Derive the central difference approximation for f′′(x) accurate to O(h4)
by applying Richardson extrapolation to the central difference approximation of O(h2).
(Note that f''(x)= 2nd derivative of x)
To derive the central difference approximation for f''(x) accurate to O(h^4) using Richardson extrapolation, we first need to determine the central difference approximation of O(h^2) for f''(x).
For the central difference approximation of the first derivative f'(x), we can use the formula:
f'(x) ≈ (f(x + h) - f(x - h)) / (2h)
We can apply the central difference approximation formula twice to obtain the central difference approximation for the second derivative f''(x):
f''(x) ≈ (f'(x + h) - f'(x - h)) / (2h)
Now, for Richardson extrapolation, we utilize a higher-order approximation by taking a linear combination of two approximations with varying step sizes. Let's denote the step size h as h1 and h2, where h1 = h and h2 = 2h.
Using the central difference approximation for f''(x) with two different step sizes, we have:
f''(x) ≈ (f'(x + h1) - f'(x - h1)) / (2h1)
And for h2:
f''(x) ≈ (f'(x + h2) - f'(x - h2)) / (2h2)
Now, we can perform Richardson extrapolation by combining the two approximations using the following formula:
f''(x) ≈ (4f'(x + h2) - 3f'(x + h1) - f'(x - h2) + 3f'(x - h1)) / (6h1)
This central difference approximation for f''(x) using Richardson extrapolation is accurate to O(h^4).
To summarize the steps:
1. Start with the central difference approximation for f''(x) using a small step size h.
2. Compute f'(x) using the central difference approximation formula for f'(x) twice, once with h1 = h and another with h2 = 2h.
3. Combine the two approximations using Richardson extrapolation formula to obtain the central difference approximation for f''(x) accurate to O(h^4).
Keep in mind that this explanation assumes that f(x) is a sufficiently smooth function and that h is small enough for the central difference approximation to be accurate.