Stability of Methods

To study stability, we use the sample equation.

The Euler method applied to this equation gives

The exact solution is

If y0 is not exact (i.e. the initial condition cannot be satisfied exactly), then use as the initial condtion the following expression.

Thus the Euler method gives the solution

The exact solution at the same time is

The global error is given by

Term A Term B

Use the following expressions from algebra.

Thus, term A is

This result is dominated for small t by the term

and it decreases as t Æ 0.

Term B, has the following behavior.

This means that

Term B oscillates in sign (i.e. the error oscillates in sign) from step to step if

To summarize, if delta-t is less than ­1/lambda, then the error in the Euler method does not oscillate in sign from step to step. If delta-t is greater than ­1/lambda but less than ­2/lambda, then the error oscillates in sign from step to step, but the method is stable. If delta-t is greater than -2/lambda, then the errors grow each step and the method is unstable.

Usually a linear, first-order problem is governed by

i.e. with the negative sign so that the eigenvalue l is positive. The above rule says for the Euler method:

A quick way to determine the stability limits is to write the solution in the form

where the function rmq is a rational polynomial (a ratio of two polynomials) and is found from inspection when the method is written in the form shown above.

The conditions for stability and to avoid oscillations are then

As an example, for Euler's method we have

This means that the Euler method has an upper bound for the stable step size. Thus when using a spreadsheet, for example, if the t is taken above this upper bound the solution error will grow from step to step and the solution will be no good. The stability limits for the different explicit methods are listed in the table. The figure shows the function rmq for different methods. The stability limits are obtained simply where the curve cuts rmq = ± 1, and the oscillation limits are where rmq = 0.

Take Home Message: To test for the stability of the method, just compare the rational polynomial rmq with the exponential.