ABSOLUTE STABILITY

So far we have considered the behaviour of numerical methods in the limit as the stepsize . However in practice we must deal with finite stepsizes. To illustrate the problems that might arise consider the mid-point method

This is a linear two-step method. In standard form the method is

thus

Checking consistency

The roots of  are given by  hence the method is both consistent and zero-stable and hence convergent.

Now consider the solution of the initial value problem

by the mid-point method using a stepsize . Using Maple we obtain the plot

Notice that the numerical solution becomes increasingly innacurate, oscillating about the exact solution, as t increases. This behaviour arises because the behaviour of the numerical solution does not mimic that of the exact solution. In this case the problem arises because of a spurious solution of the difference equation corresponding to the root  of . However the problem can also arise in one-step methods which have no spurious solutions.

The polynomial

is called the stability polynomial of the method. One of the roots  will correspond to the true solution, the other roots will lead to spurious solutions whose magnitude will have to be controlled to obtain stability.

Definition Absolute Stability

A numerical method is said to be absolutely stable for a given if all the roots of  lie within the unit circle.

A region  of the complex plane is said to be a region of absolute stability if the method is stable for all  in .

• Worked Example 7
Find and sketch the region of absolute stability for

(a) Euler's method,

(b) Trapezoidal method.

(a) For Euler's method

Thus

is shown below

(b) For the Trapezoidal method

Thus

giving the region shown below

For Runge-Kutta methods the stability polynomial has the form

where  is a polynomial for an explicit method and a rational function for an implicit method.

• Worked Example 8
Find and sketch the absolute stability region for the second order Runge-Kutta method

where

The stability polynomial is

For absolute stability we require that

In order to draw the region of absolute stability consider the boundary of . The locus of this boundary will be the set of complex numbers z such that

Thus

In order to obtain the region we need to plot the roots of the quadratic equation

for  in the range . This is best done on a computer. The resulting stability region is shown below:

The method outlined above is an example of the boundary locus method which is easily implemented for Linear Multistep methods as follows. The stability polynomial is

and hence

but on

Hence the locus of the boundary  is given by the set of complex numbers z satisfying

• Worked Example 9
Find the region of absolute stability for the Gear method

The stability polynomial is given by

Substituting  and solving

Now substitute  to obtain

which gives the plot

In order to determine whether  is the interior or exterior of the closed curve choose a point inside the curve and evaluate the roots of . In this case  consists of the exterior of the closed curve.

• Worked Example 10
Show that the mid-point method

has an empty region of absolute stability.

From above

Thus the stability polynomial is given by

Substituting  and solving

Now substitute  to obtain

which does not bound any region of the complex plane. Hence  is empty.