Second derivative central difference scheme

Derivation

Add the Taylor series of $f_{i+1}$ and $f_{i-1}$:

$$f_{i+1}+f_{i-1}=2f_i+h^2f''_i+\frac{h^4}{12}f''''_i+\cdots$$

Solve for $f''_i$:

$$\boxed{f''_i\approx\frac{f_{i+1}-2f_i+f_{i-1}}{h^2}},\qquad \text{error}=-\frac{h^2}{12}f''''_i=O(h^2)$$

It is second-order accurate, and it uses three points (stencil $i-1,\ i,\ i+1$). It is the basic formula for diffusion terms.

Numerical for $f=x^4$ at $x=1.2$

Exact: $f''=12x^2=12\times1.44=17.28$.

h = 0.2: $f(1.0)=1.0$, $f(1.2)=2.0736$, $f(1.4)=3.8416$

$$f''\approx\frac{1.0-2(2.0736)+3.8416}{0.2^2}=\frac{0.6944}{0.04}=17.36$$

h = 0.1: $f(1.1)=1.4641$, $f(1.3)=2.8561$

$$f''\approx\frac{1.4641-2(2.0736)+2.8561}{0.01}=\frac{0.173}{0.01}=17.30$$

The error falls by a factor of 4 when $h$ is halved, confirming second-order accuracy. It also matches theory: $\dfrac{h^2}{12}f''''=\dfrac{0.04}{12}\times24=0.08$ for $h=0.2$.

Answer: $f''(1.2)\approx17.36$ ($h=0.2$) and $17.30$ ($h=0.1$); exact 17.28.