Timeline for The definition of stiff ODE system
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
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| Jan 15, 2017 at 8:12 | comment | added | David Ketcheson | There is a new, better answer to this question in this paper. | |
| Jan 24, 2015 at 16:44 | comment | added | Rhei | This might be useful: blogs.mathworks.com/cleve/2014/06/09/… | |
| Feb 2, 2012 at 13:24 | history | edited | faleichik |
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| Jan 23, 2012 at 11:15 | vote | accept | faleichik | ||
| Jan 23, 2012 at 11:15 | answer | added | faleichik | timeline score: 2 | |
| Jan 22, 2012 at 1:07 | answer | added | David Ketcheson | timeline score: 2 | |
| Jan 21, 2012 at 20:15 | comment | added | faleichik | @David: In response to the note added to your previous comment. I tried to give the simplest example. In case of autonomous linear system take the second problem from my example, i. e. 2d linear system with eigenvalues {-1000000, -999999}. I do not see how to modify this problem for numerical integration with traditional explicit methods and big timesteps, although the solution is virtually constant except very thin boundary layer. And would appreciate if anyone could make it. | |
| Jan 20, 2012 at 19:50 | comment | added | Geoff Oxberry | @JedBrown I know. It's common terminology in the model reduction community, especially in combustion. | |
| Jan 20, 2012 at 19:38 | comment | added | Jed Brown | @GeoffOxberry You have to know something about the physics to classify time scales as being interesting or not. "Slow manifold" is a commonly used term in multiphysics simulation, but nobody is literally parametrizing any such beast. Instead, the term merely refers to long-term transient behavior of interest, often by nearly balancing two faster processes. A slowly moving vortex is an example in which the (fast) gravity wave is almost in equilibrium with the nonlinear convection process, giving rise to interesting long-term behavior. | |
| Jan 20, 2012 at 19:11 | comment | added | Geoff Oxberry | @JedBrown: You're right, but locating that "slow manifold" is impossible in all but the most trivial systems. Even determining its dimensionality is often nontrivial. (Otherwise, my thesis would have been a cakewalk.) | |
| Jan 20, 2012 at 18:43 | comment | added | Jed Brown | @faleichik The $\cos x$ part of your example fixes the time scale of the "slow manifold" (which is likely the time scale that you are interested in, though it's conceivable that you would be interested in much shorter time scales). I don't believe it is possible to define stiffness without choosing an observational time scale (perhaps implicitly by stating properties that you want to conserve over longer times). The stiffness ratio only quantifies the scale separation between the fastest and slowest time scales of the autonomous system. | |
| Jan 20, 2012 at 18:32 | answer | added | Jed Brown | timeline score: 15 | |
| Jan 20, 2012 at 15:16 | comment | added | faleichik | @David: I cannot agree with you. Take for example one-dimensional problem y'=-50(y-cos x). The "eigenvalue" is -50. One can not solve this problem with explicit Euler with stepsizes greater than 2/50. If we replace -50 with -50000 the restriction on the timestep becomes 2/50000. What "units" can we choose here to overcome this barrier? | |
| Jan 20, 2012 at 11:28 | history | tweeted | twitter.com/#!/StackSciComp/status/160322880360546305 | ||
| Jan 20, 2012 at 11:24 | history | edited | faleichik | CC BY-SA 3.0 |
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| Jan 20, 2012 at 10:55 | comment | added | Geoff Oxberry | I think between the comment by @DavidKetcheson and the several sources I quoted, you'll see that stiffness ratio is just a guideline. It's not perfect; that's why it's not in the definition. It happens to be a characteristic of many, but not all, stiff systems. And as for the second part, I think you'll be hard-pressed to find it unless it has special structure or arises in an application. I gave you an example of such an application where the stiffness ratio is not always large, and I encourage you to look at Hairer and Wanner's book. | |
| Jan 20, 2012 at 9:18 | answer | added | Geoff Oxberry | timeline score: 13 | |
| Jan 20, 2012 at 8:57 | comment | added | David Ketcheson | Welcome to scicomp.se. Your questions are answered thoroughly on wikipedia: en.m.wikipedia.org/wiki/Stiff_equation | |
| Jan 20, 2012 at 8:15 | history | asked | faleichik | CC BY-SA 3.0 |