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