Talk:Finite difference method

What to do about this article?

This article has some nice info missing from finite difference which is more focused on the general theory of finite differences, not necessarily applied to numerial analysis.

This article shows explicit formulas for several finite difference schemes and outlines how one could obtain other finite difference schemes.

The big question, should it be merged to finite difference, or should it stay by itself? I would be inclined towards the second, but then one would need to modify both this and finite difference so that it is clear which material belongs where and for what reason.

Eventually this may grow into a full-blown finite difference method article which we are missing, provided anybody is willing to do the work. Comments? Oleg Alexandrov (talk) 03:24, 25 July 2006 (UTC)[reply]

Or maybe finite difference scheme could be merged into finite difference, but I don't feel comfortable enough to do it. Oleg Alexandrov (talk) 03:26, 25 July 2006 (UTC)[reply]

I think a radical reorganization is in order. I propose the following. Finite-difference method can contain the standard explicit 2nd order scheme for heat equation, some analysis (consistency, stability, convergence), other schemes like implicit in time and Crank-Nicholson (up to here this is the content of Finite difference#Example: the heat equation), discussion on how to handle boundary conditions, the centered-difference scheme for the Laplace equation and the same for the wave equation. This is easily enough for a decent article. We might want to add something about FD methods for ODEs.

Finite difference should contain the definitions of the forward, backward and centred difference, how they approximate the derivative, and perhaps higher-order approximations (basically the contents of this article, finite difference schemes). This should perhaps be merged with difference operator, though that is a rather nice article so we should be careful there.

Most redirects should be changed as a result. -- Jitse Niesen (talk) 11:49, 31 July 2006 (UTC)[reply]

If

x_{1}

is my own experience with finite differences, and

x_{2}

is your experience with the same thing, then the finite difference

x_{2}-x_{1}

is actually infinite, so if you wish to reorganize things, that's fine with me. :) I also agree that reorganization is in order. Oleg Alexandrov (talk) 18:25, 31 July 2006 (UTC)[reply]

Ambiguous Terms

Does anyone have sufficient practical or theoretical experience on the closing statement:

Usually the Crank-Nicolson scheme is the most accurate scheme for small time steps. The explicit scheme is the least accurate and can be unstable, but is also the easiest to implement and the least numerically intensive. The implicit scheme works the best for large time steps.

What constitutes "small" and "large" time steps? Any boundaries on when the Crank-Nicolson scheme is or isn't the most accurate for small time steps? (When is it "usual" and when is it "unusual"?)

--KnockNrod 16:42, 3 November 2006 (UTC)[reply]