Finite Difference: Developing 2nd Order Fromms Scheme (Help)

[flyingcarpet]

how can i develope second order Fromms difference scheme by using these points:
(i-2, i-1, i , i+1) Please help me.

 

[tiny-tim]

Hi flyingcarpet!

What's a Fromms difference scheme?

I can't find Fromm on wikipedia or google …

 

[Architeuthis Dux]

Finite difference is a numerical method used to approximate the derivatives of a function at a given point by using a finite number of points around it. In order to develop a second order Fromms difference scheme, we need to consider the points (i-2, i-1, i, i+1) as given in the problem.

To begin, we first need to define the second order derivative at point i as:

f''(i) = (f(i+1) - 2f(i) + f(i-1)) / h^2

where h is the distance between the points i and i+1 (or i and i-1).

Next, we can use Taylor series expansion to approximate the values of f(i+1) and f(i-1) as:

f(i+1) = f(i) + f'(i)*h + (1/2)*f''(i)*h^2 + O(h^3)

f(i-1) = f(i) - f'(i)*h + (1/2)*f''(i)*h^2 + O(h^3)

Substituting these values in the definition of second order derivative, we get:

f''(i) = (f(i+1) - 2f(i) + f(i-1)) / h^2

= [f(i) + f'(i)*h + (1/2)*f''(i)*h^2 + O(h^3) - 2f(i) + f(i) - f'(i)*h + (1/2)*f''(i)*h^2 + O(h^3)] / h^2

= (f''(i) + O(h^2)) / h^2

= f''(i) / h^2 + O(h^2)

Therefore, the second order derivative f''(i) can be approximated as:

f''(i) = (f(i+1) - 2f(i) + f(i-1)) / h^2 + O(h^2)

This is known as the second order Fromms difference scheme.

In order to use this scheme, we need to have the values of f(i+1) and f(i-1) available. This can be achieved by using additional points (i-2, i-3) and (i+2, i+3)