Timeline for Euler method (and others) for unbounded intervals
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jun 2, 2021 at 12:27 | vote | accept | alhal | ||
| Jun 2, 2021 at 11:38 | comment | added | David Ketcheson | I have never seen such a thing written down, but it might very well have been studied somewhere. I believe that for instance if you use the implicit Euler method for the problem $u'(t) = -u(t)$ then you can show uniform convergence over $[0,\infty)$. I think that a similar result will hold for any A-stable method applied to a problem with a single globally attractive equilibrium. Probably just A(0)-stability is required and there can be weaker conditions on the flow field. | |
| Jun 2, 2021 at 9:49 | comment | added | alhal | [...] that none of the texts I consulted developed numerical methods that fit into the context (of unbounded intervals) that needed. In this regards, could you please also link any references for the implicit methods on a problem with an asymptotically stable steady state you mentioned (or any other references in this regard that you know about)? | |
| Jun 2, 2021 at 9:46 | comment | added | alhal | I guess your last comment nails it, regarding to my question 1: I am not looking for a numerical method that works for any $f$ on a finite time interval, but rather one where (even if I may have to make a series of additional assumptions on $f$,) I get convergence on unbounded time intervals. The paper you linked states in section 2. "To study numerical schemes for eq. (1) in terms of their long-time, or even infinite time,accuracy, a departure from established numerical analysis notions of convergence of numerical approximations is needed", which is in line with my finding [...] | |
| Jun 1, 2021 at 16:40 | comment | added | David Ketcheson | >"an approximation on an unbounded interval must be possible" Yes, it's possible for instance using appropriate implicit methods on a problem with an asymptotically stable steady state. But numerical methods are designed to be convergent for general problems. I do happen to know of a recent paper that tries to establish the kind of convergence you are looking for (for a PDE discretization, in fact): arxiv.org/abs/2010.15365 | |
| Jun 1, 2021 at 12:50 | comment | added | alhal | If one can somehow prove that for "step-by-step" methods indeed there always exists an example on which they fail --perhaps an interesting question in its own right--- I assume there must be other approximation methods for unbounded intervals (e.g. by constructing a suitable set of "basis functions" and then combining them in some way to approximate the original solution that is sought)? | |
| Jun 1, 2021 at 12:49 | comment | added | alhal | Only regarding 1. I'm puzzled. You said "the step-by-step nature of the approximation precludes the possibility of uniform convergence on unbounded intervals" as " the error grows without bound as the number of steps taken goes to infinity". I'm not 100% this must be true; if, for example, the error would form a convergent series, then surely an approximation on an unbounded interval must be possible? | |
| Jun 1, 2021 at 12:35 | comment | added | alhal | Thank you very much for you insight. You were right to assume i had an IVP in mind (I edited my question to clarfiy this as well as other things). I'm completely happy with 2. and 3.; in particular I saw that the book by Hairer et al you linked states in the intro "An important shift of view-point came about by ceasing to concentrate on the numerical approximation of a single solution trajectory and instead to consider a numerical method as a discrete dynamical system which approximates the flow of the differential equation". This was exactly what I had in mind! | |
| Jun 1, 2021 at 10:04 | history | edited | David Ketcheson | CC BY-SA 4.0 |
Answer third part.
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| Jun 1, 2021 at 5:31 | history | answered | David Ketcheson | CC BY-SA 4.0 |