eciv 301 programming & graphics numerical methods for engineers lecture 32 ordinary differential...
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ECIV 301
Programming & Graphics
Numerical Methods for Engineers
Lecture 32
Ordinary Differential Equations
Pendulum
W=mg
02
2
l
sinmg
dt
dm
02
2
l
sing
dt
d
OrdinaryDifferentialEquation
ODEs
02
2
l
sing
dt
dNon Linear
Linearization
Assume is small
sin 02
2
l
g
dt
d
ODEs
02
2
l
g
dt
dSecond Order
ydt
d
Systems of ODEs
0
l
g
dt
dy
ODE
15810450 234 x.xxx.y
5820122 23 .xxxdx
dy
ODE - OBJECTIVES
Cx.xxx.y 5810450 234
5820122 23 .xxxdx
dy
dx.xxxy 5820122 23
15810450 234 x.xxx.y
Undetermined
ODE- Objectives
15810450 234 x.xxx.y
Initial Conditions
10 y
ODE-Objectives
y,xfdx
dy
Given
.C.Iknowny,f 0
Calculate
xy
Runge-Kutta MethodsNew Value = Old Value + Slope X Step Size
hyy ii 1
Runge Kutta Methods
hyy ii 1
Definition of yields different Runge-Kutta Methods
Euler’s Method
hyy ii 1
y,xfdx
dy
ii y,xfLet
Sources of Error
Truncation: Caused by discretization
• Local Truncation• Propagated Truncation
Roundoff: Limited number of significant digits
Sources of Error
Propagated
Local
Euler’s Method
Heun’s Method
Predictor Corrector
2-Steps
Heun’s Method
Predict
Predictor-CorrectorSolution in 2 steps
hyy ii 10
ii y,xf
Let
Heun’s Method
Correct
Corrector
hyy ii 1
01ii y,xf
Estimate
2
01
iiii y,xfy,xfLet
Error in Heun’s Method
The Mid-Point Method
hyy ii 1
Remember:Definition of yields different Runge-Kutta Methods
Mid-Point Method
Predictor Corrector
2-Steps
Mid-Point Method
Predictor
Predict
22
1
hyy i
i
ii y,xf
Let
Mid-Point Method
Corrector
Correct
hyy ii 1
2
1
2
1 ,iiyxf
Estimate
2
1
2
1 ,iiyxf
Let
Runge Kutta – 2nd Order
hyy ii 1
21 3
2
3
1kk
y,xfdx
dy .C.Iknowny,f 0
ii y,xfk 1
hky,hxfk ii 12 4
3
4
3
Runge Kutta – 3rd Order
hyy ii 1 321 46
1kkk
y,xfdx
dy .C.Iknowny,f 0
ii y,xfk 1
hky,hxfk ii 12 2
1
2
1
hkhky,hxfk ii 213 2
Runge Kutta – 4th Order
hyy ii 1 4321 226
1kkkk
y,xfdx
dy .C.Iknowny,f 0
ii y,xfk 1
hky,hxfk ii 12 2
1
2
1
hky,hxfk ii 34
hky,hxfk ii 23 2
1
2
1
Boundary Value Problems
Fig 23.1FORWARD FINITE DIFFERENCE
Fig 23.2BACKWARD FINITE DIFFERENCE
Fig 23.3CENTERED FINITE DIFFERENCE
xo
Boundary Value Problems
x1 x2 x3 xn-1 xn...
Boundary Value Problems
xo x1 x2 x3 xn-1 xn...
),(2 112
012 yxfhyyy
Boundary Value Problems
xo x1 x2 x3 xn-1 xn...
),(2 222
123 yxfhyyy
Boundary Value Problems
xo x1 x2 x3 xn-1 xn...
),(2 332
234 yxfhyyy
Boundary Value Problems
xo x1 x2 x3 xn-1 xn...
),(2 112
21 nnnnn yxfhyyy
Boundary Value ProblemsCollect Equations:
),(2 112
012 yxfhyyy
),(2 222
123 yxfhyyy
),(2 112
21 nnnnn yxfhyyy
BOUNDARY CONDITIONS
T0 T5T0 T5
Example
02
2
TTcdx
Tda
x1 x2 x3 x4
Example
02
12012
TTc
h
TTTa
aTchTchTT 20
212 2
T0 T5T0 T5
x1 x2 x3 x4x1 x2 x3 x4
Example
02
22123
TTc
h
TTTa
aTchTchTT 21
223 2
T0 T5T0 T5
x1 x2 x3 x4x1 x2 x3 x4
Example
02
32234
TTc
h
TTTa
aTchTchTT 22
234 2
T0 T5T0 T5
x1 x2 x3 x4x1 x2 x3 x4
Example
02
42345
TTc
h
TTTa
aTchTchTT 22
234 2
T0 T5T0 T5
x1 x2 x3 x4x1 x2 x3 x4
Example
52
2
20
2
4
3
2
1
2
2
2
2
2100
1210
0121
0012
TTch
Tch
Tch
TTch
T
T
T
T
ch
ch
ch
ch
a
a
a
a
T0 T5T0 T5
x1 x2 x3 x4x1 x2 x3 x4