Stability
The boundless of the absolute stability region is hard to determine for any method.
There are some rules for linear multistep methods and Runge-Kutta methods, but for other
kind of methods the problem is more complicated. Since the absolute stability domain
contains always a segment from the real axis or entire negative real half-axis, we can
form us an opinion about the boundless of A, analyzing the boundless of
A'={hl in R | l<= 0, the method applied to the test equation y'=l y, with
constant stepsize h, produces an approximation yn goes to 0, when n goes to infinity}
since A'=A intersction with R. Note that A'' is a subset of A', where
A"={hl in R |l<= 0, the method applied to the test equation y'=l y,
with constant stepsize h, produces an approximation yn for which
|y n+1 |<= |yn|, for all n>= 0 }
Maintaining a fixed value for h, say h*=0.01, we can find the smallest value of A", varying the smallest (negative) real value of l
and computing the yn values for n=1,\ldots,N, where N is a fixed large integer. We start
with an arbitrary value for l, say l=-100. We test if, for this l-value
we get a strictly monotonically decreasing approximate solution. If the answer is yes, we replace l with 2l,
and we apply again the method to the new equation. Otherwise, we replace l by l/2, and so on.
Superior limits and inferior limits for searching l must be imposed
(there are methods with A" the empty set or A"=R - ).
If the properties of the sequence yn are changed, we have an estimation of the smallest l for which h*l in A".
Suppose the final estimation the smallest value of l is l*.
In this case, EpODE will report the boundless of the absolute stability hl>h*l*.
Example for the method DIRK4.