Regions
of absolute stability
In the study of the stability of ODE-solvers it is interesting to
visualize to region of absolute stability of a method. The region is a
set in the complex plane where a function, characteristic of the
method,
has a modulus less than one.
Here are the functions (polynomials) of two well-known Runge-Kutta
methods.
RKF45: p(h) = 1 + h + h2 / 2 + h3 / 6 + h4
/ 24 + h5 / 104
DOPRI(5, 4): p(h) = 1 + h + h2 / 2 + h3 / 6 + h4
/ 24 + h5 / 120 + h6 / 600
Just to clarify: the set of interest is the set of complex h such that
| p(h) | < 1.
Visualize the regions (in
the same plot) in a clear way.
