The centered difference discretization method can easily be constructed for higher order approximation.
Construct the 4ᵗʰ-order accurate centered difference discrete equation. Here are the hints:
Start with the Taylor's expansion for f(x + Δx):
f(x + Δx) ≈ f(x) + df/dx Δx + d²f/dx² Δx² / 2 + d³f/dx³ Δx³ / 6 + d⁴f/dx⁴ Δx⁴ / 24 + o(Δx⁵)
f(x - Δx) ≈ f(x) - df/dx Δx + d²f/dx² Δx² / 2 - d³f/dx³ Δx³ / 6 + d⁴f/dx⁴ Δx⁴ / 24 + o(Δx⁵)
1. Expand for f(x + -2Δx).
2. Calculate f(x + Δx) - f(x - Δx).
3. Calculate f(x + 2Δx) - f(x - 2Δx).
4. Compare (3) and (4). Remove the unnecessary terms.