lecture 22 :initial value problems

8/13/2019 Lecture 22 :initial value problems

1/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.1

LECTURE 22

MULTI STEP METHODS

Solve the i.v.p.

Multi step methods use information from several previous or known time levels INSERT FIGURE NO. 100

dydt ------ f y t ,( )= y t o( ) yo=

y

t

y0

t 0 t 1 t 2 t 3 t 4

y1

y2 y3

y4


2/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.2

Open Formulae (Adams-Bashforth)

explicit (non-iterative)

can have stability problems

Closed Formulae (Adams-Moulton)

implicit (iterative)

much better stability properties than open formulae

Predictor-Corrector Methods

1 cycle predictor open formula

2-3 cycles corrector closed formula

superior to either open or closed formulae separately


3/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.3

Open Formulae

Derivation

Develop a forward Taylor series of about

However by denition and etc., thus

(1)

Now replace the various derivatives of with backward difference approximations

1st Order Accurate Adams Open Formula

Retain only the rst two terms in Equation (1)

Same as the explicit or 1 st order Euler method

y t j

y j 1+ y j t + y jt ( )22!

------------- y jt ( )33!

------------- y j + + +=

y j f j= y j f j=

y j 1+ y j t f jt 2!----- f j

t ( )23!

------------- f j + + + +=

f j

y j 1+ y j t f y j t j,( )+=


4/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.4

2nd Order Accurate Adams Open Formula

Use a backward difference approximation for

Substituting we obtain:

f j

f j f j f j 1

t ----------------------- t

2----- f j O t ( )2+ +=

y j 1+ y j t f jt 2

----- f j f j 1

t ----------------------- t

2----- f j O t ( )2+ + t ( )

2

3!------------- f j+ +

+=

y j 1+ y j t 32--- f j

12--- f j 1

512------ t ( )3 f j O t ( )4+ + +=

y j 1+ y j t 32--- f j12--- f j 1 O t ( )3+ +=

y j 1+ y j t32--- f y j t j,( )

12--- f y j 1 t j 1,( ) O t( )3++=


5/24


6/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.6

Example Application of 2nd Order Adams Open Formula

Problem

i.c. gives us ,

Apply 2nd order R.K. (Improved Euler) to start the calculations

Now we know ,

dydt ------ f y t ,( )= y t o( ) yo=

yo t o

t 1 t o t +=

y1 * yo t f yo t o,( )+=

y1 yo t 12--- f yo t o,( ) f y1 * t 1,( )+[ ]+=

y1 t 1


7/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.7

From time level to ; apply 2nd order Adams Open Formula

Now we know ,

From time level to ; apply 2nd order Adams Open Formula

Now we know ,

j 1= j 1+ 2=

t 2 t 1 t +=

y2 y1 t 32--- f y1 t 1,( )

12--- f yo t o,( )+=

y2 t 2

j 2= j 1+ 3=

t 3 t 2 t +=

y3 y2 t 3

2

--- f y2 t 2,( )1

2

--- f y1 t 1,( )+=

y3 t 3


8/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.8

INSERT FIGURE NO. 102

y(t)

t

y0

t 0 t 1 t 2 t 3

y1 y2

y3

*

2nd order R-K

2nd order Adams Open

2nd order Adams Open

y1*


9/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.9

3rd Order Accurate Adams Open Formula

Substitute into Equation (1) for and using rst order accurate approximationscarrying a sufcient number of truncation terms as

Note that the local truncation term for equation (1) must be

If you use rst order accurate approximations, you must carry a sufcientnumber of truncation terms in each approximation to the derivatives of , i.e. ,

, etc.

Alternatively you must use an approximation which in of itself is accurateenough such that the leading order truncation term substituted into equation (1)leads to a truncation term.

Collecting terms we have:

f j f j

f j f j f j 1

t ----------------------- t

2----- f j O t ( )2+ +=

f j f j 2 f j 1 f j 2+

t ( )2--------------------------------------------- O t ( )+=

O t ( )4

f j

f j

O t ( )4

y j 1+ y j t2312------ f y j t j,( )

1612------ f y j 1 t j 1,( )

512------ f y j 2 t j 2,( )+ O t( )4+ +=


10/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.10

INSERT FIGURE NO. 103A

Method is third order accurate

Need to start this method with 2 steps of a 3rd order accurate R.K. method

i.c. gives ,

R.K. starter gives , and ,

Now we can use the 3rd order Adams Open Formula

y(t)

t t j-2 t j-1 t j t j+1

y j-2

y j-1

y j y j+1

yo t o

y1 t 1 y2 t 2


11/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.11

Summary of Adams Open Formulae

General form of all Adams Open formulae

Formulae are explicit: is computed in terms of slope at , ,

All higher order Adams formulae are not self starting (dont know , , etc.).

Must start method with an appropriate order Runge-Kutta formula.

Open formulae are more efcient than R.K. methods of the same order since slopecalculations at a given point are re-used for at least several time steps.

Open formulae are very easy to implement

Open formulae may have stability problems

y j 1+ y j t f y j t j,( ) f y j 1 t j 1,( ) f y j 2 t j 2,( ) f y j 3 t j 3,( ) + + + +[ ]+=

y j 1+ t j t j 1 t j 2 ,

f 1 f 2

f y t ,( )


12/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.12

Closed Formulae

Derivation

Use a backward Taylor Series expansion for about . Hence

Noting that , , , etc.

(2)

Now substitute in backward difference approximations for the various derivatives ,

, etc.

y t ( ) y t t +( )

y j y j 1+ t dy

dt ------

j 1+ t ( )

2

2!------------- d

2 y

dt 2--------

j 1+

t ( )3

3!------------- d

3 y

dt 3--------

j 1+

+ +=

dydt ------

j 1+

f j 1+= d 2 y

dt 2--------

j 1+

f j 1+= d 3 y

dt 3--------

j 1+

f j 1+=

y j y j 1+ t f j 1+t ( )22!

------------- f j 1+t ( )33!

------------- f j 1+ ++=

y j 1+

y j

t f j 1+

t 2

----- f j 1+

t 23!

-------- f j 1+

++=

f j 1+

f j 1+


13/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.13

First Order Accurate Adams Closed Formula

Only consider the rst two terms in the backward Taylor series, Equation (2)

Notes Hence we are computing our updated point using the slope of our updated point

(versus the open formula where we used the old point to compute the slope)!

Formula is implicit since computation of involves evaluating the slope at

First and second order closed formulae are self starting while higher order closedformulae are not

If is a linear function of we can solve for directly:

e.g. Solve using the rst order Adams Closed Formula

y j 1+ y j t f y j 1+ t j 1+,( ) O t( )2+ +=

y j 1+

f y j 1+ t j 1+,( )

f y t ,( ) y y

dydt ------ y t 3+=

y j 1+ y j t y j 1+ t j 1+3+( )+=


14/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.14

If is a nonlinear function of we must iterate to get a solution

e.g. Solve using the First Order Adams Closed Formula

Establish an iterative solution

Note that this 1st order closed formula is self starting

y j 1+ 1 t ( ) y j t t j 1+3+=

y j 1+1

1 t ( )------------------- y j t t j 1+3+[ ]=

f y t ,( ) ydydt ------ y2 t 3+=

y j 1+

y j

t y j 1+

2 t j 1+

3+( )+=

y j 1+k 1+( ) y j t y j 1+k ( )( )2 t j 1+3+[ ]+=


15/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.15

2nd Order Accurate Adams Closed Formulae

Approximate using a rst order backward difference approximation:

Substituting into Equation (2) and collecting terms yields a locally 3rd and globally 2ndorder method:

INSERT FIGURENO. 103B

f j 1+

f j 1+ f j 1+ f j

t ----------------------- O t ( )+=

y j 1+ y jt 2

----- f j 1+ f j+[ ] O t ( )3+ +=

y j 1+ y j t1

2--- f y j 1+ t j 1+,( ) f y j t j,( )+[ ] O t( )3+ +=

y(t)

t t j t j+1

y j

y j+1


16/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.16

Notes

Formula is implicit, i.e. involves calculating

Same as trapezoidal rule or Crank-Nicolson Slope is averaged between the old time level and the new time level


Must iterate if is nonlinear in

Formula is self starting

y j 1+ f y j 1+ t j 1+,( )

j 1+

f j+1

j+1 j

f j

y

t

f y t ,( ) y


17/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.17

3rd Order Accurate Adams Closed Formula


Notes

Formula derived by taking Equation (2) and substituting backward nite differenceapproximations for and .

y j 1+ y j t5

12------ f y j 1+ t j 1+,( )

812------ f y j t j,( )

112------ f y j 1 t j 1,( )+ O t( )4+ +=

y(t)

t t j-1 t j t j+1

y j-1

y j

y j+1

f j 1+ f j 1+


18/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.18

Difference approximations used in the derivation must be of sufcient accuracy! Youcan either:

Apply rst order approximations and carry a sufcient number of truncationterms

Apply higher order accurate approximations

Either way, all truncated terms must be consistently !

Formula is implicit since it involves computing using the slope

Must iterate if is nonlinear

Formula is third order accurate

This formula and all closed formulae of 3rd order or higher accuracy are not self starting must use R.K. methods of equal or better order accuracy as starters

It can be costly to iterate try to get a good rst guess for

try

another rst estimate could be obtained by using an open formula to computePredictor-Corrector Methods

O t ( )4

y j 1+ f y j 1+ t j 1+,( )

f y t ,( )

y j 1+0( )

y j 1+0( ) y j

y j 1+0( )


19/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.19

Predictor-Corrector Methods

Provide a rst estimate for the new solution using an open formula Predictor

Apply a closed formula using as an initial estimate to start the iteration

Corrector. Only need a few iterations since initial guess is very good!

Require that order of the corrector the order of the predictor

Advantages of P-C methods

Few iterations are required due to the excellent rst guess Stability properties are controlled by the corrector! Correctors or closed formulae

have excellent stability properties.

Error estimates are easy to obtain

If needed, use R.K. single step methods as starters

P-C methods overall are very efcient and are often used in production codes

y j 1+0( )

y j 1+0( )


20/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.20

Example

Predictor: 4th order Adams Open

Corrector: 4th order Adams Closed

When has converged specied value

Must start the method if higher than 2nd order starter

Accuracy of the starter must be equal or better than the corrector

Need to obtain values of and for the rst three steps beyond the i.c. ( , , ,with corresponding s such that the predictor can be used)

Thus, in this case we would have taken 3 starter steps to obtain , and using atleast a 4th order accurate R.K.

y j 1+0( ) y j t

5524------ f y j t j,( )

5924------ f y j 1 t j 1,( )

3724------ f y j 2 t j 2,( )

924------ f y j 3 t j 3,( )++=

y j 1+k 1+( ) y j t

924------ f y j 1+

k ( ) t j 1+,( )1924------ f y j t j,( )

524------ f y j 1 t j 1,( )

124------ f y j 2 t j 2,( )+++=

y j 1+k 1+( ) y j 1+k 1+( ) y j 1+k ( ) y j 1+ k 1+( ) y j 1+

y yo y1 y2 y3

y1 y2 y3


21/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.21

Improve the initial value for the corrector by estimating the truncation error of thepredictor modier

where

will be the value used to start the corrector

= the value from the predictor

= the nal solution (converged corrector solution) for the previous time step

= the unmodied predictor value from the previous time step

Estimate of the error in predictor comes from comparing the error series for thepredictor and the corrector

The varies depending on the method!

y j 1+0( ) y j 1+

0( ) y j y j0( )( )+=

y j 1+0( )

y j 1+0( )

y j

y j0( )


22/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.22

Estimate for the truncation error of the corrector total truncation error per time step(not total solution error)

allows modication of time step to obtain the desired accuracy

i.e. you can estimate whether to increase/decrease )

Summary of the steps in a P-C procedure

(1) Use Starter for only the rst few time steps

(2) Apply the Predictor

(3) Apply the Modier

(4) Apply the Corrector and iterate to a specied tolerance

(5) Repeat step (2)

E j 1+ 1 ( ) y j 1+ y j 1+0( )( )=

t

t


23/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.23

Implementation of Changes in Time Step

Change time step from to


If the formula contains only and no problem

If formula contains there is a problem since you dont know the value of thesolution at

Options:

Use R.K. method to start up method from the time level where you change the timestep, i.e. time level

Use a polynomial interpolator to get an actual continuous function of functionalvalues behind for some distance

t 1 t 2

j-1

t 1

t 2

j j+1

j 1+

j 1 y t j 1

t j


24/24

CE 341/441 - Lecture 22 - Fall 2004

p. 22.24

Comments Predictor-Corrector Methods

For most problems P-C method uses less computer time than R.K. methods of the sameorder

Many predictors have poor stability characteristics

Unstable predictor stability has only one time step to be unstable before it isimproved by the corrector

Hence it is important to have a stable corrector however many of the correctorshave very good stability characteristics!

lecture 22 :initial value problems

Documents