The Mathematics of Stiffness. History and Evolution of a Concept.

2016. 09. 22. 10:15
Gustaf Söderlind

Abstract: The notion of stiffness was introduced in 1952, by Curtiss and Hirschfelder. It was recognized that some well-posed initial value problems could not be solved numerically except by using dedicated implicit methods. For many years, attempts were made to characterize stiffness. These attempts were contrived, sometimes right, sometimes wrong, or otherwise flawed. A significant problem was that mathematical properties of the problem were mixed with operational criteria, such as the choice of discretization method and the accuracy requirement.

In the end, it was recognized that every numerical analyst learns what is a stiff problem by solving a few. However, it is highly unsatisfactory that a proper definition does not exist. At least, there should be a single, mathematical necessary condition for when to look out for stiffness.

In this talk we outline the history of the concept of stiffness, and end by introducing a new, unexpected criterion. This is simple in the sense that it relates a problem property (completely defined in terms of the differential equation) to a time scale. The latter is, in turn, related to the rage of integration, or to the desired time step. It turns out that the mathematically necessary condition for stiffness depends only on the divergence of the vector field of the ODE, and on the range of integration. A new theory will be introduced and explained, with numerous examples of how stiffness can be identified also in strongly nonlinear systems.