# Numerical Analysis/stability of Multistep methods

For multistep methods, the problems involved with consistence, convergence and stability are complicated because of the number of approximations involved at each step. In the one step method, the approximation depends only on the previous approximation , whereas the multistep methods use at least two of the previous approximations, and the usual method that are employed involve more.

# Stable Multistep Method

edit## Characteristic polynominals

editFor the multistep method:

- ,
- ,

the polynominal

- ,

is called the characteristic polynominal of the above mutistep method.

The stability of a multistep method with respect to round-off error is dictated by the magnitudes of the zeros of the characteristic polynominal.

## Root condition

editLet denote the (not necessarily distinct) roots of the characteristic equation

- ,

associated with the multistep difference method

- ,

If the absolute value , for each i=1,2,...,m, and all roots with abosolute value 1 are the simple roots, then the difference method is said to satisfy the root condition.

## Stability Definitions

edit- Methods that satisfy the root condition and have as the only root of the characteristic equation of the magnitude one are called strongly stable.
- Methods that satisfy the root condition and have more than one distinct root with magnitude one are called weakly stable.
- Methods that do not satisfy the root condition are called unstable.

## Stable Multistep methods

editA multistep method of the form

- ,
- ,

is stable if and only if it satisfies the root condition. Moreover, if the difference method is consistent with the differential equation, then the method is stable if and only if it is convergent.

## Examples

edit### Example1

editThe fourth-order Adams-Bashfoth method is

where The characteristic equation is

so we find the characteristic roots to be

Thus the 4th-order Adams Method satisfies the root condition and is strongly stable.

### Example 2

editMilne's Method is

where The characteristic equation is

so the characteristic roots are

Thus the 4th-order Milne's Method satisfies the root condition, but it is only weakly stable.

# Absolute Stability & Region of Absolute stability

edit## Definition

editApply a numerical method to initial-value problem

Suppose that a round off error is introduced in the initial condition for this method. At the nth step the round off error is . Provided the step size h is chosen to satisfy , the numerical method is said to be absolutely stable for the step size h. And is said to be region of absolute stability for numerical method.

## Region of absolute stability for one-step method

editIn general, when a one-step method is applied to the test equation, a function exists with the property that the difference method gives

- .

The initial round-off error and the round-off error at the ith step will satisfy which is .

The inequality will hold if . Thus, region R of absolute stability for a one-step method is

## Region of absolute stability for multistep methods

editConsider the k-step method

Applying it to

we get

The characteristic polynominal of the method is

The region R of absolute stability for a multistep method is

- .

### Example

editDetermine the region of absolute stability for two step method .

Applying this method to: , gives

The characteristic polynominal of above method is

All the zeros of the characteristic polynominal are

Thus the region of absolute stability for this method is .

# Exercises

edit## Exercise 1

editFind the region of the absolute stability of Euler's method:

Solution:

Applying Euler's method to test equation: , gives

The region R of absolute stability for Euler's method is

That is .

## Exercise 2

editFind the region R of the absolute stability of Trapezoidal method

Solution:

Applying Trapezoidal method method to test equation: , gives

which simplifies to

where , so .

The inequality will hold if . The region R of absolute stability is

## Exercise 3

editMake a table of the interval of absolute stability for four one-step methods: Euler's method, Backward- Euler, M-Euler and Trapezoidal method.

Solution:

Methods the interval Euler Backward-Euler M-Euler The trapezoidal method

## Exercise 4

editFind the Region of absolute stability of Adams-Bashforth explict Four-step Method

Solution:

Applying Adams-Bashforth explict Four-step Method to the test equation gives

The characteristic polynominal of above method is

The region R of absolute stability is

# Quiz

editThe following is a quiz covering information presented on the associated multistep method.