In order to answer the question of stiff ODEs, let's look at an example in A First Course in the Numerical Analysis of Differential Equations by Iserles [1].
Hoffman [2] states that stiffness appears for single linear and nonlinear ODEs, higher-order linear and nonlinear ODEs, and systems of linear and nonlinear ODEs. He lists several definitions for stiffness:
1. An ODE is stiff if the step size required for stability is much smaller than the step size required for accuracy.
2. An ODE is stiff if it contains some components of the solution that decay rapidly compared to other components of the solution.
3. A system of ODEs is stiff if at least one eigenvalue of the system is negative and large compared to the other eigen values of the system.
4. From a practical point of view, an ODE is stiff if the step sixe based on cost (i.e., computational time) is too large to obtain an accurate (i.e., stable) solution.
Hoffman [2] also gives a couple of examples.
In progress...to be continued.
References:
[1] A. Iserles. A First Course in the Numerical Analysis of Differential Equations. 2nd ed. Cambridge University Press, Cambridge, UK. 2009
[2] J. D. Hoffman. Numerical Methods for Engineers and Scientists. 2nd ed. Marcel Dekker, Inc., New York, NY. 2001
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