Derive the central difference approximation for f''(x) accurate to O(h^4)
by applying Richardson extrapolation to the central difference approximation of O(h^2).
where f'' = 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 start with the central difference approximation of O(h^2) for f''(x).
The central difference approximation of the second derivative of a function f(x) can be expressed as:
f''(x) ≈ [f(x + h) - 2f(x) + f(x - h)] / h^2
To apply Richardson extrapolation to this central difference approximation, we need to introduce another step size, let's call it αh, with α > 1. We will use this additional step size to refine our approximation and increase its accuracy.
Now, we can define the Richardson extrapolation as follows:
f''(x) ≈ CD1(h) + c * (CD1(αh) - CD1(h))
where CD1(h) represents the central difference approximation with step size h, and c is a constant that needs to be determined.
Next, let's find the central difference approximations CD1(h) and CD1(αh):
CD1(h) = [f(x + h) - 2f(x) + f(x - h)] / h^2
CD1(αh) = [f(x + αh) - 2f(x) + f(x - αh)] / (αh)^2
Now, let's simplify the Richardson extrapolation expression:
f''(x) ≈ CD1(h) + c * (CD1(αh) - CD1(h))
f''(x) ≈ [f(x + h) - 2f(x) + f(x - h)] / h^2 + c * ([f(x + αh) - 2f(x) + f(x - αh)] / (αh)^2 - [f(x + h) - 2f(x) + f(x - h)] / h^2)
f''(x) ≈ (α^2 * [f(x + h) - 2f(x) + f(x - h)] - [f(x + αh) - 2f(x) + f(x - αh)]) / (α^2 - 1)h^2
To achieve O(h^4) accuracy, we need the leading error term to cancel out. Therefore, we set the coefficient of the leading error term to zero:
α^2 * [f(x + h) - 2f(x) + f(x - h)] - [f(x + αh) - 2f(x) + f(x - αh)] = 0
From this equation, we can solve for c and α, which will depend on the specific function f(x) you are working with.