22 hfk

yn+1 = yn+ (k1+k2)/2

Modified Euler’s Method


tn tn + h/2 tn + h

f1

f2

f3

f4

The 4th order RK method


344

233

122

11

43211

22

22

226

1

ky,hthfhfk

ky,

hthfhfk

ky,

hthfhfk

y,thfhfk

kkkkyy

nn

nn

nn

nn

nn

The 4th order RK method


Method

Equations

Euler (Error of the order h2)

yxfhk

ky

,1

1

Modified Euler (Error of the order h3)

12

1

21

,

,2

1

kyhxfhk

yxfhk

kky

Heun (Error of the order h4)

23

12

1

31

3

2

3

2

3

1

3

1

34

1

ky,hxfhk

ky,hxfhk

y,xfhk

kky

4th order Runge Kutta (Error of the order h5)

34

23

12

1

4321

,

2

1,

2

1

2

1,

2

1

,

226

1

kyhxfhk

kyhxfhk

kyhxfhk

yxfhk

kkkky


Net Numerical Errors of RK Methods

Method-RK oforder , :nhh

~ n

)n/(n)n/( ~~h 111

:at errors Minimum


order hmin min

1 1.5 x 10-8 1.5 x 10-8

2 6.1 x 10-6 3.7 x 10-11

3 1.2 x 10-4 1.8 x 10-12

4 7.4 x 10-4 3.0 x 10-13

5 2.4 x 10-3 9.0 x 10-14

RK Methods Performanceon IBM-PC for double precision

hmin increases and min decreases as order gets larger, but needs more computational effort.


Example

tksinty

k,t?ty

kdt

dyy

tkydt

yd

t

1101

000

2

2

?txtkxdt

tyd

dt

tdx

?)t(xtxdt

)t(dy

dt

tdxdt

)t(dytx

tytx

212

22

121

2

1


Global integration errors associated with Euler's method (solid curve) and the 4th order Runge-Kutta method (dotted curve) plotted against the

step-length h. Single precision calculation.

err = yanalitic-ynumeric


Global integration errors associated with Euler's method (solid curve) and the 4th order Runge-Kutta method (dotted curve) plotted

against the step-length h. Double precision calculation.


Adaptive Integration Method

xktdt

dx

tdt

xd 22

2

41

Consider the following ODE:

Analitic solution:

2tksinx


Global integration error associated with a xed step-length (h = 0:01), 4th order RK method, plotted against the independent variable, t, for a system of o.d.e.s in which the

variation scale-length decreases rapidly with increasing t. Double precision calculation.

err = xanalitic-xnumeric


It can be seen that, although the error starts off small, it rises rapidly as the variation scale-length of the solution decreases (i.e., as t increases), and quickly becomes

unacceptably large. Of course, we could reduce the error by simply reducing the step-length, h. However, this is a very

inefficient solution. The step-length only needs to be reduced at large t. There is no need to reduce it, at all, at

small t.

Solution: h should be large at small t but needs to be reduced at large t


5

1

0

oldnew hh

The step-length h should be increased if the truncation error per step is too small, and vice versa, in such a manner that the

error per step remains relatively constant at 0.

stepper error n truncatiodesired the:0


Global integration errors associated with fixed step-length (h = 0.01), 4th order RK method (solid curve) and a corresponding adaptive method (0 = 10-8) (dotted curve), plotted against the independent variable, t, for a system of o.d.e.s in which the variation scale-length decreases rapidly with increasing t. Double precision calculation.


Differentiation

An object is moving through space, its position as a function of time x(t) is recorded in a table.

Problem: Determine the object’s velocity v(t)=dx/dt and acceleration a(t)=d2x/dt2


Method: Numeric

h

xfhxflimxf

dx

xdfh

0

1

Even a computer runs into errors with such a method because of its subtraction operations: the numerator tends to fluctuate between 0 and the machine precision as the denominator approaches zero.


Method: Forward Difference (FD)

c denotes a computed expression.

h

xfhxfxfc

1


x x+h

f(x)

FD: using two points to represent the 1st derivative function by a straight line in the interval from x to x+h

xfc1

xf 1

h


...xfh

xfh

xhfxfhxf 33

22

1

62

...xfh

xfh

xf

,h

xfhxffc

32

21

1

62

Error


Example of FD

2bxaxf bxxf 21


FD solution

This clearly becomes a good approximation only for small h, i.e., h << 2x

bhbx

xfh

xfh

xf

h

xfhxfxfc

2

...62

,

32

21

1


Method: Central Difference (CD)

h,xfDh

/hxf/hxfxf cc

221


x-h/2 x x+h/2

f(x)

CD: using two points to represent the function by a straight line in the interval from x-h/2 to x+h/2

21 /hxfc

xf 1

h


...xfh

xfh

xfh

xfh

xf

...xfh

xfh

xfh

xfh

xf

33

22

1

33

22

1

48822

48822

...xfh

xf

h

/hxf/hxfxfc

32

1

1

24

22

Error


Example of CD

2bxaxf bxxf 21


CD solution

CD Method gives the exact answer regardless of the size of h !

bx

...xfh

xf

h

/hxf/hxfxfc

224

22

32

1

1


Method:Extrapolated Difference (ED)

The error in FD ~ hThe error in CD ~ h2

The error in ED ~ h4


...xfh

xf

/h

/hxf/hxf/h,xfDc

32

1

96

2

442


x-h/2 x x+h/2

f(x)

h,xfDc

xf 1

x-h/4 x+h/4

2/h,xfDc

h

h/2


Extrapolated Difference

3

241 h,xfD/h,xfDxf cc

c


x-h/2 x x+h/2

f(x)

h,xfDc

xf 1

x-h/4 x+h/4

2/h,xfDc

h

h/2

24 /h,xfDc


xfh

xf

h,xfD/h,xfDxf cc

c

34

1

1

1641920

3

24

Error


22448

3

11 hxf

hxf

hxf

hxf

hfc

A Good Way of Computing for ED

It reduces the loss of precision that occurs when large and small numbers are added together, only to be subtracted from other large numbers.

Subtract the large number from each other and then add the difference to the small numbers !


Attention !

Regardless of the algorithm, evaluating the derivative of f(x) at x requires us to know the values of f surrounding x !

HOW ?Once we have the derivative of f(x) at x, USE the integration methods, ex., RK Method, to approximate the values of f surrounding x !


Error Analysis

The approximation/truncation errors in numerical differentiation decrease with decreasing step size h while roundoff errors increase with a smaller step size. Total error is minimum if

minimum. This occurs when

rotrunc

rotrunc


Roundoff Error

The limit of roundoff error is essentially machine precision:

h

hh

xfhxf'f

ro


Truncation Errors

Truncation Error of FD:

Truncation Error of CD:

24

223

2

hf

hf

cdtrunc

fdtrunc


Best h

The h value for which roundoff and truncation errors are equal is

Ex., for single precision 10-7 for f(x)=ex or cos(x)

hfd 0.0005 and hcd 0.01

33

22

232

242242

fh

fh

hf

h

hf

h

cdfd

cdro

fdtrunc

Komputasi Numerik: Integrasi dan Differensiasi Numerik Agus Naba Physics Dept., FMIPA-UB.

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