Date post: | 04-Apr-2018 |
Category: |
Documents |
Upload: | mohammad-rameez |
View: | 228 times |
Download: | 0 times |
of 22
7/29/2019 runge kutta.pptx
1/22
RUNGE-KUTTA 4TORDER
Presented by
M.Saravanakum
7/29/2019 runge kutta.pptx
2/22
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.
7/29/2019 runge kutta.pptx
3/22
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).
7/29/2019 runge kutta.pptx
4/22
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 ,
7/29/2019 runge kutta.pptx
5/22
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
7/29/2019 runge kutta.pptx
6/22
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)
7/29/2019 runge kutta.pptx
7/22
4th-orderRunge-Kutta Method
xi xi + h/2 xi + h
f1
f2
f3
f4
4321 226
1fffff
f
7/29/2019 runge kutta.pptx
8/22
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
7/29/2019 runge kutta.pptx
9/22
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
7/29/2019 runge kutta.pptx
10/22
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
7/29/2019 runge kutta.pptx
11/22
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
7/29/2019 runge kutta.pptx
12/22
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
7/29/2019 runge kutta.pptx
13/22
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
7/29/2019 runge kutta.pptx
14/22
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
7/29/2019 runge kutta.pptx
15/22
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
7/29/2019 runge kutta.pptx
16/22
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
7/29/2019 runge kutta.pptx
17/22
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(
7/29/2019 runge kutta.pptx
18/22
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)
7/29/2019 runge kutta.pptx
19/22
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)
7/29/2019 runge kutta.pptx
20/22
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)
7/29/2019 runge kutta.pptx
21/22
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.
7/29/2019 runge kutta.pptx
22/22
THANK YOU