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A differential equation is stiff if a numerical scheme requires a very small time-step in order to be stable for that equation.

However I don't understand why it is called stiff (sometimes rigid). Even the wikipedia page says it's more of a phenomenon than a mathematically definable property. One characterization is the stiffness ratio, defined on the wikipedia page as $$ \frac{|Re\bar{\lambda}|}{|Re\underline{\lambda}|}$$ where I think $\bar{\lambda}$ and $\underline{\lambda}$ refer to the eigenvalues with maximum and minimum real parts.

So why is it called this way? I don't know how to interpret this stiffness ratio. I was thinking of stiff as similar to "stringent" or "demanding" as a stiff person would be, which makes sense for the "requiring a small time step" definition.

How is the requirement of a small time-step related to the stiffness ratio?

blue_egg
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    See https://math.stackexchange.com/questions/3311190/i-dont-know-stiff, https://math.stackexchange.com/questions/140598/whats-the-intuition-behind-stiff, https://scicomp.stackexchange.com/questions/891/the-definition-of-stiff. Methods with a bounded stability region tend to converge to its boundary with counter-intuitive effects, just recently I described this mechanism here, step number linear in parameter. Does any of that reduce your question? – Lutz Lehmann Mar 07 '21 at 16:54
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    I'm not sure I still support all what I wrote in the first link. I was remembering having written https://math.stackexchange.com/questions/3384849/stability and https://math.stackexchange.com/questions/3599804/stiffness for a "philosophical" view on stability and stiffness. The second link above has a very clear answer to your question, can this be closed as duplicate of it, or can you enhance this question to a follow-up on that answer? – Lutz Lehmann Mar 07 '21 at 17:13
  • @LutzLehmann Yes it's a duplicate, thanks for the references! – blue_egg Mar 07 '21 at 17:15

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