7.2. Interval and Region of Absolute Stability

Consider the linear multistep method

(7.1)\[ \begin{aligned} && \sum_{i=0}^{k} \alpha_iy_{j+i} ~&=~ h \sum_{i=0}^{k} \beta_if_{j+i} & \text{with}\ \alpha_k = 1 && \end{aligned} \]

and a test differential equation,

(7.2)\[ \begin{aligned} && \frac{\dy}{\dx} ~&=~ f(x,y) ~=~ \lambda\,y\,, & y(x_0) = y_0 && \end{aligned} \]

where \(\lambda\) is a (real or complex) constant.

Definition 7.1 (Absolute Stability)

A linear multistep method is said to be absolutely stable in a region \(\,\Re\,\) of the complex plane if for all \(\,h\lambda\,\) all roots of the stability polynomial \(\,\L (z,h\lambda)\) associated with the method satisfy

(7.3)\[ \begin{aligned} && \left| z_i(h\lambda) \right| \,&<\, 1, & i &= 1,\,2,\dots,k && \end{aligned} \]

Definition 7.2 (Relative Stability)

A linear multistep method is said to be relatively stable in a region \(\,\Re\,\) of the complex plane if, for all \(\,h\lambda\,\) all roots of the stability polynomial \(\,\L (z,h\lambda)\) associated with the method satisfy

(7.4)\[ \begin{aligned} && \left| z_i(h\lambda) \right| \,&<\, \left| z_1(h\lambda) \right|\, & \qquad i &= 2,\,3,\dots,k && \end{aligned} \]

where \(z_1(h\lambda) = +1\) is a simple root (i.e. the parasitic roots are less in magnitude than the principal root).

Note

The stability polynomial of a linear multistep method is also called characteristic polynomial, and it is calculated as

\[ \L (z, h\lambda) = \sum_{i=0}^k \left(\alpha_i - h\lambda \beta_i\right) z^i = \rho(z) - h\lambda \sigma(z) \]

In some textbooks, the stability polynomial is also expressed as \(\pi (z, h\lambda)\).

Definition 7.3 (Stability Interval)

An interval \((a,b)\) of the real axis is said to be an interval of absolute/relative stability if the method is absolutely/relatively stable for all \(\,h\lambda \in (a,b)\,\). If the method is absolutely unstable for all \(\,h\lambda \in (a,b)\,\), it is said to have no interval of absolute stability.

Note

An interval of absolute stability is simply the intersection of region \(\Re\) with the real axis.

Here it should be noted that the literature in this area contains many similar definitions. Sometimes the concept of absolute stability is defined so that the roots can have magnitude 1 and sometimes strictly \(\,<1\,\); it makes little difference, provided that the root with magnitude 1 is kept in mind when considering a problem and analysing the results - for references see Lambert (1971,1991) and Shampine (1994) from the reading list.

In the following, through examples, we show two methods for finding the interval and region of absolute stability for linear multistep methods.