# What are good examples of problems which are stiff due to very long interval of integration?

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?

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.

• Thank you Arnold. The examples you've mentioned are artificial. I mentioned Becker-Doring because this is an example of real model with the properties I've described. The feature of this particular model is that it is meta-stable, i. e. the solution changes relatively fast after a very long interval of steadiness. I am looking for something similar, some practical examples. – faleichik Sep 23 '12 at 21:02
• @faleichik If you scale time appropriately, lots of standard problems, including Van der Pol and Robertson, have the same property. – Jed Brown Sep 23 '12 at 21:55
• @faleichik: I had mentioned only one explicit (and perhaps artificial) example. But I mentioned that all realistic ODE have the same property when they are dissipative. – Arnold Neumaier Sep 24 '12 at 7:50

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.