Multi-Step Methods: Explicit vs. Implicit Schemes & Adams-Bashforth/Moulton Comparison - P, Study notes of Computer Science

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Computational Methods

CMSC/AMSC/MAPL 460

Ordinary differential equations

Ramani Duraiswami,

Dept. of Computer Science

Multi-Step Methods

The principle behind a multi-step methodis to use past values,

y

and/or

dy/dx

to

construct a polynomial that approximatethe derivative function.

Multi-Step Methods

These methods are known as explicit schemes becausethe use of current and past values are used to obtain thefuture step.The method is initiated by using either a set of knownresults or from the results of a Runge-Kutta to start theinitial value problem.

Adam Bashforth Method

(4 Point)

Example

Consider

Exact Solution

The initial condition is:The step size is:

2

x

y

dy dx

−

=

x

2

2

2

e

x

x

y

−

=

( )

1

0

=

y

h

=

4 Point Adam Bashforth

The results are: Upgrade the values

(

)

(

)

(

)

( )

(

)

(

)

f

y

y

4 Point Adam Bashforth Method -

Example

The values for the Adam Bashforth

x

Adam Bashforth

f(x,y)

sum

4th order Runge-Kutta

Exact

0

1

1

1

Stability

It turns out that explicit methods are not very stable

This means that the solution may oscillate if we use largetime steps

So, if we wish to integrate over a large interval, and weneed to take many small steps to achieve accuracy, manyfunction evaluations are needed.

Implicit methods are usually more stable

Implicit Methods

There are second set of multi-step methods, which areknown as implicit methods. The implicit methods usethe future steps to modify the future steps.Since future data is used an iterative method must beusediterate an initial guess until convergenceCould use Runge-Kutta or Adams Bashforth to start theinitial value problem.

Implicit Multi-Step Methods

•The method uses what is known as a Predictor-Correctortechnique.•explicit scheme to estimate the initial guess•uses the value to guess the future y* and dy/dx= f(x,y)• Using these results, apply Adam Moulton method

Implicit Multi-Step Methods

Adams third order Predictor-Corrector scheme.Use the Adam Bashforth three point explicit scheme forthe initial guess.Use the Adam Moulton three point implicit scheme totake a second step.

[

] 2 i 1 i i i 1 i

−

−

f f f h y y

[

]

1

i

i

1

i

i

1

i

−

f f f h y y

4 Point Adam Bashforth

From the 4th order Runge KuttaThe 3 Point Adam Bashforth is:

178597 .

1

218597 .

1 ,

2 .

0

094829 .

1

104829 .

1 ,

1 .

0

0000 .

1

1 ,

0

=^ =

=

f f f

[

]

0 .

0

1 .

0

f

f

f

y

3 Point Adam Moulton

Predictor-Corrector Method

The results of explicit scheme is:The functional values are:

(

)

(

)

( )

[

]

(

)

(

)

250184 .

1

340184 .

1 ,

3 .

0

340184 .

1

121587 .

0

218597

. 1 3. 0 *

121587 .

0

1

5

094829 .

1

16

178597 .

1

23

12

1 .

0

=

=

= =

−

=

∆

f

y

y

3 Point Adam Moulton

Predictor-Corrector Method The values for the Adam Moulton Adam Moulton Three Point Predictor-Corrector Scheme x y f sum y* f* sum 0 1

• • • • • • • • • • 2. - 1. - 29. - 2. - 2. - 0. - 1. - 1. - 0. - 1. - 1. - 0. - 1. - 1. - 0. - 0. - 1. - 1. - 0. - 1. - 1. - 0. - 0. - 1. - 1. - 0. - 1. - 1. - 0. - 0. - 1. - 1. - 0. - 1. - 1. - 0. - 0. - 1. - 1. - 0. - 1. - 1. - 0. - 0. - 1. - 1. - 0. - 2. - 1. - 0. - 0. - 2. - 1. - 0. - 2. - 1. - 0. - 0. - 2. - 1. - 0. - 2. - 1. - 0. - 2. - 1. - 0. - 2. - 1. - 0.

3 Point Adam Moulton

Predictor-Corrector Method

The implicit AdamMoulton method gavesolution gives goodresults without usingmore than a three points.

Adam Moulton 3 Point Implicit Scheme

(^420) -2 -4 -6 -

0

1

2

3

4

X Value

Y Value

4th order Runge-KuttaExactAdam MoultonAdam Bashforth