# Coefficients of a difference method

• Oct 21st 2013, 09:18 AM
phys251
Coefficients of a difference method
"Find the values of alpha, beta, and gamma in the following difference method:

$w_{j+4} - w_{j+2} +\alpha (w_{j+3} - w_{j+1}) = h[\beta (f_{j+3} - 4f_{j+1}) + \gamma f_{j+2})$"

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I started by Taylor expanding all the terms in h. For example:

$w_{j+2} = w(x_j + 2h) = w_j + 2hf_j + \frac{9}{2}h^2 f_j' + \frac{9}{2}h^3 f_j'' + O(h^4)$ and

$f_{j+2} = f(x_j + 2h) = f_j + 2hf_j' + 2h^2 f_j'' + O(h^3)$

and so on for the other terms.

My two questions are:

1. What do I do with the f' and f'' terms? Should f' become (f_(i+1) - f_i)/h, etc.?
2. Once I have done all of this, is it as simple as plugging in all the w's and f's into the original difference equation, isolating like terms, and solving for alpha, beta, and gamma?
• Oct 22nd 2013, 08:22 AM
phys251
Re: Coefficients of a difference method
34 hits, 0 replies. :(

I got alpha = gamma/2 - 5/4, beta = 1/2, and gamma is free. Alternatively, gamma can be defined in terms of alpha.

Now I am to figure out the stability of this method. Gotta figure out how...