As far as I know, the concept of stiffness is hard to define rigorously, but there are plenty of handwavy descriptions and motivating examples in the literature when it comes to linear differential equations. At the same time I have never seen an explicit and straightforward definition of a stiff nonlinear differential equation. That being said, I feel like there should be one, and I just haven't seen it yet. To outline, my questions are:

Is there such thing as stiff nonlinear differential equation? If so, how is it defined?

The most straightforward approach to define one is to use linearization, but I am not sure if this is a good idea as the accuracy of linearization will probably have an decisive impact on the region of absolute stability.

  • $\begingroup$ The definition of stiff that I usually go off is from Hairer, Wanner, which explains that a stiff problem is one where implicit schemes outperform explicit ones. Using this definition, the problem lies in that unless you use some iterative scheme (Newton, Picard etc) then there isn't really a way to solve non-linear differential equations with implicit solvers. In which case the definition given above doesn't hold anymore. $\endgroup$ – Mattos Dec 21 '15 at 14:17

There is a recent paper in which the problem of defining stiffness is discussed. Söderlind, Gustaf; Jay, Laurent; Calvo, Manuel Stiffness 1952–2012: sixty years in search of a definition. BIT 55 (2015), no. 2, 531–558.


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