Lecture NumericalMethodsForODE

7/29/2019 Lecture NumericalMethodsForODE

1/6

Tony Yang, Ph.D.

Department of Civil and Environmental Engineering

University of California, Berkeley

CE 249 Experimental methods instructural engineering

Topic: Numerical Methods for ODE

2

R.L. Burden and J.D. Faires, 2001, Numerical analysis 7th

edition, Thomson Learning. M.D. Greenberg, Advance engineering mathematics 2nd

edition, 1998, Prentice-Hall. G.V. Berg, 1989, Elements of structural dynamics,

Prentice-Hall. A.K. Chopra, 2001, Dynamics of structures, Prentice-Hall. J.L. Humar, 1990, Dynamics of structures, Prentice-Hall. R.W. Clough and J. Penzien, 1975, Dynamics of

structures, McGraw-Hill. N.M. Newmark and E. Rosenblueth, 1971, Fundamentals

of earthquake engineering, Prentice-Hall. J. Strain, 2004, MATH 228a lecture notes, University of

California, Berkeley.

Text

3

Ordinary differential equation (ODE)

What is ODE? ODE is a function of one independent variable with

one or more of its derivatives.

e.g.

Initial value problem (IVP):

If we have the initial condition y(0) = y0, then wecan solve y(t).

Example:

( ) ( ) ( )( ),y t y t f t y t =

( )2 0, 0y y y y= =

( ) ( ) ( )

2

1 22

0

3 0

1 11

1 1, with 01

dyy dy dt c t c

dt y y

y t y y y tt c t y

= = + = +

= = =+ +

4

Initial value problem (IVP)

Is every initial value problem has an unique solution? No. e.g.

How do we find out if an IVP has an unique solution?

It has to satisfy the Lipschitz condition and

the solution must be continuous in a convex domain.

What does these mean?

( ) ( )2 , 0 0y t y y= =

( ) ( ) 20 ory t y t t= =

( ) ( )

( ) { }

1 2 1 2, ,

, is continuous in D = ,

ff t y f t y L y y L

y

f t y a t b y

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Example:

=> Yes there exists an unique solution to this IVP.

If fact, the solution is

Lets say we know there is an unique solution to the IVP,can we always have an general analytical solution?

Yes (if the ODE is linear).

No (if the ODE is nonlinear).

( ) sin ty t e=

( ) ( )cos ,0 1, 0 1y y t t y= =

( )

( ) { }

cos if 1

, is continuous in 0 1,

ft L L

y

f t y t y

= =

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Linear ODE:

Nonlinear ODE:

May not have a general explicit solution

What do we do?

=> Numerical solutions.

( ),y f y t=

( )y Ay g t= +

( ) ( ) ( )-At 00

=e yt A t s

y t e g s ds+


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7

Numerical methods

Equations of motion:

A general solution can be expressed as

where a, b and c are constant coefficient and R is theremainder of the expression.

In other words, the solution at n+1 step can be alinear combination of the previous displacement,velocity and acceleration.

my cy ky p+ + =

1 1

1

n n n

n l l l l l l

l n k l n k l n k u a u b u c u R

+ +

+= = = = + + +

8

Numerical methods

Explicit Euler method: ( )1 ,n n n nu u h f t u+ = +

t

y(t)

t0 t1

h

u0

t2

u1

t3

h h

u2

u3

( ),y f y t=

9

Numerical methods

Example:

Explicit Euler method ( )1 ,n n n nu u h f t u+ = +

( )

( ) ( )

2

2

1, 0 0.5,0 1

1 0.5 t

y y t y t

y t t e

= + =

= +

2.458

2.349

2.190

1.992

1.760

1.500

f(ti,ui)

2.458

1.988

1.550

1.152

0.800

0.500

ui

0.18272.6411.0

0.13872.1270.8

0.09851.6490.6

0.06211.2140.4

0.02930.8290.2

0.00000.5000.0

Erroriyiti

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Numerical methods

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

1

1.5

2

2.5

3

t

y

exact

dt = 0.1

dt = 0.2

As dt decreases, error decreases

=> u(t) arropximate y(t)

11

Numerical methods

Explicit Euler method:

Example:

Using explicit Euler method

If h>0.05

( )1 ,n n n nu u h f t u+ = +

( )

( ) 2020 , 0 0.5

0.5 t

y y y

y t e

= =

=

( ) ( )1 20 1 20n n n nu u h u h u+ = + =

u (Numerical solution became unstable)

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Numerical methods

Check convergence of numerical methods

Consistency: Error produced in each step is bounded. As step size -> 0, error -> 0 => the method isconsistent. We can also quantify how fast the

error is reducing = order of accuracy. Stability:

Check the error is bounded. Any small error in theinitial condition will not making a huge error to thefinal solution.

Consistency + Stability = Convergence.


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Taylor series expansion

General formula

Numerical method

( ) ( ) ( )( )( )( )

( ) ( )

2

0 0

0 0 0

3

0 0

2!

3!

f x x xf x f x f x x x

f x x xHOT

= + +

+ + +

"

2

12!

nn n n

y hy y h y HOT+ = + + + +

"

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Consistency


=> This shows the explicit Euler method is consistent.(as h->0, error ->0). In addition, the method is2nd order accurate. In other words, if the step sizeis reduced by , the error is reduced by .

( ) ( )( )

2

1

1 1 1

2

2

1

2!

, ,2!

2!

nn n n

n n n

nn n n n n

nn

y hy y h y HOT

e y u

y he h f t y f t u HOT

y he HOT

+

+ + +

+

= + + + +

=

= + + + +

= + +

"

"

"

( )1 ,n n n nu u h f t u+ = +

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Stability


Since

( ) ( )( )

( ) ( )( )

( )

( ) ( )( )

( )( )

2

1

1

2

1

1

1

0

, ,2!

, ,

1

1 1 1

1 11

nn n n n n n

n n n n n n n

n n n n

n

n

n

y he e h f t y f t u HOT

e e h f t y f t u

e hL e hL e

hL e hL

hLhL e

hL

+

+

++

= + + + +

+ +

+ + + +

+ + + +

+ + +

"

#

( )1 ,n n n nu u h f t u+ = +

( ) ( )21

1 12

hLhL e hL hL+ = + + +"

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Stability


This shows error associate with the initial condition isindependent of the step size, which means the methodis stable.

Stability + Consistency => Convergence. This shows theexplicit Euler method is convergence with 2nd order

accuracy.

0

0

1

1

hLnhLn

n

LTLT

ee e e

hL

ee e n

TL

+

+

( )1 ,n n n nu u h f t u+ = +

Independent of h

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Stability

Example:

This method is 4th order accurate!

( )

( )

( )

( )

( )

1 1 1

2 3 45

2 3 4

5

32 4

4 5

4 5 4 2

42 6 24

52 6 24

4 2 23

1

6

n n n n n n

n n n n n n

n n n n n

n n n n n

n

y y y h f f

h h hy y h y y y O h y

h h hy y h y y y O h

hh y y y h y h y O h

y h O h

+ = + +

= + + + + + +

+ + +

+ + +

= +

( )1 1 14 5 4 2n n n n nu u u h f f + = + + +

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Stability

Example:

check against

use the proposed numerical method

if

the numerical method is unstable!

( ) 0 1y t y = =

( )1 1 14 5 4 2n n n n nu u u h f f + = + + +

( )0, 0 1y y= =

1 15 4

n n nu u u+ =

( ) ( )

( ) ( )

( ) ( )

2

3

4

5 1 4 1 1 4

5 1 4 1 4 1 21

5 1 4 4 1 21 1 104

u

u

u

= + =

= + = +

= + =

0 11, 1u u = = +


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Stability


The error at n step is independent of the step size, thisshows the explicit Euler method is stable.

In other words, the error introduced at the initial timestep should not increase by decrease the step size (=increase number of steps).

e.g. T = 0 to 10 case A: dt = 0.1 (need 100 steps) case B: dt = 0.05 (need 200 steps) any error introduced in step 0 should not increasefrom case A to B.

0

1LTLTn

ee e e n

TL

+

( )1 ,n n n nu u h f t u+ = +

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Stability


Example:

Using explicit Euler method

If h>0.05

If h


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Runge-Kutta methods

Orders of accuracy vs. number of stages (ERK methods)

P 4 -> S = PP = 5 -> S 6

P = 6 -> S 7P = 7 -> S 9P = 10 -> S 17

This is why the 4th order accurate ERK method is mostefficient.

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Multi-step methods

Adams-Bashforth 2-step explicit method



( ) ( )( ) ( )21 , 1, 1 132

n n n n n n n

hu u f t u f t u O h+ += + =

( ) ( )

( ) ( )

( )

, 1, 1

1

2, 2 3, 3

4

1

55 59

24 37 9

n n n n

n n

n n n n

n

f t u f t uhu u

f t u f t u

O h

+

+

= + +

=

( ) ( ) ( )( )

( )

1 , 1, 1 2, 2

3

1

23 16 512

n n n n n n n n

n

hu u f t u f t u f t u

O h

+

+

= + +

=

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Multi-step methods

Adams-Moulton 2-step implicit method



( ) ( ) ( )( )

( )

1 1, 1 , 1, 1

3

1

5 812

n n n n n n n n

n

hu u f t u f t u f t u

O h

+ + +

+

= + +

=

( ) ( )

( ) ( ) ( )

( )

1, 1 ,

1

1, 1 2, 2 3, 3

51

251 646

720 264 106 19

n n n n

n n

n n n n n n

n

f t u f t uhu u

f t u f t u f t u

O h

+ +

+

+

+ = + +

=

( ) ( )

( ) ( )( )

1, 1 , 4

1 1

1, 1 2, 2

9 19

24 5

n n n n

n n n

n n n n

f t u f t uhu u O h

f t u f t u

+ +

+ +

+ = + = +

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2nd order system

Example:

=> Systems of 1st order equations.

1

2

2

2

2 1

2

2

0 1 0

2 1

n n

n n

x qlet X

x q

xqX

p x xq

X X p

= =

= =

= +

22n nq q q p + + =

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Central difference method

This method uses a finite difference approach

Substitute to the equations of motion

Rearrange the equation

Initial conditions

1 1 1 1

2

2and

2

n n n n nn n

u u u u uu u

h h

+ + += =

1 2 32 2 2

2, ,

2 2

m c m c mc c c k

h h h h h= + = =

n n n nmu cu ku p+ + =

2

0 0 00 1 0 0 0and

2

p cu ku hu u u h u u

m

= = +

2 1 31

1

n n nn

p c u c uu

c

+

=

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Central difference method

For nonlinear system

Stability of the central difference method

1

n

h

T

Lecture NumericalMethodsForODE

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