Derive the central difference approximation for f′′(x) accurate to O(h4)
by applying Richardson extrapolation to the central difference approx-
imation of O(h2).
To derive the central difference approximation for f''(x) accurate to O(h^4) using Richardson extrapolation from the central difference approximation of O(h^2), follow these steps:
1. Start with the central difference approximation for f''(x) accurate to O(h^2):
f''(x) ≈ (f(x + h) - 2f(x) + f(x - h)) / h^2
2. Define a finer step size, h/2, and calculate a new central difference approximation for f''(x) using this smaller step size:
f''(x) ≈ (f(x + h/2) - 2f(x) + f(x - h/2)) / (h/2)^2
3. Apply Richardson extrapolation by combining the two approximations obtained in steps 1 and 2. To do this, multiply the second approximation by 4 to cancel the denominator of (h/2)^2, and subtract it from the first approximation:
f''(x) ≈ (f(x + h) - 2f(x) + f(x - h)) / h^2 - 4[(f(x + h/2) - 2f(x) + f(x - h/2)) / (h/2)^2]
4. Simplify the expression:
f''(x) ≈ (16f(x + h/2) - 30f(x + h) + 16f(x - h/2) - f(x - h)) / (12h^2)
This expression represents the central difference approximation for f''(x) accurate to O(h^4) obtained by applying Richardson extrapolation to the central difference approximation of O(h^2).