Question

Derive a finite difference approximation formula for the second derivative f ′′(xi) using three points xi−1,xi, and xi+1

​ , where the spacing is such that xi−xi−1 = 2h, and xi+1−xi = h (non-uniform spacing).
Hint: use the Taylor series expansions for f(xi−1) and f(x+1). Note the non-uniform spacing!

Answer 1

To derive a finite difference approximation for the second derivative with non-uniform spacing, Taylor series expansions of the adjacent points at xi−1 and xi+1 are used, before solving to express f ′′(xi) in terms of f(xi−1), f(xi), and f(xi+1).

Step-by-step explanation:

The question involves deriving a finite difference approximation for the second derivative of a function, f ′′(xi), using non-uniformly spaced points: xi−1, xi, and xi+1. The spacing is given by xi−xi−1 = 2h, and xi+1−xi = h.

To derive the formula, one can use the Taylor series expansions around the point xi for the adjacent points xi−1 and xi+1.

To start, we expand f(xi−1) and f(xi+1) using the Taylor series in terms of h and evaluate the coefficients using f(xi) and its derivatives. Afterward, solve the resulting equations to express f ′′(xi) in terms of f(xi−1), f(xi), and f(xi+1).

The result will be an approximation that considers the non-uniform grid spacing, which is key to obtaining the correct second derivative approximation under these conditions.