Talk:Finite difference method
This article contains a good description of some aspects of finite differences but may be confusing in parts.
I think the notation for s, d, n, m in Figure 2 needs a bit more explanation here. It was possible to figure it out only from the 1 example where it is used below, but will not be clear to most readers.
The Mathematica code using PadeApproximant is nice, but there should be a reference to where this formula is derived.
The example below the code, it is not clear why this is 4th order accurate. Is there a general formula in terms of s,m,n,d?
Somewhere, in this example or in the discussion of Figure 2, it should be mentioned that in general this gives a coupled system for the unknown derivative values. How would this be used in practice? Some discussion of boundary conditions would help.
The Section titled "In meshfree settings": It seems that the main topic here is unstructured grids rather than mesh free?
In Applications, there are 3 major applications listed. Again, as above, I think there's a difference between unstructured (not lattice-based) grids and mesh free. Also, there is no mention here or below about two-point boundary value problems for ODEs, only time-dependent, the natural 1D analog of the elliptic PDEs that are discussed.
In equation (7) and below, the ODE is written as y' = f(t,y). This is quite confusing, not only because x has changed to t but even more because f(x) before was the function whose derivative is approximated, and now f represents values of the derivative rather than the function. When trying to reconcile the definitions of the LMM methods given below with Figure 2 this will be very confusing.
I think the formulas for Adams-Bashforth, etc. are not correct: shouldn't the values of d and n be switched in each case? For readers not already familiar with these methods it would help greatly to actually list some of these methods.
The quote from Richardson in the section on PDE methods seems a bit archaic and not particularly illuminating.
"For a PDE with so much viscous damping as the heat equation": I think it would be better to say "so much dissipation as the heat equation", since viscous refers to fluid viscosity whereas the heat equation has its own natural dissipation that has nothing to do with viscosity. It's true that the term "artificial viscosity" is often used in contexts outside of fluids, but for the heat equation there's nothing artificial about it.
Later in this section: " The use of very high orders of accuracy is often the best remedy (in particular for solutions that are not very smooth, and which therefore contain a relatively large amount of high modes)." This seems to imply that high order methods would be suitable for shock wave calculations, for example, whereas in fact high order methods using the simple finite differences presented in this article would be disasterous. More generally, if the solution is not very smooth in the sense that higher order derivatives may not be continuous, high order methods also won't help.
"Somewhat reminiscent of how time stepping schemes face stability conditions, both speed and reliability of elliptic solvers generally benefit if the FD approximation is diagonally dominant". I think it will be confusing to readers who don't know the theory to equate stability in time with convergence properties of iterative methods and in fact the two seem fairly unrelated to me.
I removed the link to my notes on Finite difference methods since they have been published by SIAM and should not have been available on my webpage, I didn't realize they were!
Very nice review of finite differences, by one of the leading experts in this field.
Some small comments:
The rather sudden appearance of the linear system (5) might be confusing to some. To make it more clear, it might be mentioned that we want
w_1 f(x_1) + ... + w_n f(x_n) = f'(x_c)
to hold for any polynomial of degree n (or less). As it is now, the
It is not clear why the Mathematica code works.
For convective PDEs reference to nonlinear FD schemes (flux-limiters, WENO) might be given.