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There is a class of stiff initial value problems for ODEs that have small Lipschitz constants, slowly-changing solutions, but very long interval of integration. The only practical example of such a problem that I am aware of is the Becker-Döring equation from Hairer and Wanner's book. Are there any other interesting examples of such problems? |
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A differential equation $y'=F(y)$ may be called stiff in $[0,T]$ if, for some $y$, the matrix $TF'(y)$ has some huge eigenvalues with negative real part. Thus the simplest example is $y'=-y$. Indeed, the equation $y'=-10^6 y$ for $t\in[0,1]$ is obviously stiff, but $y'=-y$ is as stiff for $t\in[0,10^6]$. An explicit method such as Euler's needs the same number of integration steps to achieve the same accuracy. Whether the example is interesting depends on your interests. But by the same argument, all dissipative ODEs become stiff when integrated over huge time intervals. As all ODEs are (by definition) dissipative if the trajectory approaches a lower-dimensional stable manifold, there is no paucity of realistic examples. |
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There is the case of production and destruction of radioactive isotopes in nuclear fuel. You typically use nuclear fuel for more than 3 years (hence, you fission isotopes into radioactive daughter products that decay with certain half-lives that can go from microseconds to years or more). So during the irradiation, if you want to track all these isotopes (to get an idea on decay heat and radioactivity level), you are solving problems that have very large differences in time-constants. And then, if you need to evaluate the impact of long term storage (or even geological storage) of this spent fuel, you want to simulate this radioactive decay for $10^2$-$10^6$ years. My apologies for the self-advertisement, but you can have a look at this paper for more information on the application of an implicit Runge-Kutta method (RADAU-IIA) to these types of problems. |
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