Talk:Explicit and implicit methods

Untitled
Oleg Alexandrov's changes to my last edit improve the article. On two minor points I disagree, and have reverted:

"mathematical" simulation: as distinct from, for example, an electrical LCR circuit to simulate a differential equation.

The "next instance of time" is just wrong; I think you mean "instant". My "interval" is also plain wrong: I was thinking in terms of a delta-T, which it isn't (it's late at night, or maybe I'm just stupid).

Pol098 03:52, 11 March 2006 (UTC) (amended)


 * Yes, "instant" is best! So we agree. Oleg Alexandrov (talk) 04:03, 11 March 2006 (UTC)


 * I like your final edit. I think the wording is much better like this. Your article I see; nice one. Pol098 04:12, 11 March 2006 (UTC)

Quadratic solution
Shouldn't the denominator of equation (4) be 2? --anon
 * Yes! Fixed, thanks. Oleg Alexandrov (talk) 15:15, 30 June 2006 (UTC)

Accuracy and CFL condition
I've never used implicit methods, but am aware of their use in avoiding the 'stiffness' of stiff systems. Could someone add to the article a comment on their accuracy regarding by how much one has violated the CFL condition? I'm told that it's something like $$O(\partial t / \lambda_{min})$$, but don't know enough about it to be sure.

...I'm also told that one only ever *uses* implicit methods when you're seeking a final steady state, since you know when you've found the right answer. This is plainly not true, but again, could someone comment on their use in strongly time-dependent systems? 7daysahead (talk) 21:51, 10 July 2010 (UTC)

No Exact Solution?
I object to the statement "In the vast majority of cases, the equation to be solved when using an implicit scheme is much more complicated than a quadratic equation, and no exact solution exists". In typical cases, an exact solution certainly exists, but there is just not a formula to compute it. The wording of the end of the sentence should to changed as to not be confusing. —Preceding unsigned comment added by 208.120.216.154 (talk) 02:26, 5 October 2010 (UTC)
 * Indeed, I made some changes now. Nico (talk) 07:49, 8 April 2014 (UTC)

Assessment comment
Substituted at 14:51, 29 April 2016 (UTC)

Some bigger edits planned
I am new to editing on Wikipedia, and just wanted to give a heads up for some major edits I intend to make on this article. Hopefully we can start a discussion.

I will introduce the standard initial value problem in the introduction, $$y'(t) = f(t,y(t)), \quad y(0) = y_0$$. Currently there are letters $$F$$, $$G$$ being used in conflicting contexts, and the letters $$F$$, $$G$$, and $$Y$$ are abandoned after this section. I'll also move the IMEX discussion till later.

In the section "Illustration using the forward and backward Euler methods", I think we should stick to the abstract IVP instead of the quadratic right hand side. While I think specific right hand sides can be useful, I don't think the entire section should be about it. Maybe ultimately the example right hand side could be it's own section.

I do agree with the pedagogical choice of sticking with simple methods such as implicit euler and explicit euler, especially for the IMEX example. However, there are also already articles about implicit and explicit Euler, so we shouldn't spend too much time repeating information about these. We should really focus on essential differences and advantages between implicit and explicit methods.

I have a problem with including Crank-Nicolson in a section all about Euler, and with the statement "Crank Nicolson can be viewed as a form of more general IMEX (Implicit-Explicit) schemes." I have never seen the interpretation of CN as an IMEX method. It is usually viewed as an implicit method, but it CAN be used as the implicit component of an IMEX method, such as Crank-Nicolson/Adams Bashforth 2 (CN/AB2), or Crank-Nicolson/Leapfrog(CNLF). We should have second order examples though, so maybe we just change the name of the section.

Let me know what you think.

Fish sounds (talk) 13:56, 2 August 2020 (UTC)