Stiff Equation - Characterization of Stiffness

Characterization of Stiffness

In this section we consider various aspects of the phenomenon of stiffness. 'Phenomenon' is probably a more appropriate word than 'property', since the latter rather implies that stiffness can be defined in precise mathematical terms; it turns out not to be possible to do this in a satisfactory manner, even for the restricted class of linear constant coefficient systems. We shall also see several qualitative statements that can be (and mostly have been) made in an attempt to encapsulate the notion of stiffness, and state what is probably the most satisfactory of these as a 'definition' of stiffness.

J. D. Lambert defines stiffness as follows:

If a numerical method with a finite region of absolute stability, applied to a system with any initial conditions, is forced to use in a certain interval of integration a steplength which is excessively small in relation to the smoothness of the exact solution in that interval, then the system is said to be stiff in that interval.

There are other characteristics which are exhibited by many examples of stiff problems, but for each there are counterexamples, so these characteristics do not make good definitions of stiffness. Nonetheless, definitions based upon these characteristics are in common use by some authors and are good clues as to the presence of stiffness. Lambert refers to these as 'statements' rather than definitions, for the aforementioned reasons. A few of these are:

1. A linear constant coefficient system is stiff if all of its eigenvalues have negative real part and the stiffness ratio is large.
2. Stiffness occurs when stability requirements, rather than those of accuracy, constrain the steplength.
3. Stiffness occurs when some components of the solution decay much more rapidly than others.