runge kutta.pptx

7/29/2019 runge kutta.pptx

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RUNGE-KUTTA 4TORDER

Presented by

M.Saravanakum


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The solution of a differential equation using higher

order derivatives of the Taylor expansion is notpractical.

Since for only the simplest functions, these higherorders are complicated. Also there is no simplealgorithm which can be developed.

This is because each series expansion is unique.


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However we have methods which use only 1st orderderivates while simulating higher order(producingequivalent results).

These one step methods are called Runge-Kuttamethods.

Approximation of the second, third and fourthorder (retaining h2, h3, h4 respectively in the Taylexpansion) require estimation at 2, 3 , 4 ptsrespectively in the interval (xi,xi+1).


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The Runge-Kutta methods have algorithms of the

form,

where is the increment function.

The increment function is a suitably chosenapproximation to on the interval

hyxhyy iiii ,,1

yxf ,


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Fourth Order Runge-Kutta Method The fourth order Runge-Kutta (RK-4) method is derived by applying th

1/3 or Simpsons 3/8 rule to integrating over the interval

formula of RK-4 based on the Simpsons 1/3 is written as

),(' tyfy ,[ nt

),(

)2

,(

)2

,(

),(where

226

1

34

221

3

121

2

1

43211

htkyhfk

htkyhfk

htkyhfk

tyhfk

kkkkyy

nn

nn

nn

nn

nn


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The 4th Order Runge-KuttaThis is a fourth order function that solvesan initial value problems using a four step

program to get an estimate of the Taylor

series through the fourth order.

This will result in a local error of O(Dh5

)and a global error of O(Dh4)


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4th-orderRunge-Kutta Method

xi xi + h/2 xi + h

f1

f2

f3

f4

4321 226

1fffff

f


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Runge-Kutta Method(4th Order) Example

Consider Exact Solution

The initial condition is:

The step size is:

2xydx

dy

x222 exxy

10 y

1.0h


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The 4th Order Runge-KuttaThe example of a single step:

104829.1226

1

109499.0104988.1,1.01.0,

10.02/.1,05.01.02

1,

2

1

10475.005.1,05.01.02

1,

2

1

1.0011.01,01.0,

4321n1n

34

223

12

21

kkkkyy

fkyhxfhk

kfkyhxfhk

fkyhxfhk

fyxfhk


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Runge-Kutta Method (4th Order)Example

The values for the 4th order Runge-Kutta method

x y f(x,y) k 1 f2 k 2 f3 k 3 f4 k 4 Exact

0 1 1 0.1 1.0475 0.10475 1.049875 0.104988 1.094988 0.109499 1

0.1 1.104829 1.094829 0.109483 1.13707 0.113707 1.139182 0.113918 1.178747 0.117875 1.104829

0.2 1.218597 1.178597 0.11786 1.215027 0.121503 1.216848 0.121685 1.250282 0.125028 1.218597

0.3 1.340141 1.250141 0.125014 1.280148 0.128015 1.281648 0.128165 1.308306 0.130831 1.340141

0.4 1.468175 1.308175 0.130817 1.331084 0.133108 1.332229 0.133223 1.351398 0.13514 1.468175

0.5 1.601278 1.351278 0.135128 1.366342 0.136634 1.367095 0.13671 1.377988 0.137799 1.601279

0.6 1.73788 1.37788 0.137788 1.384274 0.138427 1.384594 0.138459 1.38634 0.138634 1.737881

0.7 1.876246 1.386246 0.138625 1.383059 0.138306 1.382899 0.13829 1.374536 0.137454 1.876247

0.8 2.014458 1.374458 0.137446 1.360681 0.136068 1.359992 0.135999 1.340457 0.134046 2.014459

0.9 2.150396 1.340396 0.13404 1.314915 0.131492 1.313641 0.131364 1.28176 0.128176 2.150397

1 2.281717 1.281717 0.128172 1.243303 0.12433 1.241382 0.124138 1.195855 0.119586 2.281718


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Runge-Kutta Method (4th Order)Example

A comparison between the 2nd

order andthe 4th order Runge-Kutta methods show aslight difference.

Runge Kutta Comparison

-10

-8

-6

-4

-2

0

2

4

0 1 2 3 4

X Value

YValue

Exact

2nd order

4th order

Error of the Methods

0.00

2.00

4.00

6.00

8.00

10.00

0 1 2 3 4

X Value

Absolute|Error|

Error 2nd order method

Error 4th order method


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Example-1

A ball at 1200K is allowed to cool down in air at an ambienttemperature of 300K. Assuming heat is lost only due to radiation, the

differential equation for the temperature of the ball is given by

Kdt

d12000,1081102067.2 8412

Find the temperature at480

t seconds using Runge-Kutta 4

th

order method.

240h seconds.

8412 1081102067.2 dt

d

8412 1081102067.2, tf

Assume a step size of

hkkkkii 43211 226

1


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Solution

Step 1: 1200)0(,0,0 00 ti

5579.410811200102067.21200,0, 841201 ftfk o

38347.0108105.653102067.205.653,120

2405579.42

11200,240

2

10

2

1,

2

1

8412

1002

f

fhkhtfk

8954.310810.1154102067.20.1154,120

24038347.02

11200,240

2

10

2

1,

2

1

8412

2003

f

fhkhtfk

0069750.0108110.265102067.210.265,240240984.31200,2400,

8412

3004

f

fhkhtfk


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Solution Cont

1 is the approximate temperature at

240240001 httt

K65.675240 1

K

hkkkk

65.675

2401848.26

11200

240069750.08954.3238347.025579.46

11200

226

1432101


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Solution Cont

Step 2: Kti 65.675,240,1 11

44199.0108165.675102067.265.675,240, 8412111 ftfk

31372.0108161.622102067.261.622,360

24044199.02

165.675,240

2

1240

2

1,

2

1

8412

1112

f

fhkhtfk

34775.0108100.638102067.200.638,360

24031372.02

165.675,240

2

1240

2

1,

2

1

8412

2113

f

fhkhtfk

25351.0108119.592102067.219.592,48024034775.065.675,240240,

8412

3114

f

fhkhtfk


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Solution Cont

2

is the approximate temperature at

48024024012 htt

K91.594480 2

K

hkkkk

91.594

2400184.26

165.675

24025351.034775.0231372.0244199.06

165.675

2261 432112


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Solution Cont

The exact solution of the ordinary differential equation is given by tsolution of a non-linear equation as

9282.21022067.000333.0tan8519.1300

300ln92593.0 31

t

The solution to this nonlinear equation at t=480 seconds is

K57.647)480(


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Comparison with exact results

Figure . Comparison of Runge-Kutta 4th order method with exact solutio

-400

0

400

800

1200

1600

0 200 400 600

Time,t(sec)

Temperature,

h=120

Exact

h=240

h=480

(K)


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Step size, h (480) Et |t|%

480

240120

60

30

90.278

594.91646.16

647.54

647.57

737.85

52.6601.4122

0.033626

0.00086900

113.94

8.13190.21807

0.0051926

0.00013419

Effect of step size

Table 1. Temperature at 480 seconds as a function of step size

K57.647)480( (exact)


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Effects of step size on Runge-Kutta 4th

Method

Figure . Effect of step size in Runge-Kutta 4th order method

-200

0

200

400

600

800

0 100 200 300 400 500

Step size, h

Te

mperature,

(480)


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Summary Runge Kutta methods generate an accurate solution

without the need to calculate high order derivatives Second order RK have local truncation error of orde

O(h3) and global truncation error of order O(h2).

Higher order RK have better local and globaltruncation errors.

N function evaluations are needed in the Nth order Rmethod.


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THANK YOU

runge kutta.pptx

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