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 .
(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.
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
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.
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.